Solve each trigonometric equation in the interval [0,2n) by first squaring both sides. fleas x=1+sin x Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A- The solution set is .
(Simplify your answer. Use a comma to separate answers as needed. Type an exact answer, using 1: as needed. Use integers or fractions for any numbers in the expression.) 0 B. There is no solution on this interval.
An Introduction to Mis-Specification (M-S) Testingjemille6
This document provides an introduction to mis-specification (M-S) testing, which is a methodology for validating statistical models. It discusses how statistical misspecification can render statistical inference unreliable by distorting nominal error probabilities. As an example, it shows how violating the independence assumption in a normal model can increase the actual type I error rate and reduce power. It argues that model validation through M-S testing is important for ensuring reliable inference, but is often neglected due to misunderstandings. All statistical methods rely on an underlying statistical model, so any misspecification impacts reliability regardless of the method used.
This document provides lecture notes on hypothesis testing. It begins with an introduction to hypothesis testing and how it differs from estimation in its hypothetical reasoning approach. It then discusses Fisher's significance testing approach, including defining a test statistic, its sampling distribution under the null hypothesis, and calculating a p-value. It provides examples of applying this approach. Finally, it discusses some weaknesses of Fisher's approach identified by Neyman and Pearson and how their approach improved upon it by introducing the concept of alternative hypotheses and pre-data error probabilities.
This document discusses hypothesis testing, including:
- The chapter introduces hypothesis testing and defines key concepts like the null hypothesis, alternative hypothesis, type I and type II errors, and significance levels.
- It explains how to formulate and test hypotheses about population means and proportions, including how to determine critical values and p-values.
- The steps of hypothesis testing are outlined, and an example is provided to demonstrate how to test a claim about a population mean using a z-test.
- Both critical value and p-value approaches to testing hypotheses are described.
This document discusses hypothesis testing without statistics using a criminal trial as an example. It explains that in a trial, the jury must decide between a null hypothesis (H0) that the defendant is innocent, and an alternative hypothesis (H1) that the defendant is guilty based on the presented evidence. There are two possible errors - a Type I error of convicting an innocent person, and a Type II error of acquitting a guilty person. The probability of each error is inversely related to the sample size. The document provides examples to illustrate hypothesis testing concepts like rejection regions, test statistics, and interpreting p-values.
This document discusses hypothesis testing for correlation between two continuous variables. It defines correlation, outlines the steps for a hypothesis test comparing correlation to zero, and provides the technical details of calculating Pearson's correlation coefficient. A key point is that the distribution of the test statistic is t-distributed, allowing assessment of statistical significance through the p-value. The goal of the example analysis is to determine if there is a linear relationship between IL-10 and IL-6 expression levels in patients.
This document provides an overview of hypothesis testing methodology for one population. It defines key concepts like the null and alternative hypotheses, types of errors, test statistics, significance levels, and rejection regions. The two main approaches to hypothesis testing are presented: the rejection region approach and the p-value approach. Steps for conducting a hypothesis test are outlined, including stating hypotheses, choosing test criteria, collecting data, determining test statistics and p-values, and making conclusions. Examples are provided to illustrate hypothesis tests for means and proportions.
The document discusses testing of hypotheses. It defines a hypothesis as a tentative prediction about the relationship between variables. Good hypotheses are precise, testable, and consistent with known facts. Hypothesis testing involves formulating a null hypothesis (Ho) and an alternative hypothesis (H1). A significance level such as 5% is chosen. If the test statistic falls within the critical region, Ho is rejected. Type I error rejects a true Ho, while Type II error accepts a false Ho. Power refers to correctly rejecting a false Ho. The testing process determines test statistics, critical regions, and interprets results to draw conclusions.
An Introduction to Mis-Specification (M-S) Testingjemille6
This document provides an introduction to mis-specification (M-S) testing, which is a methodology for validating statistical models. It discusses how statistical misspecification can render statistical inference unreliable by distorting nominal error probabilities. As an example, it shows how violating the independence assumption in a normal model can increase the actual type I error rate and reduce power. It argues that model validation through M-S testing is important for ensuring reliable inference, but is often neglected due to misunderstandings. All statistical methods rely on an underlying statistical model, so any misspecification impacts reliability regardless of the method used.
This document provides lecture notes on hypothesis testing. It begins with an introduction to hypothesis testing and how it differs from estimation in its hypothetical reasoning approach. It then discusses Fisher's significance testing approach, including defining a test statistic, its sampling distribution under the null hypothesis, and calculating a p-value. It provides examples of applying this approach. Finally, it discusses some weaknesses of Fisher's approach identified by Neyman and Pearson and how their approach improved upon it by introducing the concept of alternative hypotheses and pre-data error probabilities.
This document discusses hypothesis testing, including:
- The chapter introduces hypothesis testing and defines key concepts like the null hypothesis, alternative hypothesis, type I and type II errors, and significance levels.
- It explains how to formulate and test hypotheses about population means and proportions, including how to determine critical values and p-values.
- The steps of hypothesis testing are outlined, and an example is provided to demonstrate how to test a claim about a population mean using a z-test.
- Both critical value and p-value approaches to testing hypotheses are described.
This document discusses hypothesis testing without statistics using a criminal trial as an example. It explains that in a trial, the jury must decide between a null hypothesis (H0) that the defendant is innocent, and an alternative hypothesis (H1) that the defendant is guilty based on the presented evidence. There are two possible errors - a Type I error of convicting an innocent person, and a Type II error of acquitting a guilty person. The probability of each error is inversely related to the sample size. The document provides examples to illustrate hypothesis testing concepts like rejection regions, test statistics, and interpreting p-values.
This document discusses hypothesis testing for correlation between two continuous variables. It defines correlation, outlines the steps for a hypothesis test comparing correlation to zero, and provides the technical details of calculating Pearson's correlation coefficient. A key point is that the distribution of the test statistic is t-distributed, allowing assessment of statistical significance through the p-value. The goal of the example analysis is to determine if there is a linear relationship between IL-10 and IL-6 expression levels in patients.
This document provides an overview of hypothesis testing methodology for one population. It defines key concepts like the null and alternative hypotheses, types of errors, test statistics, significance levels, and rejection regions. The two main approaches to hypothesis testing are presented: the rejection region approach and the p-value approach. Steps for conducting a hypothesis test are outlined, including stating hypotheses, choosing test criteria, collecting data, determining test statistics and p-values, and making conclusions. Examples are provided to illustrate hypothesis tests for means and proportions.
The document discusses testing of hypotheses. It defines a hypothesis as a tentative prediction about the relationship between variables. Good hypotheses are precise, testable, and consistent with known facts. Hypothesis testing involves formulating a null hypothesis (Ho) and an alternative hypothesis (H1). A significance level such as 5% is chosen. If the test statistic falls within the critical region, Ho is rejected. Type I error rejects a true Ho, while Type II error accepts a false Ho. Power refers to correctly rejecting a false Ho. The testing process determines test statistics, critical regions, and interprets results to draw conclusions.
This document discusses quantitative research methods and statistical inference. It covers topics like probability distributions, sampling distributions, estimation, hypothesis testing, and different statistical tests. Key points include:
- Probability distributions describe random variables and their associated probabilities. The normal distribution is important and described by its mean and standard deviation.
- Sampling distributions allow making inferences about populations based on samples. The sampling distribution of the mean approximates a normal distribution as the sample size increases.
- Statistical inference involves estimation and hypothesis testing. Estimation provides a value for an unknown population parameter based on a sample statistic. Hypothesis testing compares a null hypothesis to an alternative hypothesis based on a test statistic and can result in type 1 or type 2 errors.
High-Dimensional Methods: Examples for Inference on Structural EffectsNBER
This document describes a study that uses high-dimensional methods to estimate the effect of 401(k) eligibility on measures of accumulated assets. It begins by outlining the baseline model and notes areas for improvement, such as controlling for income. It then discusses using regularization like LASSO for variable selection in high-dimensional settings. The document explores more flexible specifications by generating many interaction and polynomial terms but notes the need for dimension reduction. It describes using LASSO to select important variables from a large set. The results select a parsimonious set of variables and estimate similar 401(k) effects as the baseline.
1) The document discusses hypothesis testing and statistical inference using examples related to coin tossing. It explains the concepts of type I and type II errors and how hypothesis tests are conducted.
2) An example is provided to test the hypothesis that the average American ideology is somewhat conservative (H0: μ = 5) using data from the National Election Study. The alternative hypothesis is that the average is less than 5 (HA: μ < 5).
3) The results of the hypothesis test show the observed test statistic is lower than the critical value, so the null hypothesis that the average is 5 is rejected in favor of the alternative that the average is less than 5.
The document discusses various probability distributions including the normal, binomial, Poisson, uniform, and chi-square distributions. It provides examples of when each distribution would be used and explains key properties such as mean, variance, and standard deviation. It also covers topics like the central limit theorem, sampling distributions, and how inferential statistics is used to generalize from samples to populations.
This document provides an overview of hypothesis testing including:
- Defining null and alternative hypotheses
- Types of errors like Type I and Type II
- Test statistics and significance levels for comparing means, proportions, and standard deviations of one and two populations
- Examples are given for hypothesis tests on population means, proportions, and comparing two population means.
This document provides an overview of hypothesis testing, analysis of variance (ANOVA), and how to properly quote references and include a bibliography. It discusses the key steps in hypothesis testing, including stating the null and alternative hypotheses, choosing a significance level, determining the sampling distribution, calculating probabilities, and deciding whether to reject or fail to reject the null hypothesis. It also outlines one-way and two-way ANOVA, explaining how to calculate variances between and within samples/groups and use an F-test statistic. Finally, it defines what a bibliography is, lists standard citation styles, and distinguishes between references cited in a work and a full bibliography.
This document discusses key concepts in research methods and biostatistics, including hypothesis testing, random error, p-values, and confidence intervals. It explains that hypothesis testing involves determining if study findings reflect chance or a true effect. The p-value represents the probability of observing results as extreme or more extreme than what was observed by chance alone. A p-value less than 0.05 indicates statistical significance. Confidence intervals provide a range of values that are likely to contain the true population parameter.
This document provides an introduction to hypothesis testing including:
1. The 5 steps in a hypothesis test: set up null and alternative hypotheses, define test procedure, collect data, decide whether to reject null hypothesis, interpret results.
2. Large sample tests for the mean involve testing if the population mean is equal to or not equal to a specified value using a test statistic that follows a normal distribution.
3. Type I and Type II errors occur when the decision made based on the hypothesis test does not match the actual truth - a Type I error rejects the null hypothesis when it is true, a Type II error fails to reject the null when it is false. The probability of each error can be minimized by choosing
The document discusses various statistical concepts related to hypothesis testing, including:
- Types I and II errors that can occur when testing hypotheses
- How the probability of committing errors depends on factors like the sample size and how far the population parameter is from the hypothesized value
- The concept of critical regions and how they are used to determine if a null hypothesis can be rejected
- The difference between discrete and continuous probability distributions and examples of each
- How an observed test statistic is calculated and compared to a critical value to determine whether to reject or not reject the null hypothesis
This document provides an overview of hypothesis testing. It defines key terms like the null hypothesis (H0), alternative hypothesis (H1), type I and type II errors, significance level, p-values, and test statistics. It explains the basic steps in hypothesis testing as testing a claim about a population parameter by collecting a sample, determining the appropriate test statistic based on the sampling distribution, and comparing it to critical values to reject or fail to reject the null hypothesis. Examples are provided to demonstrate hypothesis tests for a mean when the population standard deviation is known or unknown.
Avoiding undesired choices using intelligent adaptive systemsijaia
This document summarizes a research paper that proposes methods for avoiding undesirable choices using intelligent adaptive systems. It discusses how preferences can reverse when additional options are introduced, due to dependencies between items. A model is presented using matrices to represent utilities of items in a choice set. The model is generalized to account for multiple items. Methods are suggested for predicting when preference reversals may occur and avoiding undesired choices, such as transparently communicating the context of choices and adaptively generating choice sets.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document outlines techniques for representing uncertainty in expert systems, including Bayesian reasoning and certainty factors theory. It discusses sources of uncertain knowledge, probabilistic reasoning using Bayes' rule, and an example of computing posterior probabilities of hypotheses given observed evidence. Certainty factors theory is presented as an alternative to Bayesian reasoning that uses numerical factors between -1 and 1 to represent degrees of belief.
The document provides an overview of key statistical concepts including variance, standard deviation, the normal distribution, frequency distributions, data matrices, properties of good graphs, populations and samples, parameters and statistics, hypothesis testing, and point and interval estimation. It defines these terms and explains concepts like the null hypothesis, alternative hypothesis, critical regions, test statistics, and making decisions based on probability thresholds.
This document provides an overview of estimation and hypothesis testing. It defines key statistical concepts like population and sample, parameters and estimates, and introduces the two main methods in inferential statistics - estimation and hypothesis testing.
It explains that hypothesis testing involves setting a null hypothesis (H0) and an alternative hypothesis (Ha), calculating a test statistic, determining a p-value, and making a decision to accept or reject the null hypothesis based on the p-value and significance level. The four main steps of hypothesis testing are outlined as setting hypotheses, calculating a test statistic, determining the p-value, and making a conclusion.
Examples are provided to demonstrate left-tailed, right-tailed, and two-tailed hypothesis tests
This document discusses hypothesis testing using a single sample. It explains that a hypothesis test involves a null hypothesis (H0) which is initially assumed to be true, and an alternative hypothesis (Ha) which is the competing claim. The test aims to reject the null hypothesis in favor of the alternative. A test statistic is calculated from sample data and compared to a significance level (α) to determine whether to reject H0. Examples are provided to illustrate hypotheses about population means, proportions, and their tests.
This document provides a tutorial on principal components analysis (PCA). It begins with an introduction to PCA and its applications. It then covers the necessary background mathematical concepts, including standard deviation, covariance, and eigenvalues/eigenvectors. The tutorial includes examples throughout and recommends a textbook for further mathematical information.
Hypothesis testing refers to formal statistical procedures used to accept or reject claims about populations based on data. It involves:
1) Stating a null hypothesis that makes a claim about a population parameter.
2) Collecting sample data and computing a test statistic.
3) Determining whether to reject the null hypothesis based on the probability of obtaining the sample statistic if the null is true.
Rejecting the null supports the alternative hypothesis. Type I and Type II errors occur when the null is incorrectly rejected or not rejected. Hypothesis tests aim to minimize errors while maximizing power to detect meaningful alternative hypotheses.
- Hypothesis testing involves evaluating claims about population parameters by comparing a null hypothesis to an alternative hypothesis.
- The null hypothesis states that there is no difference or effect, while the alternative hypothesis states that a difference or effect exists.
- There are three main methods for hypothesis testing: the critical value method which separates a critical region from a noncritical region, the p-value method which calculates the probability of obtaining a test statistic at least as extreme as the sample test statistic assuming the null is true, and the confidence interval method which rejects claims not included in the confidence interval.
- The steps of hypothesis testing are to state the hypotheses, calculate the test statistic, find the critical value, make a decision to reject
Hypothesis Test _One-sample t-test, Z-test, Proportion Z-testRavindra Nath Shukla
This document discusses hypothesis testing concepts including the null and alternative hypotheses, type I and II errors, and the hypothesis testing process. It provides examples of hypothesis testing for a mean where the population standard deviation is known (z-test) and unknown (t-test). The document outlines the 6 steps in hypothesis testing and provides examples using both the critical value approach and p-value approach. It discusses the relationship between hypothesis testing and confidence intervals.
This document provides an overview of one-sample hypothesis tests. It defines key terms related to hypothesis testing such as the null and alternative hypotheses, test statistic, critical value, Type I and Type II errors, and p-value. It explains the five-step hypothesis testing procedure and how to set up hypotheses for tests involving a population mean or proportion. Examples are provided to demonstrate how to conduct hypothesis tests about a population mean when the population standard deviation is known or unknown, and how to conduct a test about a population proportion. The document also discusses how to interpret p-values and the concept of Type II errors.
This document discusses quantitative research methods and statistical inference. It covers topics like probability distributions, sampling distributions, estimation, hypothesis testing, and different statistical tests. Key points include:
- Probability distributions describe random variables and their associated probabilities. The normal distribution is important and described by its mean and standard deviation.
- Sampling distributions allow making inferences about populations based on samples. The sampling distribution of the mean approximates a normal distribution as the sample size increases.
- Statistical inference involves estimation and hypothesis testing. Estimation provides a value for an unknown population parameter based on a sample statistic. Hypothesis testing compares a null hypothesis to an alternative hypothesis based on a test statistic and can result in type 1 or type 2 errors.
High-Dimensional Methods: Examples for Inference on Structural EffectsNBER
This document describes a study that uses high-dimensional methods to estimate the effect of 401(k) eligibility on measures of accumulated assets. It begins by outlining the baseline model and notes areas for improvement, such as controlling for income. It then discusses using regularization like LASSO for variable selection in high-dimensional settings. The document explores more flexible specifications by generating many interaction and polynomial terms but notes the need for dimension reduction. It describes using LASSO to select important variables from a large set. The results select a parsimonious set of variables and estimate similar 401(k) effects as the baseline.
1) The document discusses hypothesis testing and statistical inference using examples related to coin tossing. It explains the concepts of type I and type II errors and how hypothesis tests are conducted.
2) An example is provided to test the hypothesis that the average American ideology is somewhat conservative (H0: μ = 5) using data from the National Election Study. The alternative hypothesis is that the average is less than 5 (HA: μ < 5).
3) The results of the hypothesis test show the observed test statistic is lower than the critical value, so the null hypothesis that the average is 5 is rejected in favor of the alternative that the average is less than 5.
The document discusses various probability distributions including the normal, binomial, Poisson, uniform, and chi-square distributions. It provides examples of when each distribution would be used and explains key properties such as mean, variance, and standard deviation. It also covers topics like the central limit theorem, sampling distributions, and how inferential statistics is used to generalize from samples to populations.
This document provides an overview of hypothesis testing including:
- Defining null and alternative hypotheses
- Types of errors like Type I and Type II
- Test statistics and significance levels for comparing means, proportions, and standard deviations of one and two populations
- Examples are given for hypothesis tests on population means, proportions, and comparing two population means.
This document provides an overview of hypothesis testing, analysis of variance (ANOVA), and how to properly quote references and include a bibliography. It discusses the key steps in hypothesis testing, including stating the null and alternative hypotheses, choosing a significance level, determining the sampling distribution, calculating probabilities, and deciding whether to reject or fail to reject the null hypothesis. It also outlines one-way and two-way ANOVA, explaining how to calculate variances between and within samples/groups and use an F-test statistic. Finally, it defines what a bibliography is, lists standard citation styles, and distinguishes between references cited in a work and a full bibliography.
This document discusses key concepts in research methods and biostatistics, including hypothesis testing, random error, p-values, and confidence intervals. It explains that hypothesis testing involves determining if study findings reflect chance or a true effect. The p-value represents the probability of observing results as extreme or more extreme than what was observed by chance alone. A p-value less than 0.05 indicates statistical significance. Confidence intervals provide a range of values that are likely to contain the true population parameter.
This document provides an introduction to hypothesis testing including:
1. The 5 steps in a hypothesis test: set up null and alternative hypotheses, define test procedure, collect data, decide whether to reject null hypothesis, interpret results.
2. Large sample tests for the mean involve testing if the population mean is equal to or not equal to a specified value using a test statistic that follows a normal distribution.
3. Type I and Type II errors occur when the decision made based on the hypothesis test does not match the actual truth - a Type I error rejects the null hypothesis when it is true, a Type II error fails to reject the null when it is false. The probability of each error can be minimized by choosing
The document discusses various statistical concepts related to hypothesis testing, including:
- Types I and II errors that can occur when testing hypotheses
- How the probability of committing errors depends on factors like the sample size and how far the population parameter is from the hypothesized value
- The concept of critical regions and how they are used to determine if a null hypothesis can be rejected
- The difference between discrete and continuous probability distributions and examples of each
- How an observed test statistic is calculated and compared to a critical value to determine whether to reject or not reject the null hypothesis
This document provides an overview of hypothesis testing. It defines key terms like the null hypothesis (H0), alternative hypothesis (H1), type I and type II errors, significance level, p-values, and test statistics. It explains the basic steps in hypothesis testing as testing a claim about a population parameter by collecting a sample, determining the appropriate test statistic based on the sampling distribution, and comparing it to critical values to reject or fail to reject the null hypothesis. Examples are provided to demonstrate hypothesis tests for a mean when the population standard deviation is known or unknown.
Avoiding undesired choices using intelligent adaptive systemsijaia
This document summarizes a research paper that proposes methods for avoiding undesirable choices using intelligent adaptive systems. It discusses how preferences can reverse when additional options are introduced, due to dependencies between items. A model is presented using matrices to represent utilities of items in a choice set. The model is generalized to account for multiple items. Methods are suggested for predicting when preference reversals may occur and avoiding undesired choices, such as transparently communicating the context of choices and adaptively generating choice sets.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document outlines techniques for representing uncertainty in expert systems, including Bayesian reasoning and certainty factors theory. It discusses sources of uncertain knowledge, probabilistic reasoning using Bayes' rule, and an example of computing posterior probabilities of hypotheses given observed evidence. Certainty factors theory is presented as an alternative to Bayesian reasoning that uses numerical factors between -1 and 1 to represent degrees of belief.
The document provides an overview of key statistical concepts including variance, standard deviation, the normal distribution, frequency distributions, data matrices, properties of good graphs, populations and samples, parameters and statistics, hypothesis testing, and point and interval estimation. It defines these terms and explains concepts like the null hypothesis, alternative hypothesis, critical regions, test statistics, and making decisions based on probability thresholds.
This document provides an overview of estimation and hypothesis testing. It defines key statistical concepts like population and sample, parameters and estimates, and introduces the two main methods in inferential statistics - estimation and hypothesis testing.
It explains that hypothesis testing involves setting a null hypothesis (H0) and an alternative hypothesis (Ha), calculating a test statistic, determining a p-value, and making a decision to accept or reject the null hypothesis based on the p-value and significance level. The four main steps of hypothesis testing are outlined as setting hypotheses, calculating a test statistic, determining the p-value, and making a conclusion.
Examples are provided to demonstrate left-tailed, right-tailed, and two-tailed hypothesis tests
This document discusses hypothesis testing using a single sample. It explains that a hypothesis test involves a null hypothesis (H0) which is initially assumed to be true, and an alternative hypothesis (Ha) which is the competing claim. The test aims to reject the null hypothesis in favor of the alternative. A test statistic is calculated from sample data and compared to a significance level (α) to determine whether to reject H0. Examples are provided to illustrate hypotheses about population means, proportions, and their tests.
This document provides a tutorial on principal components analysis (PCA). It begins with an introduction to PCA and its applications. It then covers the necessary background mathematical concepts, including standard deviation, covariance, and eigenvalues/eigenvectors. The tutorial includes examples throughout and recommends a textbook for further mathematical information.
Hypothesis testing refers to formal statistical procedures used to accept or reject claims about populations based on data. It involves:
1) Stating a null hypothesis that makes a claim about a population parameter.
2) Collecting sample data and computing a test statistic.
3) Determining whether to reject the null hypothesis based on the probability of obtaining the sample statistic if the null is true.
Rejecting the null supports the alternative hypothesis. Type I and Type II errors occur when the null is incorrectly rejected or not rejected. Hypothesis tests aim to minimize errors while maximizing power to detect meaningful alternative hypotheses.
- Hypothesis testing involves evaluating claims about population parameters by comparing a null hypothesis to an alternative hypothesis.
- The null hypothesis states that there is no difference or effect, while the alternative hypothesis states that a difference or effect exists.
- There are three main methods for hypothesis testing: the critical value method which separates a critical region from a noncritical region, the p-value method which calculates the probability of obtaining a test statistic at least as extreme as the sample test statistic assuming the null is true, and the confidence interval method which rejects claims not included in the confidence interval.
- The steps of hypothesis testing are to state the hypotheses, calculate the test statistic, find the critical value, make a decision to reject
Hypothesis Test _One-sample t-test, Z-test, Proportion Z-testRavindra Nath Shukla
This document discusses hypothesis testing concepts including the null and alternative hypotheses, type I and II errors, and the hypothesis testing process. It provides examples of hypothesis testing for a mean where the population standard deviation is known (z-test) and unknown (t-test). The document outlines the 6 steps in hypothesis testing and provides examples using both the critical value approach and p-value approach. It discusses the relationship between hypothesis testing and confidence intervals.
This document provides an overview of one-sample hypothesis tests. It defines key terms related to hypothesis testing such as the null and alternative hypotheses, test statistic, critical value, Type I and Type II errors, and p-value. It explains the five-step hypothesis testing procedure and how to set up hypotheses for tests involving a population mean or proportion. Examples are provided to demonstrate how to conduct hypothesis tests about a population mean when the population standard deviation is known or unknown, and how to conduct a test about a population proportion. The document also discusses how to interpret p-values and the concept of Type II errors.
InstructionDue Date 6 pm on October 28 (Wed)Part IProbability a.docxdirkrplav
This document discusses implementing a social, environmental, and economic impact measurement system within a company. It explains that measuring sustainability performance is critical for evaluating projects, the company, and its members. A proper measurement system allows companies to develop a sustainability strategy, allocate resources to support it, and evaluate trade-offs between sustainability projects. The document provides examples from Nike and P&G of measuring impacts to demonstrate the business case for sustainability. It stresses that measurement is important for linking performance to sustainability principles and facilitating continuous improvement.
8. testing of hypothesis for variable & attribute dataHakeem-Ur- Rehman
The document discusses hypothesis testing for continuous variable and attribute data. It begins by defining key concepts in statistical inference like the null and alternative hypotheses. The three types of hypotheses are explained - two-tailed, left-tailed, and right-tailed. The document then discusses hypothesis testing steps including defining the hypotheses, determining the sampling risk of type I and type II errors, calculating the p-value, and making a decision to accept or reject the null hypothesis based on the p-value and significance level. Specific parametric statistical tests are explained like the one sample t-test, two sample t-test, and ANOVA. Examples of each test are provided and how to interpret the results.
This chapter discusses fundamentals of hypothesis testing for one-sample tests. It covers:
1) Formulating the null and alternative hypotheses for tests involving a single population mean or proportion.
2) Using critical value and p-value approaches to test the null hypothesis, and defining Type I and Type II errors.
3) How to perform hypothesis tests for a single population mean when the population standard deviation is known or unknown.
Assessment 3 – Hypothesis, Effect Size, Power, and t Tests.docxcargillfilberto
Assessment 3 – Hypothesis, Effect Size, Power, and
t
Tests
Complete the following problems within this Word document. Do not submit other files. Show your work for problem sets that require calculations. Ensure that your answer to each problem is clearly visible. You may want to highlight your answer or use a different type color to set it apart.
Hypothesis, Effect Size, and Power
Problem Set 3.1: Sampling Distribution of the Mean Exercise
Criterion:
Interpret population mean and variance.
Instructions:
Read the information below and answer the questions.
Suppose a researcher wants to learn more about the mean attention span of individuals in some hypothetical population. The researcher cites that the attention span (the time in minutes attending to some task) in this population is normally distributed with the following characteristics: 20
36
. Based on the parameters given in this example, answer the following questions:
1. What is the population mean (μ)? __________________________
2. What is the population variance
? __________________________
3. Sketch the distribution of this population. Make sure you draw the shape of the distribution and label the mean plus and minus three standard deviations.
Problem Set 3.2: Effect Size and Power
Criterion:
Explain effect size and power.
Instructions:
Read each of the following three scenarios and answer the questions.
Two researchers make a test concerning the effectiveness of a drug use treatment. Researcher A determines that the effect size in the population of males is
d
= 0.36; Researcher B determines that the effect size in the population of females is
d
= 0.20. All other things being equal, which researcher has more power to detect an effect? Explain. ______________________________________________________________________
Two researchers make a test concerning the levels of marital satisfaction among military families. Researcher A collects a sample of 22 married couples (
n
= 22); Researcher B collects a sample of 40 married couples (
n
= 40). All other things being equal, which researcher has more power to detect an effect? Explain. ______________________________________________________________________
Two researchers make a test concerning standardized exam performance among senior high school students in one of two local communities. Researcher A tests performance from the population in the northern community, where the standard deviation of test scores is 110 (
); Researcher B tests performance from the population in the southern community, where the standard deviation of test scores is 60 (
). All other things being equal, which researcher has more power to detect an effect? Explain. ______________________________________________________________________
Problem Set 3.3: Hypothesis, Direction, and Population Mean
Criterion:
Explain the relationship between hypothesis, tests, and population mean.
Instructions:
Read the following and answer the questions.
This document provides an introduction to hypothesis testing. It discusses key concepts such as the null and alternative hypotheses, types of errors, levels of significance, test statistics, p-values, and decision rules. Examples are provided to demonstrate how to state hypotheses, identify the type of test, find critical values and rejection regions, calculate test statistics and p-values, and make decisions to reject or fail to reject the null hypothesis based on these concepts. The steps outlined include stating the hypotheses, specifying the significance level, determining the test statistic and sampling distribution, finding the p-value or using rejection regions to make a decision, and interpreting what the decision means for the original claim.
The document discusses hypothesis testing procedures. It describes the 7 steps of hypothesis testing which include setting the null and alternative hypotheses, choosing a statistical test, setting the significance level, establishing decision rules, collecting sample data, analyzing the data, and arriving at a statistical conclusion. Common statistical tests mentioned are F-tests, t-tests, and z-tests. The F-test compares variances by taking the ratio and uses sums of squares to develop statistics. The document provides an example of a company collecting potential customer spending data on flats in two areas to estimate differences in population means.
The document discusses hypothesis testing procedures. It describes the 7 steps of hypothesis testing which include setting the null and alternative hypotheses, choosing a statistical test, setting the significance level, establishing decision rules, collecting sample data, analyzing the data, and arriving at a statistical conclusion. Common statistical tests mentioned are F-tests, t-tests, and z-tests. The F-test compares variances by taking the ratio and uses sums of squares to develop statistics. The document provides an example of a company collecting potential customer spending data on flats in two areas to estimate differences in population means.
Introduction to hypothesis testing ppt @ bec domsBabasab Patil
This document introduces hypothesis testing, including:
- Formulating null and alternative hypotheses for tests involving population means and proportions
- Using test statistics, critical values, and p-values to test hypotheses
- Defining Type I and Type II errors and their probabilities
- Examples of hypothesis tests for means (using z-tests and t-tests) and proportions (using z-tests) are provided to illustrate the concepts.
1NOTE This is a template to help you format Project Part .docxrobert345678
This document provides a template for a student to complete a statistical analysis project involving descriptive statistics, hypothesis testing, and regression analysis. The template outlines the content and statistical analyses to be performed on two variables - sales and calls - including descriptive statistics, hypothesis tests, correlation, regression equation, and estimates. The student is instructed to input their results, analyses, and conclusions into the template for their assignment submission.
This document provides information about statistics and hypothesis testing concepts. It defines key terms like population, sample, parameters, statistics, standard error, random sampling, critical region, acceptance region, one-tailed and two-tailed tests, null and alternative hypotheses, type I and type II errors. It also describes common statistical tests like t-test, F-test, chi-square test and provides their assumptions and uses. Several examples of hypothesis testing problems and their solutions are given to illustrate statistical concepts and procedures.
I am Hannah Lucy. Currently associated with statisticshomeworkhelper.com as statistics homework helper. After completing my master's from Kean University, USA, I was in search of an opportunity that expands my area of knowledge hence I decided to help students with their homework. I have written several statistics homework till date to help students overcome numerous difficulties they face.
This document provides an overview of hypothesis testing methodology for one population. It defines key concepts like the null and alternative hypotheses, types of errors, test statistics, significance levels, and rejection regions. The two main approaches to hypothesis testing are presented: the rejection region approach and the p-value approach. Steps for conducting a hypothesis test are outlined, including stating hypotheses, choosing test criteria, collecting data, determining test statistics and p-values, and making conclusions. Examples are provided to illustrate hypothesis tests for means and proportions.
1. The document discusses hypothesis testing of claims about population parameters such as proportions, means, standard deviations, and variances from one or two samples.
2. Key concepts include hypothesis tests using z-tests, t-tests, and chi-square tests. Confidence intervals are also constructed for parameters.
3. Two examples are provided to demonstrate hypothesis testing of claims about two population proportions using z-tests. The null hypothesis is rejected in one example but not the other.
Statistics practice for finalBe sure to review the following.docxdessiechisomjj4
Statistics practice for final
Be sure to review the following and have this information handy when taking Final GHA:
· Calculating z alpha/2 and t alpha/2 on Tables II and IV
· Find sample size for estimating population mean. Formula 8.3 p. 321 OCR.
· Stating H0 and H1 claims about variation and about the mean. Chapter 9 OCR.
· Type I and Type II errors p. 345 OCR.
· Confidence Interval for difference between two population means. Chapter 10 OCR p. 428
· Pooled sample standard deviation. Chapter 10 OCR p. 397
· What do Chi-Square tests measure? How are their degrees of freedom calculated? Chapter 12 OCR
· Find F test statistic using One-Way ANOVA.xls Be sure to enable editing and change values to match your problem. One-Way ANOVA.xls
· Find equation of regression line used to predict. To do on Excel, go to a blank worksheet, enter x values in one column and their matching y values in another column. Click Insert – Select Scatterplot. Right click any one of the points (diamonds) on the graph. Left click “Add a Trendline.” Check “Display Equation on Chart” box. Regression equation will appear on chart. Try it here with No. 20 below.
Practice Problems
Chapter 8 Final Review
1) In which of the following situations is it reasonable to use the z-interval
procedure to obtain a confidence interval for the population mean?
Assume that the population standard deviation is known.
A. n = 10, the data contain no outliers, the variable under consideration is
not normally distributed.
B. n = 10, the variable under consideration is normally distributed.
C. n = 18, the data contain no outliers, the variable under consideration is
far from being normally distributed.
D. n = 18, the data contain outliers, the variable under consideration is
normally distributed.
Find the necessary sample size.
2) The weekly earnings of students in one age group are normally
distributed with a standard deviation of 10 dollars. A researcher wishes to
estimate the mean weekly earnings of students in this age group. Find the
sample size needed to assure with 95 percent confidence that the sample
mean will not differ from the population mean by more than 2 dollars.
Find the specified t-value.
3) For a t-curve with df = 6, find the two t-values that divide the area under
the curve into a middle 0.99 area and two outside areas of 0.005.
Provide an appropriate response.
4) Under what conditions would you choose to use the t-interval procedure
instead of the z-interval procedure in order to obtain a confidence
interval for a population mean? What conditions must be satisfied in
order to use the t-interval procedure?
CHAPTER 8 Answers
1) B
2) 97
3) -3.707, 3.707
4) When the population standard deviation is unknown, the t-interval procedure is used instead of the
z-interval procedure. The t-interval procedure works provided that the population is normally
distributed or the.
This document provides an overview of statistical inference. It discusses descriptive statistics, which summarize data, and inferential statistics, which are used to generalize from samples to populations. Key concepts covered include estimation, hypothesis testing, parameters, statistics, confidence intervals, significance levels, types of errors. Examples are given of how to calculate confidence intervals for means and proportions and how to perform hypothesis tests using z-tests and t-tests. Steps for conducting hypothesis tests are outlined.
1) The document discusses statistical inference and hypothesis testing. It covers topics like point and interval estimation, confidence intervals, hypothesis testing steps and terminology, tests for population means and proportions, and chi-square tests for independence.
2) An example calculates a 95% confidence interval for the mean hours students work per week based on sample data.
3) The final section discusses contingency tables and chi-square tests, providing an example to test if hand dominance and gender are associated using a contingency table. It shows calculating expected frequencies and the chi-square test statistic to evaluate the null hypothesis of independence.
Similar to IN ORDER TO IMPLEMENT A SET OF RULES / TUTORIALOUTLET DOT COM (20)
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
IN ORDER TO IMPLEMENT A SET OF RULES / TUTORIALOUTLET DOT COM
1. AC 202 Solve each trigonometric equation
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Solve each trigonometric equation in the interval [0,2n) by first
squaring both sides. fleas x=1+sin x Select the correct choice below
and, if necessary, fill in the answer box to complete your choice. O A-
The solution set is .
(Simplify your answer. Use a comma to separate answers as needed.
Type an exact answer, using 1: as needed. Use integers or fractions
for any numbers in the expression.) 0 B. There is no solution on this
interval.
Find all solutions of the equation in the interval [0,21r). tan2x=3
Select the correct choice below and, if necessary, fill in the answer
box to complete your choice. O A- The solution set is .
Solve the equation. 2sln29+sln9-1=0 What is the solution in the
interval 0 S 9 < 21:? Select the correct choice and fill in any answer
boxes in your choice below. O A- The solution set
Use trigonometric identities to solve the equation in the interval
[0,21r).
5cose= -54/§sine Select the correct choice below and, if necessary, fill
in the answer box to complete your choice. O A- The solution set is
Use trigonometric identities to solve the equation in the interval [0,
21:). sin 2x— cos 2x=1 Select the correct choice and fill in any
answer boxes in your choice below. 0A— x={D} (Type your answer
in radians. Type an exact answer, using 1: as needed. Use integers or
fractions for any numbers in the expression. Use a comma to separate
answers as needed.)
Find all solutions of the equation in the interval [0, 21:).
(tanx+1)(2 sinx—1)=0
Select the correct choice below and, if necessary, fill in the answer
box to complete your choice. 0A- x={j} (Simplify your answer. Type
2. an exact answer, using 1r, as needed. Type your answer in radians.
Use integers or fractions for any numbers in the expression. Use a
comma to separate answers as needed.)
Find all solutions of the equation in the interval [0, 21:). 8 sln 29 = 2
Select the correct choice and fill in any answer boxes in your choice
below. 0.. e={C} (Simplify your answer. Type an exact answer, using
1|: as needed. Type your answer in radians. Use integers or fractions
for any numbers in the expression. Use a comma to separate answers
as needed.)
------------------------------------------------------------------------------------
BTE 200 The following formula gives the distance between
two points
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The following formula gives the distance between two points, (x1, y1)
and (x2, y2) in the
Cartesian plane: (1‗2 — 1171)2 + (3/2 — 902 Given the center and a
point on the circle, you can use this formula to find the radius of the
circle.
Write a program that prompts the user to enter the center and a point
on the circle. The program
should then output the circle‘s radius, diameter, circumference, and
area. Your program must
have at least the following functions: a. distance: This function takes
as its parameters four numbers that represent two points in the
plane and returns the distance between them. b. radius: This function
takes as its parameters four numbers that represent the center and a
point on the circle, calls the function distance to find the radius of the
circle, and returns the
circle‘s radius. c. circumference: This function takes as its parameter
a number that represents the radius of the
circle and returns the circles circumference. (If r is the radius, the
3. circumference is 2 nr.) d. area: This function takes as its parameter a
number that represents the radius of the circle and
returns the circle‘s area. (If r is the radius, the area is 11r2.) Assume
that n = 3.1416.
------------------------------------------------------------------------------------
Chapter 8 Hypothesis Testing and Types of Errors
Business Statistics
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Chapter 8
Hypothesis Testing and Types of Errors Business Statistics January 4,
2017 ( Business Statistics) Chapter 8 January 4, 2017 1 / 48 1
Introduction to Hypothesis Testing 2 Hypothesis Testing for p 3
Hypothesis Testing for µ 4 Independent Samples: Hypothesis Testing
for µ1 − µ2 5 Dependent Samples: Hypothesis Testing for µdiff 6
Type I and Type II Errors ( Business Statistics) Chapter 8 January 4,
2017 2 / 48 Introduction to Hypothesis Testing Introduction to
Hypothesis Testing As we saw in the previous chapter, confidence
intervals estimate a
population mean or proportion by giving a range it likely falls in
using
a random sample of data.
Hypothesis testing has the same goal as a confidence interval,
informing us about a population mean or proportion, but with a
different approach.
A hypothesis test allows us to come to a conclusion about the mean
or proportion. ( Business Statistics) Chapter 8 January 4, 2017 3 / 48
Introduction to Hypothesis Testing Introduction to Hypothesis Testing
Let‘s look at some examples of when hypothesis testing would be
used.
1 After the recession, the fees charged by banks for late credit card
payments have increased. 2 The industry that promotes compact
fluorescent light bulbs claims the
4. bulbs use 75% less energy and last 10 times longer than incandescent
bulbs. 3 A Wall Street Journal article, from June 2010, titled ‗Does
the
Internet Make You Smarter or Dumber?‘ posed the possibility that
online activities turn us into shallow thinkers. Each of these examples
could be stated as a pair of opposing claims or
hypotheses. ( Business Statistics) Chapter 8 January 4, 2017 4 / 48
Introduction to Hypothesis Testing Null vs. Alternative
Looking at the three examples above, state the null and alternative
hypotheses.
General form
H0 vs. H1
1 H0 : The fees charged by banks for late credit card payments have
remained the same or decreased (not increased) after the recession.
H1 : The fees charged by banks for late credit card payments
increased after the recession. 2 H0 : Fluorescent light bulbs do not use
75% less energy and last 10
times longer than incandescent bulbs.
H1 : Fluorescent light bulbs use 75% less energy and last 10 times
longer than incandescent bulbs. 3 H0 : Online activities do not turn us
into shallow thinkers.
H1 : Online activities turn us into shallow thinkers.
( Business Statistics) Chapter 8 January 4, 2017 5 / 48 Introduction to
Hypothesis Testing Null vs. Alternative Hypothesis testing typically
begins with some claim (or belief) about
a particular parameter of a population.
Our initial assumption is known as the null hypothesis, denoted H0
(‗H naught‘). This typically is hinting that nothing is happening, the
status quo, no relationship, no difference, etc.
The null hypothesis is believed to be true unless there is
overwhelming evidence not to.
We use the sample data to see if the alternative hypothesis, denoted
Ha or H1 , is true.
The alternative hypothesis states that something is going on: a
difference, increase, decrease or relationship exists.
Typically, H1 is what the researchers hope to show. ( Business
Statistics) Chapter 8 January 4, 2017 6 / 48 Introduction to
5. Hypothesis Testing Hypothesis Testing Let‘s consider the American
justice system of ‗innocent until proven
guilty‘.
If we were to state this as a hypothesis, this would be:
H0 : The defendant is innocent.
H1 : The defendant is guilty. We make the initial assumption that the
defendant is innocent and
evidence at the trial (think sample data) either shows the claim of
guilt or it doesn‘t.
The evidence (sample data) will lead us to a verdict (conclusion). (
Business Statistics) Chapter 8 January 4, 2017 7 / 48 Introduction to
Hypothesis Testing Hypothesis Testing A hypothesis test is a five step
process.
1
2
3
4 5 State the hypotheses.
Calculate the test statistic (which is essentially a z-score)
Determine the p-value (we will discuss this in a moment).
Make a decision (where we side with either H0 or H1 ). To do this,
we
will need a significance level.
State a conclusion in terms of the problem. ( Business Statistics)
Chapter 8 January 4, 2017 8 / 48 Introduction to Hypothesis Testing
The p-value and Significance Level The p-value is the probability of
obtaining a test statistic as extreme
or more extreme than the one observed when we assume that H0 is
true.
We then compare the p-value to a significance level, denoted α, which
is similar to the confidence level for a confidence interval.
Typically, we set α = 0.05, which is equivalent to the concept of 95%
confidence intervals having a 95% success rate and a 5% failure rate.
( Business Statistics) Chapter 8 January 4, 2017 9 / 48 Introduction to
Hypothesis Testing The p-value and Significance Level In order to
make a conclusion about which hypothesis is more likely to
be correct, we have the following decision rule.
If p-value ≤ α, we declare the result significant and ‗reject H0 ‘.
6. If p-value > α, we ‗fail to reject H0 ‘, The data did not provide
significant evidence to reject H0 . Please note that we never ‗accept
H0 ‘.
The problem of ‗accepting‘ is that this wording makes it seem that we
are convinced that the null hypothesis is true. ( Business Statistics)
Chapter 8 January 4, 2017 10 / 48 Introduction to Hypothesis Testing
The p-value and Significance Level Hypothesis testing is meant to
reject the null hypothesis when the
evidence is convincingly against it.
If we collect only a small sample, we may not see convincing
evidence
for the alternative hypothesis because the sampling error is so large.
For instance, if we observed three births and observed two boys, we
would not be willing to accept the hypothesis ‗2/3 of all births are
boys‘ even though the data doesn‘t provide evidence that it is false. (
Business Statistics) Chapter 8 January 4, 2017 11 / 48 Hypothesis
Testing for p Hypothesis Testing for p The pharmaceutical company
is claiming a hypothesis, or wanting to
show, that fewer than 20% of the patients who use a particular
medication experience side effects.
In a clinical trial with 400 patients, they find 68 patients experienced
side effects.
Initially we would assume that 20% or more of patients who use a
particular medication experience side effects. ( Business Statistics)
Chapter 8 January 4, 2017 12 / 48 Hypothesis Testing for p
Hypothesis Testing for p This can be written as:
H0 : 20% or more of the patients who use a particular medication
experience side effects.
H1 : Fewer than 20% of the patients who use a particular medication
experience side effects. We can also summarize this as:
H0 : p ≥ 0.20
H1 : p < 0.20
where p is the true proportion of patients who use a particular
medication that experience side effects. ( Business Statistics) Chapter
8 January 4, 2017 13 / 48 Hypothesis Testing for p Hypothesis
Testing for p Once we have the hypotheses stated, we then need to
calculate the
7. test statistic (z-score in this case).
From our sample we see that of 400 patients, 68 patients experienced
side effects.
This is equivalent to 17% (or pˆ = 68
400 = 0.17). From the sample we are already seeing some evidence to
support the
alternative (0.17 < 0.20), but is this significant? ( Business
Statistics) Chapter 8 January 4, 2017 14 / 48 Hypothesis Testing for p
Hypothesis Testing for p Recall that the z-score for a single
proportion is
−p
Z = pˆSE
q
where SE = p(1−p)
n
Note that we are using the standard error for p in the denominator
since we are assuming H0 is true. ( Business Statistics) Chapter 8
January 4, 2017 15 / 48 Hypothesis Testing for p Hypothesis Testing
for p 0.17−0.20
0.02 0.3
0.1 Z = −1.5 0.2 Probability Z= 0.4 Using the information provided
in the problem, we obtain
q
SE = 0.2(1−0.2)
= 0.02
400 0.0 Once we obtain the test
statistics, the next step is to
calculate the p-value. −4 −2 0 2 4 Z < −1.5 Remember that the p-
value is
the probability of obtaining a
test statistic as extreme or more
extreme. In this example we are
looking for a Z = −1.5 or
smaller (i.e.
P[Z ≤ −1.5]).
P[Z ≤ −1.5] = 0.0668 ( Business Statistics) Chapter 8 January 4, 2017
16 / 48 Hypothesis Testing for p One-tailed or Two-tailed Before we
8. continue with the example, we come to an important
distinction when calculating p-values.
In this example we are calculating the p-value for the alternative
hypothesis H1 : p < 0.20.
What we will see when shading the area we wished to calculate is that
we are looking for the area less than Z = −1.5.
There are three possible alternative hypotheses, those being
H1 : p < p0
H1 : p > p0
H1 : p 6= p0
where p0 is the null value. ( Business Statistics) Chapter 8 January 4,
2017 17 / 48 Hypothesis Testing for p One-tailed or Two-tailed The
first two types of alternative hypotheses are considered
one-tailed.
------------------------------------------------------------------------------------
CMPS 12A write a C program that operates in the same
way
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write a C
program that operates in the same way, i.e. same prompts, same input
and same output. However, the
requirements for this program are relaxed somewhat from those in
pa3 in that it is not necessary to filter out
all types of bad input.
Design your program to respond to string inputs by printing
Please enter a positive integer: as specified in pa3, then scan for
another integer. Respond similarly to integer input that is negative or
zero. It is not necessary to react to double inputs according to the pa3
specifications however. Thus you
may assume that things like ―25.78‖ will not be used as input to your
program. Everything you need to do
this was explained in the lab7 project description. In particular,
9. review the explanation of the scanf()
function given in that document before you begin this program. A
sample session follows.
% GCD
Enter a positive integer: sldkfj
Please enter a positive integer: -56
Please enter a positive integer: 56
Enter another positive integer: sldkjfdlk
Please enter a positive integer: -25
Please enter a positive integer: 25
The GCD of 56 and 25 is 1
% Recall the CheckInput sequence of examples in Java whose
purpose was to learn how to filter input from
standard input. A similar sequence of examples will be posted under
Examples/lab8 on the class
webpage. Study these carefully to learn how to read and discard non-
numeric string input.
Call your program GCD.c and write a Makefile that creates an
executable file called GCD. Include a clean
utility with the Makefile that deletes the executable. Submit both files
to the assignment name lab8. 1
------------------------------------------------------------------------------------
Convert the polar coordinates
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1- Convert the polar coordinates (-3, 135º) into rectangular
coordinates. Round the rectangular coordinates to the nearest
hundredth.
a) (-2.12, -2.12) b) (-2.12, 2.12) c) (2.12, 2.12) d) (2.12, -2.12)
2-The letters x and y represent rectangular coordinates. Write the
following equation using polar coordinates (r, 0).
10. x^2 + y^2 -4x = 0
a) r=4sin0 b) r=4cos0 c) rcos^2 0 =4sin0 d) rsin^2 0= 4cos0
3- The letter x and y represents rectangular coordinates. Write the
following equation using polar coordinates (r, 0)
x^2+4y^2=4
a) cos^2 0 + 4sin^2 0 = 4r b) 4cos^2 0 + sin^2 0 =4r c) r^2(cos^2
0 + 4sin^2 0) = 4 d) r^2(4cos^2 0 + sin^2 0) = 4
4) The letters r and 0 represent polar coordinates. Write the following
equation using rectangular coordinates (x, y).
r=10sin0
a) (x+y)^2=10x b) (x+y)^2=10y c) x^2+y^2=10y d) x^2+y^2=10x
------------------------------------------------------------------------------------
CS 2400 FUNCTIONS In addition to function
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FUNCTIONS In addition to function main, a program source module
may contain one or more other functions. The execution of the
program always starts with function main. The statements of a
function (other than function main) are executed only if that function
is called in
function main or any other function that is called in function main.
After the statements of a function are executed, the next statement to
be executed is the one that
follows the statement in which that function is called. Example
The statements of the program in figure F1 are executed in the
following order: 23, 24, 26, 15, 16,
27, 28, 31, 32, 34, 15, 16, 35, 36, and 37. There are two types of
functions in the C/C++ programming language: Functions that do not
12. // to hold the perimeter
10.
11.
/*------------------------function computeAreaPeri1 ---------------------*/
12.
/*------- compute the area and the perimeter of a rectangle -----------*/
13.
void computeAreaPeri1( void )
14.
{
15.
area = len * width;
16.
peri = 2 * ( len + width );
17.
}
18.
19.
int main ()
20.
{
21.
/*-------compute and print the area and the perimeter of a rectangle
with
length 20 and width 8 ---------------------------------------------*/
22.
23.
len = 20;
24.
width = 8;
25.
26.
computeAreaPeri1( );
27.
cout << endl << ―the area of the rectangle is:t‖
<< area;
28.
16. } Exercise F4
Write a void function without parameters computeTaxNet that uses
the gross pay of an individual to
compute his tax deduction and net pay that are printed in function
main. The tax deduction is computed
as follows: if the gross pay is greater than or equal to $1000.00, then
the tax deduction is 25% of the
gross pay; otherwise, it is 18% of the gross pay. The net pay is the
gross pay minus the tax deduction.
a. Function main first calls function computeTaxNet to compute the
tax deduction and the net pay for
the gross pay $1250.
b. It then reads the gross pay of an individual and then calls function
computeTaxNet to compute his
tax deduction and net pay.
c. You must first determine and define the global variables of this
program; then write function
------------------------------------------------------------------------------------
Determine whether they form a partition for the set of
integers
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For each the following groups of sets, determine whether they form
a partition for the set of integers. Explain your answer.
a. A1 = {n Z: n > 0}
A2 = {n Z: n < 0}
A1 contains all integers greater than 0.
A2 contains all integers less than 0.
17. 0 is not covered in both cases.
A1 ∩ A2 = ∅ ; however, A 1∪A 2 ≠ Z , as 0 is missing. Therefore, it
is not
a partition of the set of all integers.
.b. B1 = {n Z : n = 2k, for some integer k}
B2 = {n Z : n = 2k + 1, for some integer k}
B3 = {n Z : n = 3k, for some integer k}
B1 is the set of all even integers, B2 is the set of all odd integers, and
B3
is the set of all integers divisible evenly by 3. E.g. B3 = {3, 9, 12, 15,
18…}
This is a partition of Z since B1 U B2 U B3 ¿ Z.
2. (10 pts) Define f: Z → Z by the rule f(x) = 6x + 1, for all integers x.
a. Is f onto?
No. For a function to be onto the codomain must equal the range. A
counter example would be f(x)=y and solve for x. We find that x = y-
1/6.
Let y=0 and x = -1/6. This means that, in order to get 0 (an integer) as
the
resulting value for f(x), we have to input -1/6 into the function. -1/6 is
not
an integer; therefore, the function is not onto.
b. Is f one-to-one?
Yes, f is one-to-one. For every x there will be a different f(x) and for
every
18. f(x) there will be a different x.
c. Is it a one-to-one correspondence?
For a function to be a one-to-one correspondence it must be both one-
to-one and
onto. It must have a codomain equal to the range, and each element of
the domain
must map to only a single element in the range so based on exercise a
we can see that
is not onto therefore is not one-to-one correspondence. d. Find the
range of f
Range = {n ∈ Z | 6n+1 ∈
= {…, -11, -5, 1, 13, 19, …} Z} 3. (10 pts) f: R → R and g: R → R
are defined by the rules:
f(x) = x2 + 2 ∀
g(y) = 2y + 3 ∀ x ϵ R y ϵ R Find f ◦ g and g ◦ f
2
2
f ◦ g = f ( g ( y ) )=(2 y+ 3) +2=4 y +12 y +11
2
f ◦ g = 4 y +12 y +11 g ◦ f = g(f(x)) = g(x2 + 2)
=> g(x2 + 2) = 2(x2 + 2) + 3
=> 2x2 + 4 + 3 = 2x2 + 7
g ◦ f = 2x2 + 7
19. 4. (10 pts) Determine whether the following binary relations are
reflexive,
symmetric, antisymmetric and transitive:
a. x R y ⇔ xy ≥ 0 ∀ x, y ϵ R Reflexive - Any relation to be reflexive,
(x,x) should belong to R.
If we consider any value of x then x*x will always be an positive
value >0. For
example X=2, Y=2 2*2 > = 0 or X= -4 Y= -4, -4*-4> = 0
therefore we can say R is
reflexive. Symmetric - any relation to be symmetric, (x,y) should
belong to R and (y,x)
should also belong to R. here for any value of x and y if (x,y) belongs
to R i.e,
x*y>=0 then y*x will also be > = 0 thus (y,x) will also belong
to R. It is also
symmetric. Not antisymmetric because it is symmetric. Transitive -
any relation to be transitive, must hold if (x,y) and (y,z) belongs to R
then (x,z) should belong to R. When x*y>=0 and y*z>=0 the
we can say x*z will
also be >=0, thus (x,z) belongs to R. Is∀an
relation.
x equivalence
,
x> 0 [ x ] = {∀ y∨ y> 0 } ,
x< 0 [ x ] = { ∀ y| y< 0 } ,
20. x=0 [ x ] = R b. x R y ⇔ x > y ∀ x, y ϵ R Not reflexive: A
counterexample to prove is not would be x=y, x=4; therefore, x
should be greater than y, but since 4 is not greater than 4, this
relationship is not
reflexive. Not Symmetric - any relation to be symmetric, (x,y) should
belong to R and (y,x)
should also belong to R. For any value of x and y if (x,y) belongs to R
i.e, x>y
then y>x is not possible so we can say that R is not symmetric
There is no x, y pairs that relate back to each other. E.g. (x, y) is
found, but not
(y, x) for all x, y in R. Therefore, it is antisymmetric. Transitive - x, y,
z are related. If x > y, and y > z, then x > z. Is not an
equivalence relation, nor partial order.
c. x R y ⇔ |x| = |y| ∀ x, y ϵ R Reflexive - Any relation to be reflexive,
(x,x) should belong to R. If we consider
any value of x then |x|=|x| will hold. R is reflexive Symmetric - It is
symmetric was for all (x, y) there is a corresponding (y, x) pair.
E.g. (-1, 1), (1, -1). Because it is symmetric cannot be antisymmetric.
Not transitive since no number is related to each other. Is not an
equivalence relation, nor partial order. 5. (10 pts) Determine whether
the following pair of statements are logically
equivalent. Justify your answer using a truth table.
p → (q → r) and p ∧ q → r p ∧ q → r p (q r)
p Q r T
T
T
24. T
T
T
T 6. (10 pts) Prove or disprove the following statement:
∀ n, m ∈ Z, If n is even and m is odd, then n + m is odd
Then write the negation of this statement and prove or disprove it.
n + m is not odd
Given that n is even and m is odd, is always going to be odd;
therefore, the negation of
the previous statement is false. 7. (10 pts) Prove the following by
induction:
n
n+1
3 n2−n
3
i
–
2=
∑
=> ∑ ( 3 i – 2 ) +(3 n+1)
2
i=1
25. i=1
2 3 (n+1) −(n+ 1)
3 n 2−n
+(3 ( n+1 ) −2)=
2
2 2 3n^2 – n / 2 + (3n+1) = 3 n +5 n+ 2
2 8. (10 pts) Use the permutation formula to calculate the number
permutations of
the set {V, W, X, Y, Z} taken three at a time. Also list these
permutations. 5P3 = 5!
5!
= =60
( 5−3 ) ! 2! permutations {V,W,X} {V,W,Y} {V,W,Z} {V,X,W}
{V,X,Y} {V,X,Z} {V,Y,W} {V,Y,X} {V,Y,Z} {V,Z,W} {V,Z,X}
{V,Z,Y} {W,V,X} {W,V,Y} {W,V,Z} {W,X,V} {W,X,Y} {W,X,Z}
{W,Y,V} {W,Y,X} {W,Y,Z}
{W,Z,V} {W,Z,X} {W,Z,Y} {X,V,W} {X,V,Y} {X,V,Z} {X,W,V}
{X,W,Y} {X,W,Z} {X,Y,V}
{X,Y,W} {X,Y,Z} {X,Z,V} {X,Z,W} {X,Z,Y} {Y,V,W} {Y,V,X}
{Y,V,Z} {Y,W,V} {Y,W,X} {Y,W,Z}
{Y,X,V} {Y,X,W} {Y,X,Z} {Y,Z,V} {Y,Z,W} {Y,Z,X} {Z,V,W}
{Z,V,X} {Z,V,Y} {Z,W,V} {Z,W,X}
{Z,W,Y} {Z,X,V} {Z,X,W} {Z,X,Y} {Z,Y,V} {Z,Y,W} {Z,Y,X} 9.
(10 pts) Translate the following English sentences into statements of
predicate
26. calculus that contain double quantifiers and explain whether it is a
true
statement.
a. Every rational number is the reciprocal of some other rational
number.
P ( x ) ∙ N ( x )=1
( ∀ P ( x ) ϵ θ ) (∃ N ( x ) ϵ θ) ¿
statement is true
Reciprocal is inversion of rational number. Dividing would equal 1.
b. Some real number is bigger than all negative integers.
x is real numbers
( ∃ x ∈ R ) , x> y
y is negative integers
statement is true.
x = 2 and y = -4
2>4
10. (10 pts) Consider the following graph: In each case, answer the
question and then write the rationale for your answer.
a. Is this graph connected? Yes, no corners are separated from the rest
of the graph
b. Is this a simple graph?
Yes, there are no multi edges. There are no loops nor parallel edges.
c. Does this graph contain any cycles?
27. It is possible if we start in the middle traveling to the left and then
come
back to the starting point.
d. Does this graph contain an Euler cycle?
This cycle requires that all edges be used in a path that starts and
stops at
the same vertex.
Is this graph a tree?
No. It does not have any open ends or a root vertex.
------------------------------------------------------------------------------------
ECON 215 Students in an introductory psychology course
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Chapter 5 Students in an introductory psychology course take five
quizzes and two exams throughout the
semester. Each year, approximately 500 students take the course. The
first quiz has 10 multiplechoice questions where each question has
four choices with only one correct answer. The
passing score on the quiz is 80%.
a. (4 pts) If a student must resort to pure guessing on each question
(selects one of the four
answer choices randomly on each question),
What is the probability the student will pass the quiz? __________
b. (3 pts) If a student knows the material well enough to be able to
eliminate 2 incorrect choices
on each question but selects the answer randomly from the 2
remaining choices,
What is the probability the student will pass the quiz? __________
c. (3 pts) If a student knows the correct answer to five of the ten
28. questions, but must resort to
pure guessing on the remaining five questions,
What is the probability the student will pass the quiz? __________
d. (6 pts) There are five sections of the introductory psychology
course and each section has 100
students. Suppose that none of the students in section 2 studied for the
first quiz. If each of the
students in section 2 answered each of the 10 questions by randomly
selecting one of the four
choices,
What is the expected number of passing quiz scores in section 2‘s
student distribution of
quiz scores? __________
What is the expected value of section 2‘s quiz scores (in
percentages)? __________
What is the standard deviation of the quiz scores (in percentages)?
______________
------------------------------------------------------------------------------------
In order to implement a DBMS
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In order to implement a DBMS, there must exist a set of rules which
state how the
database system will behave. For instance, somewhere in the DBMS
must be a set of
statements which indicate than when someone inserts data into a row
of a relation, it
has the effect which the user expects. One way to specify this is to use
words to write
an `essay' as to how the DBMS will operate, but words tend to be
imprecise and open to
interpretation. Instead, relational databases are more usually defined
using Relational
29. Algebra.
Relational Algebra is :
the formal description of how a relational database operates
an interface to the data stored in the database itself
the mathematics which underpin SQL operations
Operators in relational algebra are not necessarily the same as SQL
operators, even if
they have the same name. For example, the SELECT statement exists
in SQL, and also
exists in relational algebra. These two uses of SELECT are not the
same. The DBMS
must take whatever SQL statements the user types in and translate
them into relational
algebra operations before applying them to the database. Terminology
Relation - a set of tuples.
Tuple - a collection of attributes which describe some real world
entity.
Attribute - a real world role played by a named domain.
Domain - a set of atomic values.
Set - a mathematical definition for a collection of objects which
contains no
duplicates. Operators - Write
INSERT - provides a list of attribute values for a new tuple in a
relation. This
operator is the same as SQL.
DELETE - provides a condition on the attributes of a relation to
determine which
tuple(s) to remove from the relation. This operator is the same as
SQL.
MODIFY - changes the values of one or more attributes in one or
more tuples of
a relation, as identified by a condition operating on the attributes of
the relation.
This is equivalent to SQL UPDATE. Operators - Retrieval
There are two groups of operations:
Mathematical set theory based relations:
UNION, INTERSECTION, DIFFERENCE, and CARTESIAN
30. PRODUCT.
Special database operations:
SELECT (not the same as SQL SELECT), PROJECT, and JOIN.
Relational SELECT
SELECT is used to obtain a subset of the tuples of a relation that
satisfy a select
condition.
For example, find all employees born after 1st Jan 1950:
SELECTdob '01/JAN/1950'(employee) Relational PROJECT
The PROJECT operation is used to select a subset of the attributes of
a relation by
specifying the names of the required attributes.
For example, to get a list of all employees surnames and employee
numbers:
PROJECTsurname,empno(employee) SELECT and PROJECT
SELECT and PROJECT can be combined together. For example, to
get a list of
employee numbers for employees in department number 1: Figure :
Mapping select and project Set Operations - semantics
Consider two relations R and S. UNION of R and S
the union of two relations is a relation that includes all the tuples that
are either in
R or in S or in both R and S. Duplicate tuples are eliminated.
INTERSECTION of R and S
the intersection of R and S is a relation that includes all tuples that are
both in R
and S.
DIFFERENCE of R and S
the difference of R and S is the relation that contains all the tuples that
are in R
but that are not in S. SET Operations - requirements
For set operations to function correctly the relations R and S must be
union compatible.
Two relations are union compatible if
they have the same number of attributes
the domain of each attribute in column order is the same in both R
and S. UNION Example Figure : UNION INTERSECTION Example
31. Figure : Intersection DIFFERENCE Example Figure : DIFFERENCE
CARTESIAN PRODUCT
The Cartesian Product is also an operator which works on two sets. It
is sometimes
called the CROSS PRODUCT or CROSS JOIN.
It combines the tuples of one relation with all the tuples of the other
relation. CARTESIAN PRODUCT example Figure : CARTESIAN
PRODUCT JOIN Operator
JOIN is used to combine related tuples from two relations:
In its simplest form the JOIN operator is just the cross product of the
two
relations.
As the join becomes more complex, tuples are removed within the
cross product
to make the result of the join more meaningful.
JOIN allows you to evaluate a join condition between the attributes of
the
relations on which the join is undertaken.
The notation used is
R JOINjoin condition S JOIN Example Figure : JOIN Natural Join
Invariably the JOIN involves an equality test, and thus is often
described as an equi-join.
Such joins result in two attributes in the resulting relation having
exactly the same value.
A `natural join' will remove the duplicate attribute(s).
In most systems a natural join will require that the attributes have the
same name
to identify the attribute(s) to be used in the join. This may require a
renaming
mechanism.
If you do use natural joins make sure that the relations do not have
two attributes
with the same name by accident. OUTER JOINs
Notice that much of the data is lost when applying a join to two
relations. In some cases
this lost data might hold useful information. An outer join retains the
information that
32. would have been lost from the tables, replacing missing data with
nulls.
There are three forms of the outer join, depending on which data is to
be kept.
LEFT OUTER JOIN - keep data from the left-hand table
RIGHT OUTER JOIN - keep data from the right-hand table
FULL OUTER JOIN - keep data from both tables OUTER JOIN
example 1 Figure : OUTER JOIN (left/right) OUTER JOIN example
2 Figure : OUTER JOIN (full)
------------------------------------------------------------------------------------
MATH 6A Evaluate the line integral
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2.Evaluate the line integral ∫C7xy4ds, where C is the right half of the
circle x2+y2=36
3.Evaluate the line integral ∫CF⋅dr,
where F(x,y,z)=−sinxi−2cosyj+4xzk and C is given by the vector
function r(t)=t5i−t4j+t3k , 0≤t≤1.
4.Sketch the vector field F⃗ (x,y)=xj⃗ , the line segment
from (1,4) to (7,4), and the line segment from (4,5) to(4,7).
(a) Calculate the line integral of the vector field F⃗ along the line
segment from (1,4) to (7,4).
(b) Calculate the line integral of the vector field F⃗ along the line
segment from (4,5) to (4,7).
5.Suppose F⃗ (x,y)=−ysin(x)i⃗ +cos(x)j⃗ .
(a) Find a vector parametric equation for the parabola y=x2 from the
origin to the point (4,16) using t as a parameter.
r⃗ (t)=
(b) Find the line integral of F⃗ along the parabola y=x2 from the
origin to (4,16).
33. 6.Suppose F⃗ (x,y)=−yi⃗ +xj⃗ and C is the line segment from
point P=(5,0) to Q=(0,3).
(a) Find a vector parametric equation r⃗ (t) for the line segment C so
that points P and Q correspond to t=0 and t=1, respectively.
r⃗ (t)=
(b) Using the parametrization in part (a), the line integral
of F⃗ along C is
∫CF⃗ ⋅dr⃗ =∫baF⃗ (r⃗ (t))⋅r⃗ ′(t)dt=∫badt
with limits of integration a=and b=
(c) Evaluate the line integral in part (b).
(d) What is the line integral of F⃗ around the clockwise-
oriented triangle with corners at the origin, P, and Q? Hint: Sketch the
vector field and the triangle.
7.
Use a CAS to calculate ∫⟨ex−y,ex+y⟩⋅ds to four decimal places,
where is the curve y=sinx for 0≤x≤π9, oriented from left to right.
Answer:
8.
Evaluate the line integral∫Cydx+xdywhereCis the parameterized
pathx=t3,y=t2,2≤t≤5.
∫Cydx+xdy=
9.LetCbe the straight path from(0,0)to(5,5)and
letF⃗ =(y−x−4)i⃗ +(sin(y−x)−4)j⃗ .
(a) At each point of C, what angle does F⃗ make with a tangent vector
to C?
angle =
(Give your answer in radians.)
(b) Find the magnitude ∥F⃗ ∥ at each point of C.
∥F⃗ ∥=
34. Evaluate ∫CF⃗ ⋅dr⃗ .
∫CF⃗ ⋅dr⃗ =
10. Let C be the curve which is the union of two line segments, the
first going from (0, 0) to (-3, 1) and the second going from (-3, 1) to (-
6, 0).
Compute the line integral ∫C−3dy−1dx.
15.Suppose ∇f(x,y)=5ysin(xy)i⃗ +5xsin(xy)j⃗ , F⃗ =∇f(x,y), and C is
the segment of the parabola y=4x2from the point (3,36) to (4,64).
Then
∫CF⃗ ⋅dr⃗
16.Suppose F⃗ (x,y)=(x+4)i⃗ +(3y+2)j⃗ . Use the fundamental theorem
of line integrals to calculate the following.
(a) The line integral of F⃗ along the line segment C from the
point P=(1,0) to the point Q=(4,3).
∫CF⃗ ⋅dr⃗ =
(b) The line integral of F⃗ along the triangle C from the origin to the
point P=(1,0) to the point Q=(4,3) and back to the origin.
∫CF⃗ ⋅dr⃗ =
20.For the vector
field G⃗ =(yexy+5cos(5x+y))i⃗ +(xexy+cos(5x+y))j⃗ , find the line
integral of G⃗ along the curve C from the origin along the x-axis to
the point (5,0) and then counterclockwise around the circumference of
the circle x2+y2=25 to the point (5/2√,5/2√).∫CG⃗ ⋅dr⃗
21.
------------------------------------------------------------------------------------
MATH 201 In a public opinion poll
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35. 3) In a public opinion poll, approximately 85% of Americans felt that
having a police officer on patrol in the neighborhood made them safer
than having no police officer on patrol. The margin of error reported
was 3%. Construct an interval estimate using these figures.
4) For each of the following confidence levels, look up the critical z
values for a two-tailed test.
4a) 90% (Hint: 5% in each tail):
Work:
4b) 95% (Hint: 2.5% in each tail):
Work:
5)Remembering that a meta-analysis calculates one mean effect size
using the effect sizes of several studies, assume you are conducting a
meta-analysis over a set of five studies. The effect sizes for each study
are: d= .65, d = .19, d = .42, d = .08, d = .70
5a) Calculate the mean effect size of these studies.
5b) Use Cohen's conventions to describe the mean effect size you
calculated in part (a).
------------------------------------------------------------------------------------
MATH 333 Matrix Algebra and Complex Variables
Homework 5
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Note: show full steps to get full credit. 1. Consider function f (z) = z 3
.
(a) Use the definition of derivatives to calculate the derivative of f ,
the result should be a function
of z.
36. (b) Use Cauchy-Riemann equation to show that f is an entire function.
(c) Find the derivative of f (z) in terms of u and v.
2. At what points are the following functions not analytic
(a) z
z−3i . (b) z 2 −2iz
.
z 2 +4 3. Compute f 0 if f = 4z 3 −5z+1
2z−1 . 4. Show that the following functions are not analytic at any
point.
(a) f (z) = y + ix
(b) f (z) = z¯2
(c) f (z) = 2x2 + y + i(y 2 − x)
5. Show that (ez )0 = ez .
6. Express f (z) = e2¯z in u + iv form.
7. Solve ez−1 = −ie2 .
8. Express sin(−2i) in a + ib form.
9. Find all values of z such that cos z = −3i. 1
------------------------------------------------------------------------------------
PRINTABLE VERSION Quiz 8 Question 1 1 x + + 3
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Quiz 8
Question 1 1
πx + π + 3 .
(5
) Find the period for the following function: f (x) = 8 cos a) 2π
5 b) π
5 c) 10 d) 5 e) 10 π f) None of the above. Question 2
Find the phase shift for the following function: f (x) = 3 sin a) π units
to the left b) 3 units to the right c) 3 units to the left d) 2 units to the
left e) π units to the right f) None of the above. Question 3 1
πx + π − 2 .
(3
37. ) Give an equation of the form f (x) = A cos(Bx − C) + D which could
be used to represent the given
graph. (Note: C or D may be zero.) a) f (x) = 3 cos(2 x − π) b) f (x) =
3 cos(2 x − π) − 1 c) f (x) = 6 cos(2 x − π) + 1 d) f (x) = 3 cos(2 x − π)
+ 1 e) f (x) = 6 cos(2 x − π) − 1 f) None of the above. Question 4
Give an equation of the form f (x) = A sin(Bx − C) + D which could
be used to represent the given
graph. (Note: C or D may be zero.) a) f (x) = 3 sin(4 x − π
2) b) f (x) = 3 sin(4 x − π
−1
2) c) f (x) = 6 sin(4 x − π
+1
2) d) f (x) = 6 sin(4 x − π
−1
2) e) f (x) = 3 sin(4 x − π
+1
2) f) None of the above. Question 5
List all x -intercepts for y = −3 cos(4x + a) − π 5π
,
12 12 π
π π
on
the
interval
−
[ 6 , 2 ].
3) b) π 5π
,
12 24 c) π π
,
24 4 d) 0, e) π 7π
,
24 24 f) None of the above. 7π
24 Question 6
Write a sine function with a positive vertical displacement given the
following information: the amplitude
is 6 , the horizontal shift is 12 to the left, y − intercept is (0, 3), and
38. the period is 8 . 1
x + 3π − 3
(4
) a) f (x) = 6 sin b) f (x) = 6 sin( π
x − 3 π) − 3
4 c) f (x) = 6 sin( π
x + 3 π) + 3
8 d) f (x) = 6 sin( π
x + 3 π) + 3
4 e) f (x) = 6 sin f) None of the above. 1
x − 3π + 3
(4
) Question 7
The current I , in amperes, flowing through an AC (alternating
current) circuit at time t is I(t) = 200 sin(35 πt + a) Period: π
, where t ≥ 0 . Find the period and the horizontal shift.
6) 1
1
, shift
to the left.
35
105 b) Period: 1
1
, shift
to the right.
35
105 c) Period: 2
1
, shift
to the left.
35
420 d) Period: 2
1
, shift
to the right.
35
210 e) Period: 2
39. ------------------------------------------------------------------------------------
Prove a fundamental result about polynomials
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We will now prove a fundamental result about polynomials: every
non-zero polynomial of degreen (over a field F) has at most n roots. If
you don‘t know what a field is, you can assume in thefollowing that F
= R (the real numbers).
(a) Show that for any α ∈ F, there exists some polynomial Q(x) of
degree n−1 and some b ∈ Fsuch that P(x) = (x−α)Q(x) +b.
(b) Show that if α is a root of P(x), then P(x) = (x−α)Q(x).
(c) Prove that any polynomial of degree 1 has at most one root. This is
your base case.(d) Now prove the inductive step: if every polynomial
of degree n−1 has at most n−1 roots, thenany polynomial of degree n
has at most n roots
------------------------------------------------------------------------------------
QNT 295 The H2 Hummer limousine has eight tires on it
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The H2 Hummer limousine has eight tires on it. A fleet of 1230 H2
limos was fit with a batch of tires that mistakenly passed quality
testing. The following table lists the frequency distribution of the
number of defective tires on the 1230 H2 limos.
Number of defective tires
0 1 2 3 4 5 6 7 8
40. Number of H2 limos 50 228 333 328 195
76 16 3 1
Construct a probability distribution table for the numbers of defective
tires on these limos.
Round your answers to three decimal places.
x P(x)
0 0
1 228
2 666
3 904
4 780
5 380
6 96
7 21
8 8
Calculate the mean and standard deviation for the probability
distribution you developed for the number of defective tires on all
1230 H2 Hummer limousines. Round your answers to three decimal
places. There is an average of 342.55 defective tires per limo, with a
standard deviation of 18.50 tires.
The H2 Hummer limousine has eight tires on it. A fleet of 1230 H2
limos was fit with a batch of tires that mistakenly passed quality
testing. The following table lists the frequency distribution of the
number of defective tires on the 1230 H2 limos.
Number of defective tires
41. 0 1 2 3 4 5 6 7 8
Number of H2 limos 50 228 333 328 195
76 16 3 1
Construct a probability distribution table for the numbers of defective
tires on these limos. Round your answers to three decimal places.
x P(x)
0 0
1 228
2 666
3 904
4 780
5 380
6 96
7 21
8 8
Calculate the mean and standard deviation for the probability
distribution you developed for the number of defective tires on all
1230 H2 Hummer limousines. Round your answers to three decimal
places. There is an average of 342.55 defective tires per limo, with a
standard deviation of 18.50 tires.
------------------------------------------------------------------------------------
Question numbers refer to the exercises at the end of each
section of the textbook
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42. www.tutorialoutlet.com
Question numbers refer to the exercises at the end of each section of
the textbook (1St edition numbers are given if they are different). You
must show your work to get full marks. For
some questions, I have given hints, clarifications, or extra instructions.
Be sure to follow these! Note: Only a selection of exercises may be
graded. 1) A reflection in R2 across a line I. through the origin is a
linear transformation that
takes a vector 56 to its ―mirror image‖ on the opposite side of L.
Suppose that M is the standard matrix for reflection across L. L
a. What happens to any 56 if you apply the reflection ‗- g}
transformation twice?
b. Explain why this tells you that M 2 = I, must be true. a 3 c. Show
that it follows that M '1 = M . d. Bonus: It is a remarkable fact that
doing a reflection across one line followed by a reflection across
another line always ends up being a rotation. This can be proven with
matrix multiplication, as follows. The matrix for a reflection across
the line that is inclined at angle a is given _ cos(2a) sin(2a)
by M― ' sin(2a) -cos(2a) ‘ and the matrix for a rotation by angle 6' is
given by R9 = sin 6 cos 6 cos!) — sing ]_
Use matrix multiplication and trigonometric identities to prove that
the composition of a reflection across the line with angle a with
another reflection across the line with angle 5 is a rotation. What is
the angle of this rotation? 2) Do textbook question 3.3 #54. a
3) Let S be the subset of R4 consisting of all vectors that have the
form 3; 3g , b where a and b are any scalars.
a. Find four different vectors that are in S. You should show how you
know they are in S. Then write down one vector in R4 that is not in S,
with an explanation
of how you know that. b. Prove that S is a subspace of R4 by proving
that it satisfies all three
conditions of a subspace. Then find a basis for the subspace (with
some explanation of how you know it is a basis) and the dimension of
the subspace
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