The document discusses significance testing and how to carry out tests to evaluate claims. It introduces the basics of significance testing including stating hypotheses, checking assumptions, finding test statistics and p-values, and making decisions. An example is provided to demonstrate how to conduct a z-test for a population mean to evaluate a claim about average blood pressure. The analysis finds no evidence to reject the null hypothesis that the mean blood pressure is different than the national average.
The document discusses significance testing and how to carry out tests to evaluate claims. It explains the basics of significance testing including stating hypotheses, checking assumptions, choosing a test, finding test statistics and p-values, making decisions, and summarizing. An example tests the claim that the average blood pressure of male executives is different than the national average. The document also covers type I and type II errors, and how to increase the power of significance tests.
This document introduces key concepts in hypothesis testing. It defines the null and alternative hypotheses (H0 and HA), which are competing statements about a population parameter. H0 is assumed true unless rejected. Type I and II errors occur when the wrong hypothesis is chosen. A test statistic quantifies how far the sample result is from what H0 predicts; if it falls in the "critical region", H0 is rejected. The critical region and significance level α determine the decision rule for each test.
Lecture6 Applied Econometrics and Economic Modelingstone55
The manager of a pizza restaurant conducted an experiment to determine if customers prefer a new baking method for pepperoni pizzas. He provided 100 randomly selected customers with both an old-style and new-style pizza and had them rate the difference on a scale from -10 to 10. Based on the customer ratings, the manager wants to use hypothesis testing to determine if he should switch to the new baking method. The null hypothesis is that customers are indifferent between the methods, while the alternative hypothesis is that customers prefer the new method. The results of the experiment provide strong statistical evidence to reject the null hypothesis and support switching to the new baking method.
This document provides an overview of hypothesis testing concepts and procedures. It begins with definitions of hypothesis tests and their purpose in determining if a sample value occurs by random chance or is statistically significant. It then covers the nature of hypothesis tests by first supposing the null hypothesis is true and then seeing if data provides evidence against it. The document outlines the steps of hypothesis testing including assumptions, hypotheses statements, calculations, and conclusions. It provides examples of writing hypotheses, calculating p-values, and writing conclusions. It also covers key concepts like significance levels, statistically significant results, and facts about p-values. Finally, it discusses matched pairs tests and provides an example to determine if a situation represents matched pairs. (179 words)
This document discusses significance tests for population means and proportions using Student's t-distribution and the normal distribution. It provides examples of hypothesis testing for a population mean using a paired t-test and for a population proportion using a single-sample z-test. It also discusses the assumptions, test statistics, and interpretations for these tests. Confidence intervals are presented as complementary to significance tests for estimating population parameters.
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 discusses hypothesis testing, including:
1) The steps of hypothesis testing are stating the null and alternative hypotheses, calculating a test statistic and p-value, and drawing a conclusion about whether to reject the null hypothesis.
2) A hypothesis test is used to assess evidence against a claim, while a confidence interval estimates a population parameter.
3) The null hypothesis is the initial claim and the alternative hypothesis is what the test is assessing evidence for. A small p-value provides evidence against the null hypothesis.
4) The conclusion is to reject the null hypothesis if the p-value is below the significance level, otherwise fail to reject the null hypothesis due to insufficient evidence against it.
Stuck with your hypothesis testing Assignment. Get 24/7 help from tutors with Phd in the subject. Email us at support@helpwithassignment.com
Reach us at http://www.HelpWithAssignment.com
The document discusses significance testing and how to carry out tests to evaluate claims. It explains the basics of significance testing including stating hypotheses, checking assumptions, choosing a test, finding test statistics and p-values, making decisions, and summarizing. An example tests the claim that the average blood pressure of male executives is different than the national average. The document also covers type I and type II errors, and how to increase the power of significance tests.
This document introduces key concepts in hypothesis testing. It defines the null and alternative hypotheses (H0 and HA), which are competing statements about a population parameter. H0 is assumed true unless rejected. Type I and II errors occur when the wrong hypothesis is chosen. A test statistic quantifies how far the sample result is from what H0 predicts; if it falls in the "critical region", H0 is rejected. The critical region and significance level α determine the decision rule for each test.
Lecture6 Applied Econometrics and Economic Modelingstone55
The manager of a pizza restaurant conducted an experiment to determine if customers prefer a new baking method for pepperoni pizzas. He provided 100 randomly selected customers with both an old-style and new-style pizza and had them rate the difference on a scale from -10 to 10. Based on the customer ratings, the manager wants to use hypothesis testing to determine if he should switch to the new baking method. The null hypothesis is that customers are indifferent between the methods, while the alternative hypothesis is that customers prefer the new method. The results of the experiment provide strong statistical evidence to reject the null hypothesis and support switching to the new baking method.
This document provides an overview of hypothesis testing concepts and procedures. It begins with definitions of hypothesis tests and their purpose in determining if a sample value occurs by random chance or is statistically significant. It then covers the nature of hypothesis tests by first supposing the null hypothesis is true and then seeing if data provides evidence against it. The document outlines the steps of hypothesis testing including assumptions, hypotheses statements, calculations, and conclusions. It provides examples of writing hypotheses, calculating p-values, and writing conclusions. It also covers key concepts like significance levels, statistically significant results, and facts about p-values. Finally, it discusses matched pairs tests and provides an example to determine if a situation represents matched pairs. (179 words)
This document discusses significance tests for population means and proportions using Student's t-distribution and the normal distribution. It provides examples of hypothesis testing for a population mean using a paired t-test and for a population proportion using a single-sample z-test. It also discusses the assumptions, test statistics, and interpretations for these tests. Confidence intervals are presented as complementary to significance tests for estimating population parameters.
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 discusses hypothesis testing, including:
1) The steps of hypothesis testing are stating the null and alternative hypotheses, calculating a test statistic and p-value, and drawing a conclusion about whether to reject the null hypothesis.
2) A hypothesis test is used to assess evidence against a claim, while a confidence interval estimates a population parameter.
3) The null hypothesis is the initial claim and the alternative hypothesis is what the test is assessing evidence for. A small p-value provides evidence against the null hypothesis.
4) The conclusion is to reject the null hypothesis if the p-value is below the significance level, otherwise fail to reject the null hypothesis due to insufficient evidence against it.
Stuck with your hypothesis testing Assignment. Get 24/7 help from tutors with Phd in the subject. Email us at support@helpwithassignment.com
Reach us at http://www.HelpWithAssignment.com
This presentation includes detailed information on Hypothesis testing for large and small samples, for two sample means. Briefed computational procedure with various case studies.
The document discusses hypothesis testing and outlines the steps to conduct hypothesis testing. It defines key terms like the null hypothesis, alternative hypothesis, and level of significance. It also explains how to determine the rejection region and calculate the test statistic. An example is provided where the null hypothesis that the mean number of TVs in US homes is at least 3 is tested using sample data and the steps of hypothesis testing.
The document provides an overview of hypothesis testing, including defining the null and alternative hypotheses, types of errors, significance levels, critical values, test statistics, and conducting hypothesis tests using both the traditional and p-value methods. Examples are provided for z-tests, t-tests, and tests of proportions to demonstrate applications of hypothesis testing methodology.
A one-sample z-test is used to compare a sample proportion to a population proportion. The document provides an example where a survey claims 90% of doctors recommend aspirin, and a sample of 100 doctors found 82% recommend aspirin. The z-test is calculated to determine if this difference is statistically significant. The null hypothesis is the sample and population proportions are the same. If the calculated z-statistic falls outside the critical values of -1.96 and 1.96, the null will be rejected, meaning the proportions are significantly different.
This document provides an overview of biostatistics. It defines biostatistics and discusses variables that can be studied, including discrete and continuous variables. It describes common software used for analysis and summarizes typical descriptive measures like mean, median, standard deviation, etc. The document outlines common types of comparisons between continuous and categorical variables, including t-tests, ANOVA, and chi-square tests. It also discusses concepts like alpha, beta, power, and cautions around hypothesis testing and interpreting statistical significance.
This document describes how to perform a one-sample z-test to compare a sample proportion to a population proportion. It provides an example where a survey claims 90% of doctors recommend aspirin, and a sample of 100 doctors found 82% recommend aspirin. It outlines calculating the z-statistic to determine if this difference is statistically significant using a 95% confidence level. The z-statistic is calculated to be -1.08, which falls within the acceptable range so the null hypothesis that the population and sample proportions are the same is retained.
The document discusses hypothesis testing, which involves testing a hypothesis about a population using a sample of data. It explains that a hypothesis test has four main steps: 1) stating the null and alternative hypotheses, where the null hypothesis asserts there is no difference between the sample and population, 2) setting the significance level, 3) determining the test statistic and critical region for rejecting the null hypothesis, and 4) making a decision to reject or fail to reject the null hypothesis based on whether the test statistic falls in the critical region. Type I and type II errors are also defined. The document provides examples of null and alternative hypotheses using mathematical symbols and discusses how to determine if a p-value is statistically significant.
Calculating a single sample z test by handKen Plummer
The document explains how to calculate a single-sample z-test. It provides an example of testing a claim that 9 out of 10 doctors recommend aspirin by taking a random sample of 100 doctors, of which 82 recommend aspirin. It defines the null and alternative hypotheses, identifies the critical z-value of -1.96 and +1.96, and shows the step-by-step calculations to find the z-statistic of -2.67, which falls outside the critical values. This indicates the sample result is statistically significant and differs from the claimed population value.
This document provides an overview of hypothesis testing basics:
A) Hypothesis testing involves stating a null hypothesis (H0) and alternative hypothesis (Ha) based on a research question. H0 assumes no effect or difference, while Ha claims an effect.
B) A test statistic is calculated from sample data and compared to a theoretical distribution to evaluate H0. For a one-sample z-test with known standard deviation, the test statistic is a z-score.
C) The p-value represents the probability of observing the test statistic or a more extreme value if H0 is true. Small p-values provide evidence against H0. Conventionally, p ≤ 0.05 is considered significant
This document provides an introduction to probability and important concepts in probability theory. It defines probability as a measure of the likelihood of an event occurring based on chance. Probability can be estimated empirically by calculating the relative frequency of outcomes in a series of trials, or estimated subjectively based on experience. Classical probability uses an a priori approach to assign probabilities to outcomes that are considered equally likely, such as outcomes of rolling dice or drawing cards. The document provides examples and definitions of key probability terms and concepts such as sample space, events, axioms of probability, and approaches to calculating probability.
Meaning of Probability, Experiment. Events – Simple and Compound, Sample Space, Probability of Events, Event Independent and Dependent Events, Probability Laws Bayes Theorem
This document outlines the key steps and concepts in hypothesis testing. It introduces hypothesis testing as a method to test conjectures about population parameters. The null hypothesis states that there is no difference or effect, while the alternative hypothesis specifies an expected difference or effect. Steps in hypothesis testing include stating hypotheses, determining critical values, defining critical and noncritical regions, and making a decision to reject or fail to reject the null hypothesis. Type I and type II errors are also defined. The document provides examples and outlines objectives for understanding hypothesis testing of means, proportions, variances and using confidence intervals.
The document describes how to calculate a single-sample z-test. It explains that the z-critical value is determined by the significance level, often 0.05 corresponding to z-values of -1.96 and 1.96. An example calculates the z-statistic to test if sample data matches a population claim. It finds the z-statistic is -2.67, which is outside the critical values and considered rare, so the null hypothesis that the sample matches the claim is rejected.
The document provides an overview of key concepts in probability theory including:
- Definitions of probability, experiments, sample space, and events
- Approaches to probability including classical, statistical, and subjective
- Rules of probability including addition, multiplication, and conditional probability
- Bayes' theorem and how it differs from conditional probability
- Random variables and their probability distributions
The document is intended to introduce students to probability concepts and their applications in decision making under uncertainty.
This document provides instructions for conducting two statistical tests: Spearman's Rank Correlation Coefficient and Chi-Square. Spearman's Rank is used to analyze the relationship between two variables like distance and environmental quality. It involves ranking values, calculating differences between ranks, and using a formula to determine if the relationship is statistically significant. Chi-Square analyzes relationships between categorical variables like opinions and demographics. It involves creating a results table, calculating expected values, applying a formula, and determining statistical significance based on degrees of freedom. Both tests are used to evaluate a null hypothesis of no relationship between variables.
This document provides an introduction to hypothesis testing, including:
1. Defining hypotheses as claims about population parameters and the distinction between the null and alternative hypotheses.
2. Explaining the hypothesis testing process, including specifying the significance level, determining the rejection region, calculating test statistics, and making a decision.
3. Providing examples of one-sample z-tests and t-tests for the mean when the population standard deviation is known and unknown.
4. Discussing type I and type II errors and how significance levels influence the probability of each.
Detection theory involves making decisions based on observations. There are two main hypotheses: the null hypothesis (H0, e.g. target is absent) and alternative hypothesis (H1, e.g. target is present). A likelihood ratio test compares the likelihood ratio to a threshold to determine which hypothesis is chosen. There are two types of errors: Type I errors (false alarms) occur when H0 is rejected when it is true, and Type II errors (misses) occur when H1 is rejected when it is true. Bayesian decision theory aims to minimize the average risk by choosing the hypothesis with minimum cost based on prior probabilities and costs of different errors.
This document discusses inferential statistics and epidemiological research. It introduces concepts like the central limit theorem, standard error, confidence intervals, hypothesis testing, and different statistical tests. Specifically, it covers:
- The central limit theorem states that sample means will follow a normal distribution, even if the population is not normally distributed.
- Standard error is used to measure sampling variation and determine confidence intervals around sample statistics to estimate population parameters.
- Hypothesis testing involves a null hypothesis of no difference and an alternative hypothesis of a significant difference.
- Common tests discussed include chi-square tests to compare proportions between groups and determine if differences are significant.
The document discusses comparing two population parameters using sample data. It covers comparing two means using a two-sample t-test or z-test, and comparing two proportions using a two-sample z-test. Key assumptions for these tests include independent random samples from each population and sample sizes large enough for the sampling distributions to be approximately normal. An example compares systolic blood pressure for two groups, one taking calcium and one placebo, and finds no significant difference. A second example finds preschool significantly reduces the proportion needing later social services.
AP Stats Procedures for Two Independent SamplesJune Patton
The document discusses procedures for comparing two independent means, including:
1) The three conditions necessary are simple random samples from both populations, normally distributed sampling distributions, and sample sizes less than 1/10 the population size.
2) The test statistic takes the form of the difference between the two sample means divided by the standard error of the difference between the means.
3) The calculator should be used to conduct the hypothesis test and compute the confidence interval since the calculations are complex. Pooling the variances is not recommended.
This document discusses key concepts for collecting data and conducting research studies. It defines variables, data sets, and types of bias that can occur in data collection. Common sampling methods like simple random sampling, stratified sampling, and cluster sampling are described. The document also distinguishes between observational studies and experiments, noting that experiments allow researchers to control variables and determine causal effects. Key aspects of experimental design like treatments, placebos, and control groups are also explained.
This presentation includes detailed information on Hypothesis testing for large and small samples, for two sample means. Briefed computational procedure with various case studies.
The document discusses hypothesis testing and outlines the steps to conduct hypothesis testing. It defines key terms like the null hypothesis, alternative hypothesis, and level of significance. It also explains how to determine the rejection region and calculate the test statistic. An example is provided where the null hypothesis that the mean number of TVs in US homes is at least 3 is tested using sample data and the steps of hypothesis testing.
The document provides an overview of hypothesis testing, including defining the null and alternative hypotheses, types of errors, significance levels, critical values, test statistics, and conducting hypothesis tests using both the traditional and p-value methods. Examples are provided for z-tests, t-tests, and tests of proportions to demonstrate applications of hypothesis testing methodology.
A one-sample z-test is used to compare a sample proportion to a population proportion. The document provides an example where a survey claims 90% of doctors recommend aspirin, and a sample of 100 doctors found 82% recommend aspirin. The z-test is calculated to determine if this difference is statistically significant. The null hypothesis is the sample and population proportions are the same. If the calculated z-statistic falls outside the critical values of -1.96 and 1.96, the null will be rejected, meaning the proportions are significantly different.
This document provides an overview of biostatistics. It defines biostatistics and discusses variables that can be studied, including discrete and continuous variables. It describes common software used for analysis and summarizes typical descriptive measures like mean, median, standard deviation, etc. The document outlines common types of comparisons between continuous and categorical variables, including t-tests, ANOVA, and chi-square tests. It also discusses concepts like alpha, beta, power, and cautions around hypothesis testing and interpreting statistical significance.
This document describes how to perform a one-sample z-test to compare a sample proportion to a population proportion. It provides an example where a survey claims 90% of doctors recommend aspirin, and a sample of 100 doctors found 82% recommend aspirin. It outlines calculating the z-statistic to determine if this difference is statistically significant using a 95% confidence level. The z-statistic is calculated to be -1.08, which falls within the acceptable range so the null hypothesis that the population and sample proportions are the same is retained.
The document discusses hypothesis testing, which involves testing a hypothesis about a population using a sample of data. It explains that a hypothesis test has four main steps: 1) stating the null and alternative hypotheses, where the null hypothesis asserts there is no difference between the sample and population, 2) setting the significance level, 3) determining the test statistic and critical region for rejecting the null hypothesis, and 4) making a decision to reject or fail to reject the null hypothesis based on whether the test statistic falls in the critical region. Type I and type II errors are also defined. The document provides examples of null and alternative hypotheses using mathematical symbols and discusses how to determine if a p-value is statistically significant.
Calculating a single sample z test by handKen Plummer
The document explains how to calculate a single-sample z-test. It provides an example of testing a claim that 9 out of 10 doctors recommend aspirin by taking a random sample of 100 doctors, of which 82 recommend aspirin. It defines the null and alternative hypotheses, identifies the critical z-value of -1.96 and +1.96, and shows the step-by-step calculations to find the z-statistic of -2.67, which falls outside the critical values. This indicates the sample result is statistically significant and differs from the claimed population value.
This document provides an overview of hypothesis testing basics:
A) Hypothesis testing involves stating a null hypothesis (H0) and alternative hypothesis (Ha) based on a research question. H0 assumes no effect or difference, while Ha claims an effect.
B) A test statistic is calculated from sample data and compared to a theoretical distribution to evaluate H0. For a one-sample z-test with known standard deviation, the test statistic is a z-score.
C) The p-value represents the probability of observing the test statistic or a more extreme value if H0 is true. Small p-values provide evidence against H0. Conventionally, p ≤ 0.05 is considered significant
This document provides an introduction to probability and important concepts in probability theory. It defines probability as a measure of the likelihood of an event occurring based on chance. Probability can be estimated empirically by calculating the relative frequency of outcomes in a series of trials, or estimated subjectively based on experience. Classical probability uses an a priori approach to assign probabilities to outcomes that are considered equally likely, such as outcomes of rolling dice or drawing cards. The document provides examples and definitions of key probability terms and concepts such as sample space, events, axioms of probability, and approaches to calculating probability.
Meaning of Probability, Experiment. Events – Simple and Compound, Sample Space, Probability of Events, Event Independent and Dependent Events, Probability Laws Bayes Theorem
This document outlines the key steps and concepts in hypothesis testing. It introduces hypothesis testing as a method to test conjectures about population parameters. The null hypothesis states that there is no difference or effect, while the alternative hypothesis specifies an expected difference or effect. Steps in hypothesis testing include stating hypotheses, determining critical values, defining critical and noncritical regions, and making a decision to reject or fail to reject the null hypothesis. Type I and type II errors are also defined. The document provides examples and outlines objectives for understanding hypothesis testing of means, proportions, variances and using confidence intervals.
The document describes how to calculate a single-sample z-test. It explains that the z-critical value is determined by the significance level, often 0.05 corresponding to z-values of -1.96 and 1.96. An example calculates the z-statistic to test if sample data matches a population claim. It finds the z-statistic is -2.67, which is outside the critical values and considered rare, so the null hypothesis that the sample matches the claim is rejected.
The document provides an overview of key concepts in probability theory including:
- Definitions of probability, experiments, sample space, and events
- Approaches to probability including classical, statistical, and subjective
- Rules of probability including addition, multiplication, and conditional probability
- Bayes' theorem and how it differs from conditional probability
- Random variables and their probability distributions
The document is intended to introduce students to probability concepts and their applications in decision making under uncertainty.
This document provides instructions for conducting two statistical tests: Spearman's Rank Correlation Coefficient and Chi-Square. Spearman's Rank is used to analyze the relationship between two variables like distance and environmental quality. It involves ranking values, calculating differences between ranks, and using a formula to determine if the relationship is statistically significant. Chi-Square analyzes relationships between categorical variables like opinions and demographics. It involves creating a results table, calculating expected values, applying a formula, and determining statistical significance based on degrees of freedom. Both tests are used to evaluate a null hypothesis of no relationship between variables.
This document provides an introduction to hypothesis testing, including:
1. Defining hypotheses as claims about population parameters and the distinction between the null and alternative hypotheses.
2. Explaining the hypothesis testing process, including specifying the significance level, determining the rejection region, calculating test statistics, and making a decision.
3. Providing examples of one-sample z-tests and t-tests for the mean when the population standard deviation is known and unknown.
4. Discussing type I and type II errors and how significance levels influence the probability of each.
Detection theory involves making decisions based on observations. There are two main hypotheses: the null hypothesis (H0, e.g. target is absent) and alternative hypothesis (H1, e.g. target is present). A likelihood ratio test compares the likelihood ratio to a threshold to determine which hypothesis is chosen. There are two types of errors: Type I errors (false alarms) occur when H0 is rejected when it is true, and Type II errors (misses) occur when H1 is rejected when it is true. Bayesian decision theory aims to minimize the average risk by choosing the hypothesis with minimum cost based on prior probabilities and costs of different errors.
This document discusses inferential statistics and epidemiological research. It introduces concepts like the central limit theorem, standard error, confidence intervals, hypothesis testing, and different statistical tests. Specifically, it covers:
- The central limit theorem states that sample means will follow a normal distribution, even if the population is not normally distributed.
- Standard error is used to measure sampling variation and determine confidence intervals around sample statistics to estimate population parameters.
- Hypothesis testing involves a null hypothesis of no difference and an alternative hypothesis of a significant difference.
- Common tests discussed include chi-square tests to compare proportions between groups and determine if differences are significant.
The document discusses comparing two population parameters using sample data. It covers comparing two means using a two-sample t-test or z-test, and comparing two proportions using a two-sample z-test. Key assumptions for these tests include independent random samples from each population and sample sizes large enough for the sampling distributions to be approximately normal. An example compares systolic blood pressure for two groups, one taking calcium and one placebo, and finds no significant difference. A second example finds preschool significantly reduces the proportion needing later social services.
AP Stats Procedures for Two Independent SamplesJune Patton
The document discusses procedures for comparing two independent means, including:
1) The three conditions necessary are simple random samples from both populations, normally distributed sampling distributions, and sample sizes less than 1/10 the population size.
2) The test statistic takes the form of the difference between the two sample means divided by the standard error of the difference between the means.
3) The calculator should be used to conduct the hypothesis test and compute the confidence interval since the calculations are complex. Pooling the variances is not recommended.
This document discusses key concepts for collecting data and conducting research studies. It defines variables, data sets, and types of bias that can occur in data collection. Common sampling methods like simple random sampling, stratified sampling, and cluster sampling are described. The document also distinguishes between observational studies and experiments, noting that experiments allow researchers to control variables and determine causal effects. Key aspects of experimental design like treatments, placebos, and control groups are also explained.
This document provides examples and explanations of various graphical methods for describing data, including frequency distributions, bar charts, pie charts, stem-and-leaf diagrams, histograms, and cumulative relative frequency plots. It demonstrates how to construct these graphs using sample data on student weights, grades, ages, and other examples. The goal is to help readers understand different ways to visually represent data distributions and patterns.
This document discusses various numerical methods for describing data, including measures of central tendency (mean, median), variability (range, variance, standard deviation), and graphical representations (boxplots). It provides examples and formulas for calculating the mean, median, quartiles, interquartile range, variance, standard deviation, and constructing boxplots. Outliers are defined as observations more than 1.5 times the interquartile range from the quartiles.
This document discusses simulations of multinomial randomized response models. It describes how the Warner randomized response model was extended to allow for multiple mutually exclusive categories. The key points are:
1) Abul-Ela et al. (1967) defined the multinomial randomized response model which allows sampling of multiple groups during a single experiment.
2) Simulations of the trinomial randomized response model are discussed, including calculating variances, biases, and mean squared errors to compare to direct response models.
3) Optimal values of the randomization probabilities (p-values) need to be determined to minimize variance while maximizing compliance for the trinomial model.
1. A study examined survival times of patients with advanced cancers in different organs (stomach, bronchus, colon, ovary, or breast) treated with ascorbate.
2. An analysis of variance (ANOVA) was used to determine if survival times differed based on the affected organ. ANOVA compares the means of multiple groups and tests if they are equal.
3. The ANOVA test statistic, F, compares the variation between groups (mean square for treatments) to the variation within groups (mean square for error). If F exceeds a critical value, then at least one group mean is significantly different from the others.
1. The document discusses probability and chance experiments. It provides examples to illustrate key concepts such as sample space, events, and how to calculate probabilities.
2. One example examines student food preferences in a cafeteria, with the sample space consisting of all possible combinations of student gender and food line choice.
3. The document also covers conditional probability, explaining how to calculate the probability of an event given that another event has occurred. An example calculates the probability of nausea given being seated in the front of a bus.
The document describes multiple regression models and their applications. It begins by defining a general multiple regression model that relates a dependent variable to multiple predictor variables. It then discusses key aspects of multiple regression models like regression coefficients, the regression function, polynomial regression models, and qualitative predictor variables. The document provides examples of applying multiple regression to model lung capacity based on variables like height, age, gender, and activity level. It describes building different regression models and evaluating their fit and significance.
This document provides an overview of confidence intervals. It defines key terms like statistical inference, confidence level, and margin of error. It explains how to construct confidence intervals for means using the z-distribution when the population standard deviation is known, and using the t-distribution when it is unknown. It also covers how to estimate population proportions using the normal distribution. Examples are provided to demonstrate how to use the PANIC method to set up and calculate confidence intervals.
Chapter 20 and 21 combined testing hypotheses about proportions 2013calculistictt
This document discusses hypotheses testing and the reasoning behind it. It explains that hypotheses testing involves proposing a null hypothesis and an alternative hypothesis based on a parameter of interest. Data is then analyzed to either reject or fail to reject the null hypothesis. Specifically:
1) The null hypothesis proposes a baseline model or value for a parameter.
2) Statistics are calculated based on the data and compared to what we would expect if the null hypothesis is true.
3) If the results are inconsistent enough with the null hypothesis, we can reject it in favor of the alternative hypothesis. Otherwise we fail to reject the null hypothesis.
The goal is to quantify how unlikely the results would be if the null hypothesis is true,
This document provides an overview of hypothesis testing concepts that were covered in Lecture 2, including:
- Introduction to hypothesis testing and the concepts of testing hypotheses about population parameters.
- Examples of hypothesis testing in different contexts like determining if a new drug is effective or if a defendant is guilty or innocent.
- The key concepts of hypothesis testing including the null and alternative hypotheses, types of errors, rejection regions, test statistics, and p-values.
- Worked examples demonstrating how to conduct hypothesis tests about a population mean when the population standard deviation is known, including using the rejection region and p-value methods.
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 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.
We are a growth marketing agency, that helps startups and well establish companies to achieve rapid and sustainable growth. We use various types of marketing and product iterations — content marketing, social media marketing, paid ads, email marketing, SEO and viral strategies, among others, with a purpose to increase the conversion rate and achieve rapid growth of the user base.
This document discusses test of significance and summarizes key aspects of hypothesis testing. It defines the null hypothesis as the statistical hypothesis of no difference that provides a reference for examining how data departs from what is expected. The alternative hypothesis is any other hypothesis that would be accepted when the null hypothesis is rejected. The document outlines the steps in conducting a significance test, including stating the problem, formulating hypotheses, deciding on a significance level, choosing a test statistic, comparing results to critical values, making a decision, and concluding. It also discusses concepts like p-values, calculated versus tabulated values, and provides an example of a t-test and chi-square test.
This document discusses p-values and their significance in statistical hypothesis testing. It defines a p-value as the probability of obtaining a result equal to or more extreme than what was observed assuming the null hypothesis is true. Lower p-values indicate stronger evidence against the null hypothesis. The document outlines the steps in hypothesis testing which include stating hypotheses, determining acceptable type I and type II error rates, selecting a statistical test to calculate a test statistic, determining the p-value, making inferences, and forming conclusions. It emphasizes that statistical significance does not necessarily imply real-world significance.
Testing of Hypothesis, p-value, Gaussian distribution, null hypothesissvmmcradonco1
This document provides an overview of key concepts in statistical hypothesis testing. It defines what a hypothesis is, the different types of hypotheses (null, alternative, one-tailed, two-tailed), and statistical terms used in hypothesis testing like test statistics, critical regions, significance levels, critical values, type I and type II errors. It also explains the decision making process in hypothesis testing, such as rejecting or failing to reject the null hypothesis based on whether the test statistic falls within the critical region or if the p-value is less than the significance level.
The document provides an introduction to hypothesis testing, including its real-life applications, key definitions, and structure. It defines hypothesis testing as the process of testing the validity of a statistical hypothesis based on a random sample from a population. The document outlines the common steps in hypothesis testing: 1) stating the null and alternative hypotheses, 2) choosing a significance level, 3) determining the test statistic and decision criteria, 4) rejecting or failing to reject the null hypothesis, and 5) drawing a conclusion. It also defines important terminology like population mean, null and alternative hypotheses, test statistic, significance level, critical region, and p-value. Real-life examples from pharmaceutical testing and legal cases are provided to illustrate the motivation for hypothesis
This document provides an overview of key statistical analysis techniques used in research methods, including descriptive statistics, validity testing, reliability testing, hypothesis testing, and techniques for comparing means such as t-tests and ANOVA. Descriptive statistics like mean and standard deviation are used to summarize variables measured on interval/ratio scales, while frequency and percentage summarize nominal/ordinal scales. Validity is assessed through exploratory factor analysis (EFA) to establish underlying dimensions. Reliability is measured using Cronbach's alpha. Hypothesis testing involves stating null and alternative hypotheses and making decisions based on statistical tests and p-values. T-tests compare two means and ANOVA compares three or more means, both assuming equal variances based on Levene
Chapter 6 part2-Introduction to Inference-Tests of Significance, Stating Hyp...nszakir
Mathematics, Statistics, Introduction to Inference, Tests of Significance, The Reasoning of Tests of Significance, Stating Hypotheses, Test Statistics, P-values, Statistical Significance, Test for a Population Mean, Two-Sided Significance Tests and Confidence Intervals
This document discusses hypothesis testing and various statistical tests used for hypothesis testing including t-tests, z-tests, chi-square tests, and ANOVA. It provides details on the general steps for conducting hypothesis testing including setting up the null and alternative hypotheses, collecting and analyzing sample data, and making a decision to reject or fail to reject the null hypothesis. It also discusses types of errors, required distributions, test statistics, p-values and choosing parametric or non-parametric tests based on the characteristics of the data.
importance of P value and its uses in the realtime SignificanceSukumarReddy43
This document discusses p-values and their significance in statistical hypothesis testing. It defines a p-value as the probability of obtaining a result equal to or more extreme than what was actually observed. A smaller p-value indicates stronger evidence against the null hypothesis. The document outlines the steps in significance testing: stating the research question, determining the probability of erroneous conclusions, choosing a statistical test to calculate a test statistic, obtaining the p-value, making an inference, and forming conclusions. It explains the concepts of type I and type II errors and how a p-value below 0.05 is typically considered statistically significant.
Hypothesis Testing techniques in social research.pptSolomonkiplimo
1) This document discusses hypothesis testing and comparing populations. It covers developing null and alternative hypotheses, types of errors, significance levels, and approaches using p-values and critical values.
2) Key steps in hypothesis testing include specifying the null and alternative hypotheses, choosing a significance level, calculating a test statistic, and determining whether to reject the null based on the p-value or critical value.
3) Comparing two populations involves testing whether their means are equal or different. The standard deviations play a role in determining if sample means are close enough to indicate the true population means are probably the same or different.
This document describes how to perform a chi-square test to determine if two genes are independently assorting or linked. It explains that for a two-point testcross of a heterozygote individual, you expect a 25% ratio for each of the four possible offspring genotypes if the genes are independent. The chi-square test compares observed vs. expected offspring ratios. It notes that the standard test assumes equal segregation of alleles, which may not always be true.
The document discusses confidence intervals and hypothesis testing. It provides examples of constructing 95% confidence intervals for a population mean using a sample mean and standard deviation. It also demonstrates how to identify the null and alternative hypotheses, determine if a test is right-tailed, left-tailed, or two-tailed, and calculate p-values to conclude whether to reject or fail to reject the null hypothesis based on a significance level of 0.05. Examples include testing claims about population proportions and means.
The document discusses confidence intervals and hypothesis testing. It provides examples of constructing 95% confidence intervals for a population mean using a sample mean and standard deviation. It also demonstrates how to identify the null and alternative hypotheses, determine if a test is right-tailed, left-tailed, or two-tailed, and calculate p-values to conclude whether to reject or fail to reject the null hypothesis based on a significance level of 0.05. Examples include testing claims about population proportions and means.
The document discusses confidence intervals and hypothesis testing. It provides examples of constructing 95% confidence intervals for a population mean and proportion. It also demonstrates identifying the null and alternative hypotheses and interpreting the results of hypothesis tests, including calculating p-values.
This document provides an overview of statistical inference and hypothesis testing. It discusses key concepts such as the null and alternative hypotheses, type I and type II errors, one-tailed and two-tailed tests, test statistics, p-values, confidence intervals, and parametric vs non-parametric tests. Specific statistical tests covered include the t-test, z-test, ANOVA, chi-square test, and correlation analyses. The document also addresses how sample size affects test power and significance.
This document discusses sampling distributions and the central limit theorem. It defines key terms like population, statistic, and sampling distribution. It shows examples of how sampling distributions become more normal and less variable as the sample size increases. The central limit theorem states that for large sample sizes, the sampling distribution of the sample mean will be approximately normally distributed even if the population is not. It provides properties and rules for the sampling distributions of the sample mean and sample proportion.
This document discusses the importance of statistics and introduces key concepts. It explains that statistics involves collecting, analyzing, and drawing conclusions from data. It also defines important statistical terms like population, sample, variable, and different types of data. Frequency distributions are introduced as a way to organize categorical data by displaying the categories and associated frequencies or relative frequencies. An example frequency distribution is provided using vision correction data from a classroom example.
This document provides an overview of random variables and probability distributions. It defines discrete and continuous random variables and gives examples of each. Discrete random variables have probabilities associated with each possible value, while continuous random variables are defined by probability density functions where the area under the curve equals the probability. The document discusses how to calculate the mean, variance and standard deviation of discrete random variables from their probability distributions. It also covers how the mean and variance are affected for linear transformations of random variables.
This document summarizes bivariate data and linear regression analysis. It introduces scatterplots and the Pearson correlation coefficient as ways to examine relationships between two variables. A positive correlation indicates that as one variable increases, so does the other, while a negative correlation means one variable increases as the other decreases. The least squares line provides the best fit linear relationship between two variables by minimizing the sum of squared residuals. Calculating the slope and y-intercept of this line allows predicting y-values from x-values. Examples using bus fare and distance data demonstrate these concepts.
This document provides information on estimating population characteristics from sample data, including:
- Point estimates are single numbers based on sample data that represent plausible values of population characteristics.
- Confidence intervals provide a range of plausible values for population characteristics with a specified degree of confidence.
- Formulas are given for constructing confidence intervals for population proportions and means using large sample approximations or t-distributions.
- Guidelines for determining necessary sample sizes to estimate population values within a specified margin of error are also outlined.
This document discusses methods for comparing two population or treatment means, including notation, hypothesis tests, and confidence intervals. Key points covered include:
1) Notation for comparing two means includes the sample size, mean, variance, and standard deviation for each population or treatment.
2) Hypothesis tests for comparing two means can use a z-test if the population standard deviations are known, or a two-sample t-test if the standard deviations are unknown.
3) Confidence intervals can be constructed for the difference between two population means using a t-distribution, assuming independent random samples of sufficient size or approximately normal populations.
1. The document discusses categorical data analysis and goodness-of-fit tests. It introduces concepts such as univariate categorical data, expected counts, the chi-square test statistic, and assumptions of the chi-square test.
2. An example analyzes faculty status data from a university using a goodness-of-fit test to determine if the proportions are equal across categories. The test fails to reject the null hypothesis that the proportions are equal.
3. Tests for homogeneity and independence in two-way tables are described. Examples calculate expected counts and perform chi-square tests to compare populations' category proportions.
1. The document discusses hypothesis testing using a single sample. It outlines the formal structure of hypothesis tests including the null and alternative hypotheses.
2. Common hypothesis tests are presented including tests of a population proportion, mean, and variability. Examples of hypotheses and solutions are provided.
3. The key steps in a hypothesis testing analysis are defined including stating the hypotheses, selecting the significance level, computing the test statistic and p-value, and making a conclusion. Large sample and small sample tests are described.
This document provides an overview of simple linear regression and correlation. It defines key concepts such as the population regression line, the simple linear regression model equation, and assumptions of the model. Examples are provided to demonstrate calculating the least squares regression line, interpreting the slope and intercept, and evaluating goodness of fit using r-squared. Formulas are given for computing sums of squares, estimating the standard deviation of residuals, and constructing confidence intervals for the slope of the population regression line.
This document provides instructions for adding grades to a Google Site by hosting the grades on Dropbox. It explains how to sign up for a Dropbox account, export grades from EasyGrade Pro and save them to the Dropbox public folder, and embed the grades on a new "Grades" page on the Google Site using an iframe. The process is then tested and instructions are provided for uploading updated grades by exporting new reports from EasyGrade Pro and allowing Dropbox to automatically sync the changes.
This document provides an overview of how to perform chi-square tests for goodness of fit and tests of homogeneity using categorical data. It explains how to set up and carry out chi-square tests through defining hypotheses, calculating test statistics, determining p-values, and making decisions. Examples are provided for chi-square goodness of fit tests to determine if observed count data fits an expected distribution, as well as chi-square tests of homogeneity to assess if the distribution of one categorical variable is the same across categories of another variable. Calculator instructions are also given for performing the relevant calculations and statistical analyses on a TI-83/84 graphing calculator.
The document discusses regression analysis and constructing confidence intervals and conducting significance tests for the slope (β) of the regression line. It provides guidance on checking the assumptions of the regression model, outlines the steps for constructing a confidence interval for β which involves calculating the standard error of the slope (SEb) and the appropriate t-statistic. It also outlines the steps for a significance test on β, which involves defining the null and alternative hypotheses, checking assumptions, and determining whether to reject or fail to reject the null based on the calculated p-value. An example problem is presented to demonstrate applying these procedures.
This document discusses sampling distributions and their properties. It defines key terms like parameter, statistic, sampling variability, and sampling distribution. It explains that sampling distributions describe the distribution of all possible sample statistics from repeated sampling. The document then discusses sampling distributions for proportions and means specifically. It provides the formulas for the standard deviation of the sampling distribution and conditions for using the normal approximation. An example problem demonstrates how to calculate the probability of a sample proportion being more than 2% from the population parameter using the normal approximation.
The document summarizes key concepts about the binomial and geometric distributions:
The binomial distribution models the number of successes in a fixed number of yes/no trials where the probability of success is constant. The geometric distribution models the number of trials until the first success. Both have calculators functions and follow patterns for mean, standard deviation, and normal approximations. Formulas for probability mass and cumulative distribution functions are provided.
The document summarizes key concepts about the binomial and geometric distributions:
The binomial distribution models the number of successes in a fixed number of yes/no trials where the probability of success is constant. The geometric distribution models the number of trials until the first success. Both have calculators functions and follow patterns for mean, standard deviation, and normal approximations. Formulas for probability mass and cumulative distribution functions are provided.
This document summarizes key concepts about random variables. It defines discrete and continuous random variables and explains how to represent their probability distributions. Discrete variables have countable outcomes and are represented by probability mass functions, while continuous variables have uncountable outcomes and are represented by density curves. The mean and variance are introduced as measures of central tendency and spread for random variables. Formulas are provided for calculating the mean and variance of discrete and continuous random variables. Transformations of random variables are also discussed.
1) Simulation involves defining a scenario with known probabilistic outcomes, running the scenario many times to model likely outcomes, and comparing the results to alternative models.
2) There are 5 steps to simulation: state the problem, make assumptions, create a mathematical model, run many repetitions, and state conclusions.
3) Probability models describe random phenomena using a sample space (all possible outcomes) and assigning probabilities to each outcome or event.
This document provides an overview of podcasting and how to create a basic podcast using the free audio editing software Audacity. It explains that podcasts allow for on-demand listening of automatically downloaded audio files. It then demonstrates how to use Audacity to edit audio clips, record voiceovers, and mix elements into a final podcast file. The document concludes by explaining how to publish the finished podcast on the hosting site Pod-O-Matic, which provides subscribers an RSS feed to access new episodes.
This document provides instructions for adding grades to a Google Site by hosting the grades on a free web hosting service. It explains how to sign up for a free Tripod account, create a "Grades" page on the Google Site that embeds an iframe of the hosted grades, export grades from EasyGrade Pro as HTML files and zip them, upload the zipped grades file to Tripod, test that the grades display correctly, and update the hosted grades as needed by replacing the zipped files. The process allows teachers to securely share student grades online through their Google Site without exceeding Google's file hosting limits.
This document provides an overview of experimental design and sampling techniques in statistics. It defines key terms like population, sample, census, bias, and experimental units. It discusses different sampling methods like simple random sampling, stratified random sampling, cluster sampling, and multistage sampling. It also covers principles of experimental design like control, replication, and randomization. Finally, it describes different types of experimental designs including completely randomized design, block design, and matched pairs design.
3. The Pizza Problem Let us suppose that a certain pizza company claims that they deliver their pizza in an average of 20 minutes Now, we are told “average time” so it’s possible that they’ve delivered a pizza in 5 minutes, and it’s also possible that they delivered a pizza in 30 minutes If we order pizza 10 times, what average time will convince you that they’re claim is wrong? Welcome to significance testing!
4. How significance testing works Assume that a claim about an average or proportion is true Compute the average or prop of a sample Compare the sample with the sampling distribution for the claim and sample size. If the probability of obtaining the sample avg or prop is too low, we conclude that our claim is improbable, and reject it.
5. How significance testing works In all cases, we are comparing the sample with the sampling distribution for the claim and sample size
6. PHANTOMS (a framework) As with Confidence Intervals, there is an acronym to help you remember the steps of a significance test State the Parameter State the Hypothesis pair Check the Assumptions State the Name of the test Find the value of the Test Statistic Obtain a p-value Make a decision Summarize
7. State Parameters Parameters work the same way they did in Confidence Intervals = The true average of the Pizza Company’s delivery times x-bar = the average delivery time for a sample of 10 deliveries from the Pizza Company p = the proportion of all deliveries from the Pizza Company that are delivered in less than 20 minutes p-hat = the proportion of sample of 10 deliveries from the Pizza Company that are delivered in less than 20 minutes
8. Stating Hypotheses Hypotheses come in pairs: “the null hypothesis” H0 “H naught” This is the presumed claim For our purposes, our null hypothesis will always be in the forms: “= __” “p = ___”
9. Stating Hypotheses Hypotheses come in pairs: “the alternative hypothesis” Ha This is the suspicion of the researcher There are 3 alt hyps that we can test “ ≠ ___” (two-sided alternative) “p > ___” (one-sided alternative) “ < ___” (one-sided alternative)
10. Stating Hypotheses Notice: Hypotheses are always about the parameter ( or p, neverxbar or phat) Written Examples “H0: = 20 minutes Ha: > 20 minutes” “H0: p = 0.5 Ha: p < 0.5”
11. Checking the Assumptions Since we are comparing our samples to a sampling distribution (just like the last chapter), the assumptions are the same We will review them now:
12. Checking the Assumptions Assumptions for mean SRS IndependenceN > 10n Normality (a, b, or c must be true) (a) population is Normal, or (b) n > 30; Central Limit Theorem, or (c) Sample is approximately normal: (1) histogram single peak and symmetric, (2) Normal probability plot is linear, (3) no Outliers
13. Checking the Assumptions Assumptions for proportions SRS IndependenceN > 10n Normality np > 10nq > 10
14. Name of Test “one-sided z test for means” “two-sided z test for means” “one-sided t test for means” “two-sided t test for means” “one-sided z test for proportions” “two-sided z test for proportions” More on these later
15. Test Statistics Test Statistics are always of the form: Standard Deviation of the sampling distribution depends on the characteristic tested
16. Test Statistics Std Dev for mean ( known): Std Dev for mean ( unknown): Std Dev for proportions: Notice that we use ‘p’ and not ‘p-hat’
17. P-values The P-value is the probability of obtaining a measurement as extreme as the test statistic At its most basic, computing the P-Value is the same as computing area from a Normal curve or Student’s t-distribution Computation varies slightly when using 2-sided alternative vs. 1-sided alternative
18. P-values Two sided alternatives For these alt hyps, we calculate a p-val based on area “from two tails”
19. P-values Example: Let’s assume our sample of 24 has:x-bar = 22 and s = 1.53 H0: = 20Ha: 20 “2-sided t-test for means”
34. Making a decision The P-value serves as the indicator If the test statistic is likely under the presumed sampling distribution (i.e. the p-value is large), then we have no reason to reject the null-hypothesis If the test statistic is unlikely (i.e. the p-value is small), then we have reason to reject the null-hypothesis. “If the p-value is low, reject the Hoe”
35. Making a decision Significance level (‘alpha’ ) This is the probability level at which we will reject H0 Typical sig levels = 0.10, 0.05, 0.01 If no significance level is given, we will generally reject at the = 0.05 level. When p-val < , then we “reject H0 at the = __ level” When H0 is rejected, we say the data is “statistically significant at the = __ level”
36. Making a decision “Reject or Fail to Reject” When p val > alpha, we “fail to reject H0” This means that we do not have evidence to show H0 is incorrect This does not mean, H0 is “correct” When p val < alpha we “reject H0” This means that H0 is unlikely The new estimate for or p is our sample data (x-bar or p-hat)
37. WOW That was a lot of information! We will be going over this information again at a slower pace in the coming weeks. We’ll work out the mechanics later Understanding the basics and the “whys” right now will help you in the future!
42. Example 11.10 The mean systolic blood pressure for males 35 to 44 years is 128, and the standard deviation in this population is 15. The medical records of 72 male executives in this age group finds the mean systolic blood pressure is 129.93. Is this evidence that the mean blood pressure for all the company’s younger male executives is different than the national average?
43. Example 11.10 We are going to check to see if our sample comes from a population with the same and sigma as the national population. Because of this, our parameter will come from the national averages. The null hypothesis will assume that younger male executives have the same mean blood pressure as the national average. The null hypothesis will always assume “things are equal”
44. Example 11.10 Parameter “Let = average blood pressure of all younger male executives in the company” “Let x-bar = average blood pressure in the sample of 72 younger male executives from the company”
46. Example 11.10 Assumptions Simple Random Sample “We are not told that our sample is from an SRS. We should check how this sample was chosen. We will proceed as though this sample was an SRS” Independence “We are not told the size the population of young male executives. We should check that the population is greater than 10(72) = 720.” Normality “Because we have a large sample, the Central Limit Theorem guarantees that the sampling distribution is approximately Normal”
47. Example 11.10 Assumptions (cont.) The preceding example illustrates ‘what to do’ if you think that an assumption is not met. If you believe that an assumption is not met: (1) state the condition that must be qualified, (2) mention that it “needs to checked,” and (3) state you will “proceed as though this assumption was met” Always try to carry out the significance test.
48. Example 11.10 Name of the Test “We will conduct a z-test for a population mean”
53. Example 11.10 Make a Decision We are not given an in this examplewe should use the standard 0.05 significance level. The p-value is larger than our , so we should reject the null hypothesis Note: nothing needs to be written for this part of PHANTOMS
54. Example 11.10 Summarize “Approximately 27% of the time, a sample of size n =72 will produce an average at least as extreme as 129.93. Since this p-value is larger than a presumed = 0.05, we cannot reject our null hypothesis. We have no evidence to suggest that the mean systolic blood pressure of young executives is not 128.”
55. Example 11.10 Summarize (cont.) Note that the summary contains 3 parts: (1) Interpret the p-value
56. Approximately 27% of the time, a sample of size n =72 will produce an average at least as extreme as 129.93. Example 11.10 Summarize (cont.) Note that the summary contains 3 parts: (1) Interpret the p-value
57. Example 11.10 Summarize (cont.) Note that the summary contains 3 parts: (1) Interpret the p-value (2) Compare the p-value with
58. Example 11.10 Summarize (cont.) Note that the summary contains 3 parts: (1) Interpret the p-value (2) Compare the p-value with Since this p-value is larger than a presumed = 0.05, we cannot reject our null hypothesis.
59. Example 11.10 Summarize (cont.) Note that the summary contains 3 parts: (1) Interpret the p-value (2) Compare the p-value with (3) Interpret the conclusion in context
60. Example 11.10 Summarize (cont.) Note that the summary contains 3 parts: (1) Interpret the p-value (2) Compare the p-value with (3) Interpret the conclusion in context We have no evidence to suggest that the mean systolic blood pressure of young executives is not 128.
61. Tests and Confidence Intervals A “two-sided alternative” and the “confidence interval” are the same test. A test will reject the null hypothesis of a two-sided alternative when the test statistic is outside the confident interval with CL = 1 - The link between confidence intervals and a two-sided test is called “duality” Refer to example 11.12
64. More on Significance Levels The significance level for a test is informed by the plausibility of H0. If H0 is particularly “strong” or has a many years behind it, then the evidence must also be “strong” (small ) If we were trying to disprove the gravitational constant, the would have to be very, very small!
65. More on Significance Levels What are the consequences of rejecting H0? There will always be a cost/benefit to rejecting H0 If it is more expensive to reject than it is to fail to reject, then the evidence must be strong (small ) Consider the Toyota brake recall 2009
66. More on Significance Levels There is no “hard line” between reject and fail to reject There isn’t a real difference between = 0.10 and = 0.11 There is no sharp border between “statistically significant” and “statistically insignificant” As P-value decreases, the strength of the evidence increases Although = 0.05 is ‘handy rule of thumb,’ it is not a universal rule
67. Cautions Don’t forget to examine the data The presence of outliers can affect whether the significance tests are plausible “Statistically Significant” is not the same thing as “Important” Lack of significance may signal an important conclusion A Test of Significance is not appropriate for all data sets
69. “What if” we made the wrong decision? There are two kinds of wrong decisions: Reject a H0 that was actually true This is a “TYPE I ERROR” Fail to reject H0 that was false This is a “TYPE II ERROR” Some students find it helpful to think: “You can reject one hoe, but who can fail to reject two hoes” whatever floats your boat, eh?
70. “What if” we made the wrong decision? TYPE I ERROR The null hypothesis was true! The probability that we made this error will be same as (since H0 was true) You will need to know how to recognize this error in context and You will need to know the probability of making a Type I error
71. “What if” we made the wrong decision? TYPE II ERROR In this case, the null hypothesis was incorrect, but we failed to reject it The probability of making a Type II error is a “what if” calculation “What if is actually 42- what’s the probability that I fail to reject?” The probability of making a Type II error is known as
78. Type II Error is the area of the tail for the sampling distribution of the “what if” parameter value H0: = 5, xbar = 5.8, = 0.7, n = 40 Calculation of when a = 6 Since 0 > we need to calculate the left tail area .
79. Type II Error Mercifully, the AP exam will never ask you to compute You will be asked to interpret Remember that is always dependent on an alternative value of the parameter .
80. Power The probability that the significance test will reject H0 at an level for an alternative value of the parameter is the power of the test against the alternative. Power = 1- Power is the probability of not making a TYPE II error Lots of power is a good thing!
81. How to increase power Increase the significance level () Consider an alternative parameter that is further away from the null hypothesis Increase the sample size Decrease All the above have the effect of decreasing . Less = More power