同济大学：《应用统计学》课程电子教案（课件讲稿）Chapter 8 Fundamentals of Hypothesis Testing（One-Sample Tests）

Statistics for Managers Using Microsoft Excel Chapter 8 Fundamentals of Hypothesis Testing:One-Sample Tests Statistics for Managers Using Microsoft Excel Chap 8-1

Statistics for Managers Using Microsoft Excel Chap 8-1 Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests Statistics for Managers Using Microsoft Excel

Learning Goals After completing this chapter,you should be able to: Formulate null(Ho)and alternative hypotheses (H1)for applications involving one sample population mean or proportion Formulate a decision rule for testing a hypothesis Know how to use the critical value and p-value approaches to test the null hypothesis(for both mean and proportion problems) Know what Type I and Type ll errors are Statistics for Ma agers Using Microsoft Excel Chap 8-2

Statistics for Managers Using Microsoft Excel Chap 8-2 Learning Goals After completing this chapter, you should be able to:  Formulate null (H0) and alternative hypotheses (H1) for applications involving one sample population mean or proportion  Formulate a decision rule for testing a hypothesis  Know how to use the critical value and p-value approaches to test the null hypothesis (for both mean and proportion problems)  Know what Type I and Type II errors are

What is a Hypothesis? A hypothesis is a claim (assumption)about a population parameter: population mean Example:The mean monthly cell phone bill of this city isμ=$42 population proportion Example:The proportion of adults in this city with cell phones is p=.68 Statistics for Ma Using Microsoft Excel Chap8-3

Statistics for Managers Using Microsoft Excel Chap 8-3 What is a Hypothesis?  A hypothesis is a claim (assumption) about a population parameter:  population mean  population proportion Example: The mean monthly cell phone bill of this city is μ = $42 Example: The proportion of adults in this city with cell phones is p = .68

What is Hypothesis testing? A step-by-step methodology that enables one to make inferences about a population parameter by analyzing differences between the results observed (the sample statistic)and the results one is expected to obtain if some underlying hypothesis is actually true. For example,is the mean weight of the cereal boxes produced at a company consistent with what would be expected if the mean of the entire population of cereal boxes is 368 grams?Or,can you infer that the population mean is not equal to 368 grams because the sample mean is significantly different from 368 grams Statistics Using Microsoft Excel Chap 8-4

Statistics for Managers Using Microsoft Excel Chap 8-4 What is Hypothesis testing?  A step-by-step methodology that enables one to make inferences about a population parameter by analyzing differences between the results observed (the sample statistic) and the results one is expected to obtain if some underlying hypothesis is actually true.  For example, is the mean weight of the cereal boxes produced at a company consistent with what would be expected if the mean of the entire population of cereal boxes is 368 grams? Or, can you infer that the population mean is not equal to 368 grams because the sample mean is significantly different from 368 grams

The Null Hypothesis,Ho States the assumption(numerical)to be tested (the first step of HT) Example:The average number of TV sets in U.S. Homes is equal to three（H,:μ=3) Is always about a population parameter, not about a sample statistic H:μ=3 H岁 3 Statistics for Mar agers Using Microsoft Excel Chap8-5

Statistics for Managers Using Microsoft Excel Chap 8-5 The Null Hypothesis, H0 H0 :μ  3 H0 :μ  3 H0 : X  3  States the assumption (numerical) to be tested (the first step of HT) Example: The average number of TV sets in U.S. Homes is equal to three ( )  Is always about a population parameter, not about a sample statistic

The Null Hypothesis,Ho continued) Begin with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty Refers to the status quo "Always contains“=”,“≤”or“≥”sign May or may not be rejected Statistics for Ma agers Using Microsoft Excel Chap 8-6

Statistics for Managers Using Microsoft Excel Chap 8-6 The Null Hypothesis, H0  Begin with the assumption that the null hypothesis is true  Similar to the notion of innocent until proven guilty  Refers to the status quo  Always contains “=” , “≤” or “” sign  May or may not be rejected (continued)

The Alternative Hypothesis,H1 Is the opposite of the null hypothesis e.g.,The average number of TV sets in U.S. homes is not equal to3(H1:μ≠3) Challenges the status quo ·Never contains the“="”,“≤”or"≥”sign May or may not be proven Is generally the hypothesis that the researcher is trying to prove Statistics for Mar agers Using Microsoft Excel Chap 8-7

Statistics for Managers Using Microsoft Excel Chap 8-7 The Alternative Hypothesis, H1  Is the opposite of the null hypothesis  e.g., The average number of TV sets in U.S. homes is not equal to 3 ( H1: μ ≠ 3 )  Challenges the status quo  Never contains the “=” , “≤” or “” sign  May or may not be proven  Is generally the hypothesis that the researcher is trying to prove

A Summary of the Null and Alternative Hypothesis The Null Hypothesis Ho,is the hypothesis that is always tested. The Alternative Hypothesis,H1,is set up as the opposite of the null hypothesis and represents the conclusion supported if the null hypothesis is rejected The Null Hypothesis always refers to a specified value of the population parameter (such as u),not to a sample statistic (such as X) Statistics for agers Using Microsoft Excel Chap 8-8

Statistics for Managers Using Microsoft Excel Chap 8-8 A Summary of the Null and Alternative Hypothesis  The Null Hypothesis H0, is the hypothesis that is always tested.  The Alternative Hypothesis, H1, is set up as the opposite of the null hypothesis and represents the conclusion supported if the null hypothesis is rejected  The Null Hypothesis always refers to a specified value of the population parameter (such as u), not to a sample statistic (such as ) X

Hypothesis Testing Process Claim:the population mean age is 50. (Null Hypothesis: Population H0:μ=50) Now select a random sample Is X-20 likely if u 50? If not likely, Suppose 亦亦7 the sample REJECT mean age Sample Null Hypothesis is20:X=20 Statistics for Managers Using Microsoft Excel Chap8-9

Statistics for Managers Using Microsoft Excel Chap 8-9 Population Claim: the population mean age is 50. (Null Hypothesis: REJECT Suppose the sample mean age is 20: X = 20 Sample Null Hypothesis Is 20 likely if μ = 50? Hypothesis Testing Process If not likely, Now select a random sample H0: μ = 50 ) X

Reason for Rejecting Ho Sampling Distribution of X 20 μ=50 If Ho is true If it is unlikely that ..then we we would get a ↑ reject the null sample mean of if in fact this were hypothesis that this value... the population mean... μ=50. Statistics for Ma agers Using Microsoft Excel Chap 8-10

Statistics for Managers Using Microsoft Excel Chap 8-10 Sampling Distribution of X μ = 50 If H0 is true If it is unlikely that we would get a sample mean of this value ... ... then we reject the null hypothesis that μ = 50. Reason for Rejecting H0 20 ... if in fact this were the population mean… X