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This document provides an overview of hypothesis testing and the steps involved. It discusses: 1) Defining the null and alternative hypotheses based on the research question. The null hypothesis represents "no difference" while the alternative hypothesis claims the null is false. 2) Calculating the test statistic, which is used to test the null hypothesis. For a one-sample z-test, this involves calculating the z-score when the population standard deviation is known. 3) Computing the p-value, which is the probability of observing a test statistic as extreme or more extreme than what was observed, assuming the null hypothesis is true. Small p-values provide strong evidence against the null. 4) Interpre

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Basic of Hypothesis Testing TEKU QM

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

Nonparametric and Distribution- Free Statistics

This document summarizes several nonparametric statistical tests that do not rely on assumptions about the population distribution, including the sign test, Wilcoxon signed-rank test, median test, Mann-Whitney test, Kolmogorov-Smirnov goodness-of-fit test, and Kruskal-Wallis one-way analysis of variance. It provides details on the theoretical background, test statistics, assumptions, and procedures for the sign test for single samples and paired data. An example application of the paired sign test to a dental study is shown.

hypothesisTestPPT.pptx

The document provides information about hypothesis testing and the steps involved. It discusses:
1. Formulating the null and alternative hypotheses, specifying the significance level, and choosing the appropriate test statistic.
2. Examples of hypothesis tests involving means, including the z-test for large samples when the population variance is known, and the t-test for small samples when the population variance is unknown.
3. The calculations and decisions involved in conducting hypothesis tests, such as computing the test statistic, comparing it to the critical value, and determining whether to reject or fail to reject the null hypothesis.

Hypothesis testing1

This document provides an overview of hypothesis testing. It begins by defining hypothesis testing and listing the typical steps: 1) formulating the null and alternative hypotheses, 2) computing the test statistic, 3) determining the p-value and interpretation, and 4) specifying the significance level. It then discusses different types of hypothesis tests for claims about a mean when the population standard deviation is known or unknown, as well as tests for claims about a population proportion. Examples are provided for each type of test to demonstrate how to apply the steps. The document aims to explain the concept and process of hypothesis testing for making data-driven decisions about statistical claims.

Hmisiri nonparametrics book

This document provides an overview of nonparametric statistical methods for analyzing ranked data. It discusses the Wilcoxon rank-sum test and sign test, which are nonparametric alternatives to the t-test that do not assume a normal distribution. The document explains how to rank data values and handle ties. It also provides examples of using the sign test to compare a sample mean to a hypothesized value and interpreting the results.

TEST OF SIGNIFICANCE.pptx

This document discusses hypothesis testing and significance tests. It defines key terms like parameters, statistics, sampling distribution, standard error, null and alternative hypotheses, type I and type II errors. It explains how to set up a hypothesis test, including choosing a significance level and critical value. Both one-tailed and two-tailed tests are described. Finally, it provides an overview of different types of significance tests for both large and small sample sizes.

hypothesis test

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

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.

Basic of Hypothesis Testing TEKU QM

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

Nonparametric and Distribution- Free Statistics

This document summarizes several nonparametric statistical tests that do not rely on assumptions about the population distribution, including the sign test, Wilcoxon signed-rank test, median test, Mann-Whitney test, Kolmogorov-Smirnov goodness-of-fit test, and Kruskal-Wallis one-way analysis of variance. It provides details on the theoretical background, test statistics, assumptions, and procedures for the sign test for single samples and paired data. An example application of the paired sign test to a dental study is shown.

hypothesisTestPPT.pptx

The document provides information about hypothesis testing and the steps involved. It discusses:
1. Formulating the null and alternative hypotheses, specifying the significance level, and choosing the appropriate test statistic.
2. Examples of hypothesis tests involving means, including the z-test for large samples when the population variance is known, and the t-test for small samples when the population variance is unknown.
3. The calculations and decisions involved in conducting hypothesis tests, such as computing the test statistic, comparing it to the critical value, and determining whether to reject or fail to reject the null hypothesis.

Hypothesis testing1

This document provides an overview of hypothesis testing. It begins by defining hypothesis testing and listing the typical steps: 1) formulating the null and alternative hypotheses, 2) computing the test statistic, 3) determining the p-value and interpretation, and 4) specifying the significance level. It then discusses different types of hypothesis tests for claims about a mean when the population standard deviation is known or unknown, as well as tests for claims about a population proportion. Examples are provided for each type of test to demonstrate how to apply the steps. The document aims to explain the concept and process of hypothesis testing for making data-driven decisions about statistical claims.

Hmisiri nonparametrics book

This document provides an overview of nonparametric statistical methods for analyzing ranked data. It discusses the Wilcoxon rank-sum test and sign test, which are nonparametric alternatives to the t-test that do not assume a normal distribution. The document explains how to rank data values and handle ties. It also provides examples of using the sign test to compare a sample mean to a hypothesized value and interpreting the results.

TEST OF SIGNIFICANCE.pptx

This document discusses hypothesis testing and significance tests. It defines key terms like parameters, statistics, sampling distribution, standard error, null and alternative hypotheses, type I and type II errors. It explains how to set up a hypothesis test, including choosing a significance level and critical value. Both one-tailed and two-tailed tests are described. Finally, it provides an overview of different types of significance tests for both large and small sample sizes.

hypothesis test

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

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.

Hypothesis Testing Assignment Help

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

Hypothesis testing an introduction

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

Lec12-Probability (1).ppt

The document discusses probability models and how they can be used to make inferences from data. It introduces concepts like random variables, probability distributions, conditional probability, Bayes' rule, and Markov models. It provides examples of how to compute probabilities both mathematically and through simulation. The goal is to choose a probability model that best explains the data and make predictions from that model.

Lec12-Probability.ppt

The document discusses probability models and how they can be used to make inferences from data. It introduces concepts like random variables, probability distributions, conditional probability, Bayes' rule, and Markov models. It provides examples of how to compute probabilities both mathematically and through simulation. The goal is to choose a probability model that best explains the data and make predictions from that model.

Lec12-Probability.ppt

The document discusses probability models and how they can be used to make inferences from data. It introduces concepts like random variables, probability distributions, conditional probability, Bayes' rule, and Markov models. It provides examples of how to compute probabilities both mathematically and through simulation. The goal is to choose a probability model that best explains the data and make predictions from that model.

Lec12-Probability.pptThe document discusses probability models and how they can be used to make inferences from data. It introduces concepts like random variables, probability distributions, conditional probability, Bayes' rule, and Markov models. It provides examples of how to compute probabilities both mathematically and through simulation. The goal is to choose a probability model that best explains the data and make predictions from that model.

Nonparametric and Distribution- Free Statistics _contd

This document summarizes the Wilcoxon signed-rank test, a nonparametric test used to test hypotheses about the median or location of a population when the assumptions of the t-test are not met. It provides an example applying the test to data on cardiac output measurements from 15 patients. The test calculates the differences between each observation and the hypothesized median, ranks the absolute values of the differences, and sums the ranks with positive and negative signs. The smaller of the two sums is the test statistic, which is compared to a critical value to determine if the null hypothesis that the population median equals the hypothesized value can be rejected.

08 test of hypothesis large sample.ppt

The document discusses quantitative methods and hypothesis testing. It covers key concepts like null and alternative hypotheses, types of hypothesis tests, test statistics, rejection regions, and p-values. Examples are provided to illustrate hypothesis testing for population means, proportions, differences between means and proportions. The goal is to introduce the mechanics and general procedure of hypothesis testing and how to report results through p-values.

Testing of hypothesis

This document discusses the process of testing hypotheses. It begins by defining hypothesis testing as a way to make decisions about population characteristics based on sample data, which involves some risk of error. The key steps are outlined as:
1) Formulating the null and alternative hypotheses, with the null hypothesis stating no difference or relationship.
2) Computing a test statistic based on the sample data and selecting a significance level, usually 5%.
3) Comparing the test statistic to critical values to either reject or fail to reject the null hypothesis.
Examples are provided to demonstrate hypothesis testing for a single mean, comparing two means, and testing a claim about population characteristics using sample data and statistics.

Bayesian statistics using r intro

This document provides an introduction to Bayesian statistics using R. It discusses key Bayesian concepts like the prior, likelihood, and posterior distributions. It assumes familiarity with basic probability and probability distributions. Examples are provided to demonstrate Bayesian estimation and inference for binomial and normal distributions. Specifically, it shows how to estimate the probability of success θ in a binomial model and the mean μ in a normal model using different prior distributions and calculating the resulting posterior distributions in R.

슬로우캠퍼스: scikit-learn & 머신러닝 (강박사)

This document discusses the normal distribution and its key properties. It also discusses sampling distributions and the central limit theorem. Some key points:
- The normal distribution is bell-shaped and symmetric. It is defined by its mean and standard deviation. Approximately 68% of values fall within 1 standard deviation of the mean.
- Sample statistics like the sample mean follow sampling distributions. When samples are large and random, the sampling distributions are often normally distributed according to the central limit theorem.
- Correlation and regression analyze the relationship between two variables. Correlation measures the strength and direction of association, while regression finds the best-fitting linear relationship to predict one variable from the other.

HYPOTHESIS TESTING.ppt

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.

Bayesian decision making in clinical research

The document discusses the applications of Bayesian statistics in clinical research and decision making. It provides 3 case studies that demonstrate how Bayesian and frequentist analyses can lead to different conclusions from the same data. The case studies show that Bayesian analysis provides the probability that a hypothesis is true given the data, while frequentist analysis does not. Overall, the document argues that Bayesian statistics is better aligned with clinical decision making compared to conventional frequentist statistics.

Day 3 SPSS

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.

Lec13_Bayes.pptx

(1) Bayes' theorem provides a framework for updating probabilities based on new information or evidence.
(2) It allows calculating the conditional probability of a hypothesis being true given observed data by combining the prior probability of the hypothesis with the likelihood of making the observation under that hypothesis.
(3) Bayesian inference uses Bayes' theorem to update beliefs as new data becomes available, representing uncertainty about unknown parameters as a probability distribution that is continually refined as more evidence is accumulated.

Hypothesis testing

This document discusses hypothesis testing, which involves drawing inferences about a population based on a sample from that population. It outlines the key elements of a hypothesis test, including the null and alternative hypotheses, test statistics, critical regions, significance levels, critical values, and p-values. Type I and Type II errors are explained, where a Type I error involves rejecting the null hypothesis when it is true, and a Type II error involves failing to reject the null when it is false. The power of a hypothesis test is defined as the probability of correctly rejecting the null hypothesis when it is false. Controlling type I and II errors involves considering the significance level, sample size, and population parameters in the null and alternative hypotheses.

HW1_STAT206.pdfStatistical Inference II J. Lee Assignment.docx

HW1_STAT206.pdf
Statistical Inference II: J. Lee Assignment 1
Problem 1. Suppose the day after the Drexel-Northeastern basketball game, a poll of 1000 Drexel students
was conducted and it was determined that 850 out of the 1000 watched the game (live or on television).
Assume that this was a simple random sample and that the Drexel undergraduate population is 20000.
(a) Generate an unbiased estimate of the true proportion of Drexel undergraduate students who watched
the game.
(b) What is your estimated standard error for the proportion estimate in (a)?
(c) Give a 95% confidence interval for the true proportion of Drexel undergraduate students who watched
the game.
Problem 2. (Exercise 18 in Chapter 7 of Rice) From independent surveys of two populations, 90% con-
fidence intervals for the population means are conducted. What is the probability that neither interval
contains the respective population mean? That both do?
Problem 3. (Exercise 23 in Chapter 7 of Rice)
(a) Show that the standard error of an estimated proportion is largest when p = 1/2.
(b) Use this result and Corollary B of Section 7.3.2 (also, on Page 17 of the lecture notes) to conclude that
the quantity
1
2
√
N − n
N(n − 1)
is a conservative estimate of the standard error of p̂ no matter what the value of p may be.
(c) Use the central limit theorem to conclude that the interval
p̂ ±
√
N − n
N(n − 1)
contains p with probability at least .95.
HW2_STAT206.pdf
Statistical Inference II: J. Lee Assignment 2
Problem 1. The following data set represents the number of NBA games in January 2016, watched by 10
randomly selected student in STAT 206.
7, 0, 4, 2, 2, 1, 0, 1, 2, 3
(a) What is the sample mean?
(b) Calculate sample variance.
(c) Estimate the mean number of NBA games watched by a student in January 2016.
(d) Estimate the standard error of the estimated mean.
Problem 2. True or false? Tell me why for the false statements.
(a) The center of a 95% confidence interval for the population mean is a random variable.
(b) A 95% confidence interval for µ contains the sample mean with probability .95.
(c) A 95% confidence interval contains 95% of the population.
(d) Out of one hundred 95% confidence intervals for µ, 95 will contain µ.
Problem 3. An investigator quantifies her uncertainty about the estimate of a population mean by reporting
X ± sX . What size confidence interval is?
Problem 4. For a random sample of size n from a population of size N, consider the following as an
estimate of µ:
Xc =
n∑
i=1
ciXi,
where the ci are fixed numbers and X1, . . . ,Xn are the sample. Find a condition on the ci such that the
estimate is unbiased.
Problem 5. A sample of size 100 has the sample mean X = 10. Suppose the we know that the population
standard deviation σ = 5. Find a 95% confidence interval for the population mean µ.
Problem 6. Suppose the we know that the population standard deviation σ = 5. Then how large should a
sample be to estimate the popula.

Statistical tests of significance and Student`s T-Test

Statistical tests of significance is explained along with steps involve in Statistical tests of significance and types of significance test are also mentioned. Student`s T-Test is explained

TEST of hypothesis

This document discusses significance testing and hypothesis testing procedures, including:
1) The objectives of hypothesis testing are to test claims about population parameters by stating hypotheses (H0 and Ha), calculating test statistics, determining p-values, and considering significance levels.
2) Examples of one-sample and two-sample hypothesis tests are provided, including a test of whether a population gained weight on average and a test comparing the means of two samples.
3) Statistical tests discussed include the t-test (for comparing two means), z-test (when population standard deviation is unknown), and chi-square test (for testing goodness of fit to an expected distribution). Formulas and procedures for conducting these tests are outlined.

Parametric test

This document provides an overview of sampling and hypothesis testing in MATLAB. It discusses simple random sampling, random sampling functions in MATLAB, and the basics of hypothesis testing. It then describes various parametric tests that can be performed in MATLAB, including one sample t-test, paired t-test, two sample t-test, z-test, and F-test. Examples of how to perform each test in MATLAB are also provided.

sesion1.ppt

Este documento presenta una introducción a la econometría. Explica que la econometría ayuda a combinar la teoría económica y los datos económicos para tomar mejores decisiones económicas. También describe los modelos econométricos básicos, incluido el modelo de regresión lineal simple, y los supuestos subyacentes a estos modelos. Finalmente, introduce conceptos clave como el término de error y los mínimos cuadrados ordinarios.

Hypothesis Testing Assignment Help

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

Hypothesis testing an introduction

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

Lec12-Probability (1).pptThe document discusses probability models and how they can be used to make inferences from data. It introduces concepts like random variables, probability distributions, conditional probability, Bayes' rule, and Markov models. It provides examples of how to compute probabilities both mathematically and through simulation. The goal is to choose a probability model that best explains the data and make predictions from that model.

Nonparametric and Distribution- Free Statistics _contd

This document summarizes the Wilcoxon signed-rank test, a nonparametric test used to test hypotheses about the median or location of a population when the assumptions of the t-test are not met. It provides an example applying the test to data on cardiac output measurements from 15 patients. The test calculates the differences between each observation and the hypothesized median, ranks the absolute values of the differences, and sums the ranks with positive and negative signs. The smaller of the two sums is the test statistic, which is compared to a critical value to determine if the null hypothesis that the population median equals the hypothesized value can be rejected.

08 test of hypothesis large sample.ppt

The document discusses quantitative methods and hypothesis testing. It covers key concepts like null and alternative hypotheses, types of hypothesis tests, test statistics, rejection regions, and p-values. Examples are provided to illustrate hypothesis testing for population means, proportions, differences between means and proportions. The goal is to introduce the mechanics and general procedure of hypothesis testing and how to report results through p-values.

Testing of hypothesis

This document discusses the process of testing hypotheses. It begins by defining hypothesis testing as a way to make decisions about population characteristics based on sample data, which involves some risk of error. The key steps are outlined as:
1) Formulating the null and alternative hypotheses, with the null hypothesis stating no difference or relationship.
2) Computing a test statistic based on the sample data and selecting a significance level, usually 5%.
3) Comparing the test statistic to critical values to either reject or fail to reject the null hypothesis.
Examples are provided to demonstrate hypothesis testing for a single mean, comparing two means, and testing a claim about population characteristics using sample data and statistics.

Bayesian statistics using r intro

This document provides an introduction to Bayesian statistics using R. It discusses key Bayesian concepts like the prior, likelihood, and posterior distributions. It assumes familiarity with basic probability and probability distributions. Examples are provided to demonstrate Bayesian estimation and inference for binomial and normal distributions. Specifically, it shows how to estimate the probability of success θ in a binomial model and the mean μ in a normal model using different prior distributions and calculating the resulting posterior distributions in R.

슬로우캠퍼스: scikit-learn & 머신러닝 (강박사)

This document discusses the normal distribution and its key properties. It also discusses sampling distributions and the central limit theorem. Some key points:
- The normal distribution is bell-shaped and symmetric. It is defined by its mean and standard deviation. Approximately 68% of values fall within 1 standard deviation of the mean.
- Sample statistics like the sample mean follow sampling distributions. When samples are large and random, the sampling distributions are often normally distributed according to the central limit theorem.
- Correlation and regression analyze the relationship between two variables. Correlation measures the strength and direction of association, while regression finds the best-fitting linear relationship to predict one variable from the other.

HYPOTHESIS TESTING.ppt

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.

Bayesian decision making in clinical research

The document discusses the applications of Bayesian statistics in clinical research and decision making. It provides 3 case studies that demonstrate how Bayesian and frequentist analyses can lead to different conclusions from the same data. The case studies show that Bayesian analysis provides the probability that a hypothesis is true given the data, while frequentist analysis does not. Overall, the document argues that Bayesian statistics is better aligned with clinical decision making compared to conventional frequentist statistics.

Day 3 SPSS

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.

Lec13_Bayes.pptx

(1) Bayes' theorem provides a framework for updating probabilities based on new information or evidence.
(2) It allows calculating the conditional probability of a hypothesis being true given observed data by combining the prior probability of the hypothesis with the likelihood of making the observation under that hypothesis.
(3) Bayesian inference uses Bayes' theorem to update beliefs as new data becomes available, representing uncertainty about unknown parameters as a probability distribution that is continually refined as more evidence is accumulated.

Hypothesis testing

This document discusses hypothesis testing, which involves drawing inferences about a population based on a sample from that population. It outlines the key elements of a hypothesis test, including the null and alternative hypotheses, test statistics, critical regions, significance levels, critical values, and p-values. Type I and Type II errors are explained, where a Type I error involves rejecting the null hypothesis when it is true, and a Type II error involves failing to reject the null when it is false. The power of a hypothesis test is defined as the probability of correctly rejecting the null hypothesis when it is false. Controlling type I and II errors involves considering the significance level, sample size, and population parameters in the null and alternative hypotheses.

HW1_STAT206.pdfStatistical Inference II J. Lee Assignment.docx

HW1_STAT206.pdf
Statistical Inference II: J. Lee Assignment 1
Problem 1. Suppose the day after the Drexel-Northeastern basketball game, a poll of 1000 Drexel students
was conducted and it was determined that 850 out of the 1000 watched the game (live or on television).
Assume that this was a simple random sample and that the Drexel undergraduate population is 20000.
(a) Generate an unbiased estimate of the true proportion of Drexel undergraduate students who watched
the game.
(b) What is your estimated standard error for the proportion estimate in (a)?
(c) Give a 95% confidence interval for the true proportion of Drexel undergraduate students who watched
the game.
Problem 2. (Exercise 18 in Chapter 7 of Rice) From independent surveys of two populations, 90% con-
fidence intervals for the population means are conducted. What is the probability that neither interval
contains the respective population mean? That both do?
Problem 3. (Exercise 23 in Chapter 7 of Rice)
(a) Show that the standard error of an estimated proportion is largest when p = 1/2.
(b) Use this result and Corollary B of Section 7.3.2 (also, on Page 17 of the lecture notes) to conclude that
the quantity
1
2
√
N − n
N(n − 1)
is a conservative estimate of the standard error of p̂ no matter what the value of p may be.
(c) Use the central limit theorem to conclude that the interval
p̂ ±
√
N − n
N(n − 1)
contains p with probability at least .95.
HW2_STAT206.pdf
Statistical Inference II: J. Lee Assignment 2
Problem 1. The following data set represents the number of NBA games in January 2016, watched by 10
randomly selected student in STAT 206.
7, 0, 4, 2, 2, 1, 0, 1, 2, 3
(a) What is the sample mean?
(b) Calculate sample variance.
(c) Estimate the mean number of NBA games watched by a student in January 2016.
(d) Estimate the standard error of the estimated mean.
Problem 2. True or false? Tell me why for the false statements.
(a) The center of a 95% confidence interval for the population mean is a random variable.
(b) A 95% confidence interval for µ contains the sample mean with probability .95.
(c) A 95% confidence interval contains 95% of the population.
(d) Out of one hundred 95% confidence intervals for µ, 95 will contain µ.
Problem 3. An investigator quantifies her uncertainty about the estimate of a population mean by reporting
X ± sX . What size confidence interval is?
Problem 4. For a random sample of size n from a population of size N, consider the following as an
estimate of µ:
Xc =
n∑
i=1
ciXi,
where the ci are fixed numbers and X1, . . . ,Xn are the sample. Find a condition on the ci such that the
estimate is unbiased.
Problem 5. A sample of size 100 has the sample mean X = 10. Suppose the we know that the population
standard deviation σ = 5. Find a 95% confidence interval for the population mean µ.
Problem 6. Suppose the we know that the population standard deviation σ = 5. Then how large should a
sample be to estimate the popula.

Statistical tests of significance and Student`s T-Test

Statistical tests of significance is explained along with steps involve in Statistical tests of significance and types of significance test are also mentioned. Student`s T-Test is explained

TEST of hypothesis

This document discusses significance testing and hypothesis testing procedures, including:
1) The objectives of hypothesis testing are to test claims about population parameters by stating hypotheses (H0 and Ha), calculating test statistics, determining p-values, and considering significance levels.
2) Examples of one-sample and two-sample hypothesis tests are provided, including a test of whether a population gained weight on average and a test comparing the means of two samples.
3) Statistical tests discussed include the t-test (for comparing two means), z-test (when population standard deviation is unknown), and chi-square test (for testing goodness of fit to an expected distribution). Formulas and procedures for conducting these tests are outlined.

Parametric test

This document provides an overview of sampling and hypothesis testing in MATLAB. It discusses simple random sampling, random sampling functions in MATLAB, and the basics of hypothesis testing. It then describes various parametric tests that can be performed in MATLAB, including one sample t-test, paired t-test, two sample t-test, z-test, and F-test. Examples of how to perform each test in MATLAB are also provided.

Hypothesis Testing Assignment Help

Hypothesis Testing Assignment Help

Hypothesis testing an introduction

Hypothesis testing an introduction

Lec12-Probability (1).ppt

Lec12-Probability (1).ppt

Lec12-Probability.ppt

Lec12-Probability.ppt

Lec12-Probability.ppt

Lec12-Probability.ppt

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- 1. April 23 Chapter 9: Basics of Hypothesis Testing
- 2. In Chapter 9: 9.1 Null and Alternative Hypotheses 9.2 Test Statistic 9.3 P-Value 9.4 Significance Level 9.5 One-Sample z Test 9.6 Power and Sample Size
- 3. Terms Introduce in Prior Chapter • Population all possible values • Sample a portion of the population • Statistical inference generalizing from a sample to a population with calculated degree of certainty • Two forms of statistical inference – Hypothesis testing – Estimation • Parameter a characteristic of population, e.g., population mean µ • Statistic calculated from data in the sample, e.g., sample mean ( ) x
- 4. Distinctions Between Parameters and Statistics (Chapter 8 review) Parameters Statistics Source Population Sample Notation Greek (e.g., μ) Roman (e.g., xbar) Vary No Yes Calculated No Yes
- 6. Sampling Distributions of a Mean (Introduced in Ch 8) The sampling distributions of a mean (SDM) describes the behavior of a sampling mean n SE SE N x x x where , ~
- 7. Hypothesis Testing • Is also called significance testing • Tests a claim about a parameter using evidence (data in a sample • The technique is introduced by considering a one-sample z test • The procedure is broken into four steps • Each element of the procedure must be understood
- 8. Hypothesis Testing Steps A. Null and alternative hypotheses B. Test statistic C. P-value and interpretation D. Significance level (optional)
- 9. §9.1 Null and Alternative Hypotheses • Convert the research question to null and alternative hypotheses • The null hypothesis (H0) is a claim of “no difference in the population” • The alternative hypothesis (Ha) claims “H0 is false” • Collect data and seek evidence against H0 as a way of bolstering Ha (deduction)
- 10. Illustrative Example: “Body Weight” • The problem: In the 1970s, 20–29 year old men in the U.S. had a mean μ body weight of 170 pounds. Standard deviation σ was 40 pounds. We test whether mean body weight in the population now differs. • Null hypothesis H0: μ = 170 (“no difference”) • The alternative hypothesis can be either Ha: μ > 170 (one-sided test) or Ha: μ ≠ 170 (two-sided test)
- 11. §9.2 Test Statistic n SE H SE x x x and true is assuming mean population where z 0 0 0 stat This is an example of a one-sample test of a mean when σ is known. Use this statistic to test the problem:
- 12. Illustrative Example: z statistic • For the illustrative example, μ0 = 170 • We know σ = 40 • Take an SRS of n = 64. Therefore • If we found a sample mean of 173, then 5 64 40 n SEx 60 . 0 5 170 173 0 stat x SE x z
- 13. Illustrative Example: z statistic If we found a sample mean of 185, then 00 . 3 5 170 185 0 stat x SE x z
- 14. Reasoning Behinµzstat 5 , 170 ~ N x Sampling distribution of xbar under H0: µ = 170 for n = 64
- 15. §9.3 P-value • The P-value answer the question: What is the probability of the observed test statistic or one more extreme when H0 is true? • This corresponds to the AUC in the tail of the Standard Normal distribution beyond the zstat. • Convert z statistics to P-value : For Ha: μ > μ0 P = Pr(Z > zstat) = right-tail beyond zstat For Ha: μ < μ0 P = Pr(Z < zstat) = left tail beyond zstat For Ha: μ μ0 P = 2 × one-tailed P-value • Use Table B or software to find these probabilities (next two slides).
- 16. One-sided P-value for zstat of 0.6
- 17. One-sided P-value for zstat of 3.0
- 18. Two-Sided P-Value • One-sided Ha AUC in tail beyond zstat • Two-sided Ha consider potential deviations in both directions double the one- sided P-value Examples: If one-sided P = 0.0010, then two-sided P = 2 × 0.0010 = 0.0020. If one-sided P = 0.2743, then two-sided P = 2 × 0.2743 = 0.5486.
- 19. Interpretation • P-value answer the question: What is the probability of the observed test statistic … when H0 is true? • Thus, smaller and smaller P-values provide stronger and stronger evidence against H0 • Small P-value strong evidence
- 20. Interpretation Conventions* P > 0.10 non-significant evidence against H0 0.05 < P 0.10 marginally significant evidence 0.01 < P 0.05 significant evidence against H0 P 0.01 highly significant evidence against H0 Examples P =.27 non-significant evidence against H0 P =.01 highly significant evidence against H0 * It is unwise to draw firm borders for “significance”
- 21. α-Level (Used in some situations) • Let α ≡ probability of erroneously rejecting H0 • Set α threshold (e.g., let α = .10, .05, or whatever) • Reject H0 when P ≤ α • Retain H0 when P > α • Example: Set α = .10. Find P = 0.27 retain H0 • Example: Set α = .01. Find P = .001 reject H0
- 22. (Summary) One-Sample z Test A. Hypothesis statements H0: µ = µ0 vs. Ha: µ ≠ µ0 (two-sided) or Ha: µ < µ0 (left-sided) or Ha: µ > µ0 (right-sided) B. Test statistic C. P-value: convert zstat to P value D. Significance statement (usually not necessary) n SE SE x x x where z 0 stat
- 23. §9.5 Conditions for z test • σ known (not from data) • Population approximately Normal or large sample (central limit theorem) • SRS (or facsimile) • Data valid
- 24. The Lake Wobegon Example “where all the children are above average” • Let X represent Weschler Adult Intelligence scores (WAIS) • Typically, X ~ N(100, 15) • Take SRS of n = 9 from Lake Wobegon population • Data {116, 128, 125, 119, 89, 99, 105, 116, 118} • Calculate: x-bar = 112.8 • Does sample mean provide strong evidence that population mean μ > 100?
- 25. Example: “Lake Wobegon” A. Hypotheses: H0: µ = 100 versus Ha: µ > 100 (one-sided) Ha: µ ≠ 100 (two-sided) B. Test statistic: 56 . 2 5 100 8 . 112 5 9 15 0 stat x x SE x z n SE
- 26. C. P-value: P = Pr(Z ≥ 2.56) = 0.0052 P =.0052 it is unlikely the sample came from this null distribution strong evidence against H0
- 27. • Ha: µ ≠100 • Considers random deviations “up” and “down” from μ0 tails above and below ±zstat • Thus, two-sided P = 2 × 0.0052 = 0.0104 Two-Sided P-value: Lake Wobegon
- 28. §9.6 Power and Sample Size Truth Decision H0 true H0 false Retain H0 Correct retention Type II error Reject H0 Type I error Correct rejection α ≡ probability of a Type I error β ≡ Probability of a Type II error Two types of decision errors: Type I error = erroneous rejection of true H0 Type II error = erroneous retention of false H0
- 29. Power • β ≡ probability of a Type II error β = Pr(retain H0 | H0 false) (the “|” is read as “given”) • 1 – β “Power” ≡ probability of avoiding a Type II error 1– β = Pr(reject H0 | H0 false)
- 30. Power of a z test where • Φ(z) represent the cumulative probability of Standard Normal Z • μ0 represent the population mean under the null hypothesis • μa represents the population mean under the alternative hypothesis n z a | | 1 0 1 2
- 31. Calculating Power: Example A study of n = 16 retains H0: μ = 170 at α = 0.05 (two-sided); σ is 40. What was the power of test’s conditions to identify a population mean of 190? 5160 . 0 04 . 0 40 16 | 190 170 | 96 . 1 | | 1 0 1 2 n z a
- 32. Reasoning Behind Power • Competing sampling distributions Top curve (next page) assumes H0 is true Bottom curve assumes Ha is true α is set to 0.05 (two-sided) • We will reject H0 when a sample mean exceeds 189.6 (right tail, top curve) • The probability of getting a value greater than 189.6 on the bottom curve is 0.5160, corresponding to the power of the test
- 34. Sample Size Requirements Sample size for one-sample z test: where 1 – β ≡ desired power α ≡ desired significance level (two-sided) σ ≡ population standard deviation Δ = μ0 – μa ≡ the difference worth detecting 2 2 1 1 2 2 z z n
- 35. Example: Sample Size Requirement How large a sample is needed for a one-sample z test with 90% power and α = 0.05 (two-tailed) when σ = 40? Let H0: μ = 170 and Ha: μ = 190 (thus, Δ = μ0 − μa = 170 – 190 = −20) Round up to 42 to ensure adequate power. 99 . 41 20 ) 96 . 1 28 . 1 ( 40 2 2 2 2 2 1 1 2 2 z z n
- 36. Illustration: conditions for 90% power.

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