Chapter 8 Introduction to Hypothesis Testing

Transcription

1 Chapter 8 Student Lecture Notes 8-1 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 p-value approaches to test the null hypothesis Fall 26 Fundamentals of Business Statistics 2

2 Chapter 8 Student Lecture Notes 8-2 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 8-3 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 8-4 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 8-5 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 8-6 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 (n-1) 1. Use Z~N(,1)

7 Chapter 8 Student Lecture Notes 8-7 Calculating the Test Statistic: Z Two-Sided: 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 One-Sided Upper Tail: H : μ μ ; H A : μ > μ Reject H if Z* > Z (.5 α), otherwise do not reject H One-Sided 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 Two-Sided: 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 One-Sided Upper Tail: H : μ μ ; H A : μ > μ * ( 1) Reject H if t > t( n 1 α ), otherwise do not reject H One-Sided 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 8-8 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 8-9 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 8-1 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 p-value Approach to Testing p-value: 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 8-11 p-value Approach to Testing Convert Sample Statistic to Test Statistic ( Z* or t* statistic ) Obtain the p-value from a table or computer Compare the p-value with α If p-value < α, reject H If p-value α, do not reject H Fall 26 Fundamentals of Business Statistics 21 P-Value Calculation Z test statistic Two-Sided: 2 min {P(Z Z*,Z Z*)} One-Sided Upper Tail P(Z Z*) One-Sided Lower Tail P(Z Z*) T test statistic Two-Sided: 2 min {P(t t*,t t*)} One-Sided Upper Tail P(t t*) One-Sided Lower Tail P(t t*) Fall 26 Fundamentals of Business Statistics 22

12 Chapter 8 Student Lecture Notes 8-12 p-value 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 8-13 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 8-14 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 p-value and compare to α (continued) Do not reject H Reject H Fall 26 Fundamentals of Business Statistics 28

15 Chapter 8 Student Lecture Notes 8-15 Example: Two-Tail 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