Chapter 8 Introduction to Hypothesis Testing


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1 Chapter 8 Student Lecture Notes 81 Chapter 8 Introduction to Hypothesis Testing Fall 26 Fundamentals of Business Statistics 1 Chapter Goals After completing this chapter, you should be able to: Formulate null and alternative hypotheses for applications involving a single population mean Formulate a decision rule for testing a hypothesis Know how to use the test statistic, critical value, and pvalue approaches to test the null hypothesis Fall 26 Fundamentals of Business Statistics 2
2 Chapter 8 Student Lecture Notes 82 Testing Theories Hypotheses Competing theories that we want to test about a population are called Hypotheses in statistics. Specifically, we label these competing theories as Null Hypothesis (H ) and Alternative Hypothesis (H 1 or H A ). H : The null hypothesis is the status quo or the prevailing viewpoint. H A : The alternative hypothesis is the competing belief. It is the statement that the researcher is hoping to prove. Fall 26 Fundamentals of Business Statistics 3 The Null Hypothesis, H Begin with the assumption that the null hypothesis is true Refers to the status quo Always contains =, or sign May or may not be rejected (continued) Fall 26 Fundamentals of Business Statistics 4
3 Chapter 8 Student Lecture Notes 83 The Alternative Hypothesis, H A Challenges the status quo Never contains the =, or sign Is generally the hypothesis that is believed (or needs to be supported) by the researcher Provides the direction of extreme Fall 26 Fundamentals of Business Statistics 5 Hypothesis Testing Process Claim: the population mean age is 5. (Null Hypothesis: H : μ = 5 ) Is x=2 likely if μ = 5? If not likely, REJECT Null Hypothesis Fall 26 Fundamentals of Business Statistics Suppose the sample mean age is 2: x = 2 Population Now select a random sample Sample
4 Chapter 8 Student Lecture Notes 84 Deciding Which Theory to Support Decision making is based on the rare event concept. Since the null hypothesis is the status quo, we assume that it is true unless the observed result is extremely unlikely (rare) under the null hypothesis. Definition: If the data were indeed unlikely to be observed under the assumption that H is true, and therefore we reject H in favor of H A, then we say that the data are statistically significant. Fall 26 Fundamentals of Business Statistics 7 Reason for Rejecting H Sampling Distribution of x μ = 5 If H is true x Fall 26 Fundamentals of Business Statistics 8
5 Chapter 8 Student Lecture Notes 85 Level of Significance, α Defines unlikely values of sample statistic if null hypothesis is true Defines rejection region of the sampling distribution Is designated by α, (level of significance) Is selected by the researcher at the beginning Provides the critical value(s) of the test Fall 26 Fundamentals of Business Statistics 9 Level of Significance and the Rejection Region Level of significance = α H : μ 3 H A : μ < 3 H : μ 3 H A : μ > 3 Lower tail test Upper tail test α α Represents critical value Rejection region is shaded H : μ = 3 H A : μ 3 Two tailed test α/2 Fall 26 Fundamentals of Business Statistics 1 α /2
6 Chapter 8 Student Lecture Notes 86 Critical Value Approach to Testing Convert sample statistic (e.g.: statistic ( Z* or t* statistic ) x ) to test Determine the critical value(s) for a specified level of significance α from a table or computer If the test statistic falls in the rejection region, reject H ; otherwise do not reject H Fall 26 Fundamentals of Business Statistics 11 Critical Value Approach to Testing Convert sample statistic ( x ( Z* or t* statistic ) Is X ~ N? Yes No ) to a test statistic Sample Size? Yes Is σ known? Large (n 1) No, use sample standard deviation s Small 2. Use T~t (n1) 1. Use Z~N(,1)
7 Chapter 8 Student Lecture Notes 87 Calculating the Test Statistic: Z TwoSided: H : μ = μ ; H A : μ μ Reject H if Z* > Z (.5 α/2) or Z* < Z (.5 α/2), otherwise do not reject H OneSided Upper Tail: H : μ μ ; H A : μ > μ Reject H if Z* > Z (.5 α), otherwise do not reject H OneSided Lower Tail: H : μ z * = x μ σ n μ ; H A : μ < μ Reject H if Z* < Z (.5 α), otherwise do not reject H Fall 26 Fundamentals of Business Statistics 13 T test Statistic TwoSided: H : μ = μ ; H A : μ μ * Reject H if ( n 1) * ( n 1) t > t(, otherwise do not 1 α / 2) or t < t( 1 α / 2) reject H OneSided Upper Tail: H : μ μ ; H A : μ > μ * ( 1) Reject H if t > t( n 1 α ), otherwise do not reject H OneSided Lower Tail: H : μ μ ; H A : μ < μ * ( n 1) Reject H if t < t( 1 α ), otherwise do not reject H t * = x μ s n Fall 26 Fundamentals of Business Statistics 14
8 Chapter 8 Student Lecture Notes 88 Review: Steps in Hypothesis Testing 1. Specify the population value of interest 2. Formulate the appropriate null and alternative hypotheses 3. Specify the desired level of significance 4. Determine the rejection region 5. Obtain sample evidence and compute the test statistic 6. Reach a decision and interpret the result Fall 26 Fundamentals of Business Statistics 15 Hypothesis Testing Example Test the claim that the true mean # of TV sets in US homes is less than 3. Assume that σ =.8 1. Specify the population value of interest 2. Formulate the appropriate null and alternative hypotheses 3. Specify the desired level of significance
9 Chapter 8 Student Lecture Notes 89 Hypothesis Testing Example 4. Determine the rejection region (continued) α = Reject H Do not reject H Reject H if Z* test statistic < otherwise do not reject H Fall 26 Fundamentals of Business Statistics 17 Hypothesis Testing Example 5. Obtain sample evidence and compute the test statistic A sample is taken with the following results: n = 1, x = 2.84 (σ =.8 is assumed known) Then the test statistic is: Z * = x μ σ = n Fall 26 Fundamentals of Business Statistics 18
10 Chapter 8 Student Lecture Notes 81 Hypothesis Testing Example (continued) 6. Reach a decision and interpret the result α = Reject H Do not reject H z Since Z* = 2. <, Fall 26 Fundamentals of Business Statistics 19 pvalue Approach to Testing pvalue: Probability of obtaining a test statistic more extreme than the observed sample value given H is true Also called observed level of significance Smallest value of α for which H can be rejected Fall 26 Fundamentals of Business Statistics 2
11 Chapter 8 Student Lecture Notes 811 pvalue Approach to Testing Convert Sample Statistic to Test Statistic ( Z* or t* statistic ) Obtain the pvalue from a table or computer Compare the pvalue with α If pvalue < α, reject H If pvalue α, do not reject H Fall 26 Fundamentals of Business Statistics 21 PValue Calculation Z test statistic TwoSided: 2 min {P(Z Z*,Z Z*)} OneSided Upper Tail P(Z Z*) OneSided Lower Tail P(Z Z*) T test statistic TwoSided: 2 min {P(t t*,t t*)} OneSided Upper Tail P(t t*) OneSided Lower Tail P(t t*) Fall 26 Fundamentals of Business Statistics 22
12 Chapter 8 Student Lecture Notes 812 pvalue example Fall 26 Fundamentals of Business Statistics 23 Example: Upper Tail z Test for Mean (σ Known) A phone industry manager thinks that customer monthly cell phone bill have increased, and now average over $52 per month. The company wishes to test this claim. (Assume σ = 1 is known) Form hypothesis test: H : μ 52 H A : μ > 52 the average is not over $52 per month the average is greater than $52 per month (i.e., sufficient evidence exists to support the manager s claim) Fall 26 Fundamentals of Business Statistics 24
13 Chapter 8 Student Lecture Notes 813 Example: Find Rejection Region (continued) Reject H α = Do not reject H Reject H Fall 26 Fundamentals of Business Statistics 25 Example: Test Statistic Obtain sample evidence and compute the test statistic A sample is taken with the following results: n = 64, x = 53.1 (σ=1 was assumed known) (continued) Then the test statistic is: Z * = x μ σ n = Fall 26 Fundamentals of Business Statistics 26
14 Chapter 8 Student Lecture Notes 814 Example: Decision (continued) Reach a decision and interpret the result: Reject H α = Do not reject H Reject H Fall 26 Fundamentals of Business Statistics 27 p Value Solution Calculate the pvalue and compare to α (continued) Do not reject H Reject H Fall 26 Fundamentals of Business Statistics 28
15 Chapter 8 Student Lecture Notes 815 Example: TwoTail Test (σ Unknown) The average cost of a hotel room in New York is said to be $168 per night. A random sample of 25 hotels resulted in X = $172.5 and s = $15.4. Test at the α =.5 level. (Assume the population distribution is normal) H : μ = 168 H A : μ 168 Fall 26 Fundamentals of Business Statistics 29 Outcomes and Probabilities Possible Hypothesis Test Outcomes Key: Outcome (Probability) Decision Do Not Reject H Reject H State of Nature H True No error (1  ) α Type I Error ( ) α H False Type II Error ( β ) No Error ( 1  β ) Fall 26 Fundamentals of Business Statistics 3
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