Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-1 Business Statistics: A Decision-Making Approach 7 th Edition Chapter 9 Introduction to Hypothesis Testing

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-2 Chapter Goals After completing this chapter, you should be able to: Formulate null and alternative hypotheses for applications involving a single population mean or proportion 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 Know what Type I and Type II errors are Compute the probability of a Type II error

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-3 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 π =.68

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-4 The Null Hypothesis, H 0 States the assumption (numerical) to be tested Example: The average number of TV sets in U.S. Homes is at least three ( ) Is always about a population parameter, not about a sample statistic

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-5 The Null Hypothesis, H 0 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)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-6 The Alternative Hypothesis, H A Is the opposite of the null hypothesis e.g.: The average number of TV sets in U.S. homes is less than 3 ( H A : µ < 3 ) Challenges the status quo Never contains the “=”, “≤” or “  ” sign May or may not be accepted Is generally the hypothesis that is believed (or needs to be supported) by the researcher – a research hypothesis

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-7 Formulating Hypotheses Example 1: Ford motor company has worked to reduce road noise inside the cab of the redesigned F150 pickup truck. It would like to report in its advertising that the truck is quieter. The average of the prior design was 68 decibels at 60 mph. What is the appropriate hypothesis test?

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-8 Formulating Hypotheses Example 1: Ford motor company has worked to reduce road noise inside the cab of the redesigned F150 pickup truck. It would like to report in its advertising that the truck is quieter. The average of the prior design was 68 decibels at 60 mph. What is the appropriate test? H 0 : µ ≥ 68 (the truck is not quieter) status quo H A : µ < 68 (the truck is quieter) wants to support If the null hypothesis is rejected, Ford has sufficient evidence to support that the truck is now quieter.

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-9 Formulating Hypotheses Example 2: The average annual income of buyers of Ford F150 pickup trucks is claimed to be $65,000 per year. An industry analyst would like to test this claim. What is the appropriate hypothesis test?

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-10 Formulating Hypotheses Example 1: The average annual income of buyers of Ford F150 pickup trucks is claimed to be $65,000 per year. An industry analyst would like to test this claim. What is the appropriate test? H 0 : µ = 65,000 (income is as claimed) status quo H A : µ ≠ 65,000 (income is different than claimed) The analyst will believe the claim unless sufficient evidence is found to discredit it.

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Population Claim: the population mean age is 50. Null Hypothesis: REJECT Suppose the sample mean age is 20: x = 20 Sample Null Hypothesis Is x = 20 likely if µ = 50? Hypothesis Testing Process If not likely, Now select a random sample: H 0 : µ = 50

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-12 Sampling Distribution of x μ = 50 If H 0 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 H if in fact this were the population mean… x

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-13 Errors in Making Decisions Type I Error Reject a true null hypothesis Considered a serious type of error The probability of Type I Error is  Called level of significance of the test Set by researcher in advance

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-14 Errors in Making Decisions Type II Error Fail to reject a false null hypothesis The probability of Type II Error is β β is a calculated value, the formula is discussed later in the chapter (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-15 Outcomes and Probabilities State of Nature Decision Do Not Reject H 0 No error (1 - )  Type II Error ( β ) Reject H 0 Type I Error ( )  Possible Hypothesis Test Outcomes H 0 False H 0 True Key: Outcome (Probability) No Error ( 1 - β )

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-16 Type I & II Error Relationship  Type I and Type II errors cannot happen at the same time  Type I error can only occur if H 0 is true  Type II error can only occur if H 0 is false If Type I error probability (  ), then Type II error probability ( β )

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-17 Factors Affecting Type II Error All else equal, β when the difference between hypothesized parameter and its true value β when  β when σ β when n The formula used to compute the value of β is discussed later in the chapter

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-18 Level of Significance,  Defines unlikely values of sample statistic if null hypothesis is true Defines rejection region of the sampling distribution Is designated by , (level of significance) Typical values are.01,.05, or.10 Is selected by the researcher at the beginning Provides the critical value(s) of the test

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-19 Hypothesis Tests for the Mean σ Knownσ Unknown Hypothesis Tests for  Assume first that the population standard deviation σ is known

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap Specify population parameter of interest 2. Formulate the null and alternative hypotheses 3. Specify the desired significance level, α 4. Define the rejection region 5. Take a random sample and determine whether or not the sample result is in the rejection region 6. Reach a decision and draw a conclusion Process of Hypothesis Testing

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-21 Level of Significance and the Rejection Region H 0 : μ ≥ 3 H A : μ < 3 0 H 0 : μ ≤ 3 H A : μ > 3 H 0 : μ = 3 H A : μ ≠ 3   /2 Lower tail test Level of significance =  0 /2  Upper tail testTwo tailed test 0  -z α zαzα -z α/2 z α/2 Reject H 0 Do not reject H 0 Example:

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-22 Reject H 0 Do not reject H 0 The cutoff value, or, is called a critical value  -z α xαxα -zα-zα xαxα 0 µ=3 H 0 : μ ≥ 3 H A : μ < 3 Critical Value for Lower Tail Test

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-23 Reject H 0 Do not reject H 0 Critical Value for Upper Tail Test  zαzα xαxα 0 H 0 : μ ≤ 3 H A : μ > 3 µ=3 The cutoff value, or, is called a critical value zαzα xαxα

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-24 Do not reject H 0 Reject H 0 There are two cutoff values (critical values): or Critical Values for Two Tailed Tests  /2 -z α/2 x α/2 ± z α/2 x α/2 0 H 0 : μ = 3 H A : μ  3 z α/2 x α/2 Lower Upper x α/2 LowerUpper  /2 µ=3

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-25 The Rejection Region H 0 : μ ≥ 3 H A : μ < 3 0 H 0 : μ ≤ 3 H A : μ > 3 H 0 : μ = 3 H A : μ ≠ 3   /2 Lower tail test 0 /2  Upper tail testTwo tailed test 0  -z α zαzα -z α/2 z α/2 Reject H 0 Do not reject H 0 Example: Reject H 0 if z < -z α i.e., if x < x α Reject H 0 if z > z α i.e., if x > x α Reject H 0 if z z α/2 i.e., if x x α/2(U)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-26 z-units: For given , find the critical z value(s): -z α, z α,or ±z α/2 Convert the sample mean x to a z test statistic: Reject H 0 if z is in the rejection region, otherwise do not reject H 0 x units: Given , calculate the critical value(s) x α, or x α/2(L) and x α/2(U) The sample mean is the test statistic. Reject H 0 if x is in the rejection region, otherwise do not reject H 0 Two Equivalent Approaches to Hypothesis Testing

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-27 Hypothesis Testing Example Test the claim that the true mean # of TV sets in US homes is at least 3. (Assume σ = 0.8) 1. Specify the population value of interest The mean number of TVs in US homes 2.Formulate the appropriate null and alternative hypotheses H 0 : μ  3 H A : μ < 3 (This is a lower tail test) 3. Specify the desired level of significance Suppose that  =.05 is chosen for this test

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-28 Reject H 0 Do not reject H 0 4. Determine the rejection region  =.05 -z α = This is a one-tailed test with  =.05. Since σ is known, the cutoff value is a z value: Reject H 0 if z < z  = ; otherwise do not reject H 0 Hypothesis Testing Example (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap Obtain sample evidence and compute the test statistic Suppose a sample is taken with the following results: n = 100, x = 2.84 (  = 0.8 is assumed known) Then the test statistic is: Hypothesis Testing Example

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-30 Reject H 0 Do not reject H 0  = Reach a decision and interpret the result -2.0 Since z = -2.0 < , we reject the null hypothesis that the mean number of TVs in US homes is at least 3. There is sufficient evidence that the mean is less than 3. Hypothesis Testing Example (continued) z

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-31 Reject H 0  = Do not reject H 0 3 An alternate way of constructing rejection region: 2.84 Since x = 2.84 < , we reject the null hypothesis Hypothesis Testing Example (continued) x Now expressed in x, not z units

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-32 p-Value Approach to Testing Convert Sample Statistic ( ) to Test Statistic (a z value, if σ is known) Determine the p-value from a table or computer Compare the p-value with  If p-value < , reject H 0 If p-value  , do not reject H 0 x

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-33 p-Value Approach to Testing p-value: Probability of obtaining a test statistic more extreme ( ≤ or  ) than the observed sample value given H 0 is true Also called observed level of significance Smallest value of  for which H 0 can be rejected (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-34 p-value =.0228  =.05 p-value example Example: How likely is it to see a sample mean of 2.84 (or something further below the mean) if the true mean is  = 3.0? x z

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-35 Compare the p-value with  If p-value < , reject H 0 If p-value  , do not reject H 0 Here: p-value =.0228  =.05 Since.0228 <.05, we reject the null hypothesis (continued) p-value example p-value =.0228  =

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-36 Example: Upper Tail z Test for Mean (  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  = 10 is known) H 0 : μ ≤ 52 the average is not over $52 per month H A : μ > 52 the average is greater than $52 per month (i.e., sufficient evidence exists to support the manager’s claim) Form hypothesis test:

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-37 Reject H 0 Do not reject H 0 Suppose that  =.10 is chosen for this test Find the rejection region:  =.10 z α = Reject H 0 Reject H 0 if z > 1.28 Example: Find Rejection Region (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-38 Review: Finding Critical Value - One Tail Z z Standard Normal Distribution Table (Portion) What is z given  = 0.10?  =.10 Critical Value =

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-39 Obtain sample evidence and compute the test statistic Suppose a sample is taken with the following results: n = 64, x = 53.1 (  =10 was assumed known) Then the test statistic is: Example: Test Statistic (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-40 Reject H 0 Do not reject H 0 Example: Decision  = Reject H 0 Do not reject H 0 since z = 0.88 ≤ 1.28 i.e.: there is not sufficient evidence that the mean bill is over $52 z =.88 Reach a decision and interpret the result: (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-41 Reject H 0  =.10 Do not reject H Reject H 0 z =.88 Calculate the p-value and compare to  (continued) p-value =.1894 p -Value Solution Do not reject H 0 since p-value =.1894 >  =.10

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-42 Critical Value Approach to Testing When σ is known, convert sample statistic ( ) to a z test statistic x  Known  Unknown Hypothesis Tests for  The test statistic is:

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-43 Critical Value Approach to Testing When σ is unknown, convert sample statistic ( ) to a t test statistic x  Known  Unknown Hypothesis Tests for  The test statistic is: (The population must be approximately normal)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-44 Hypothesis Tests for μ, σ Unknown 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 (critical values are from the t-distribution with n-1 d.f.) 5. Obtain sample evidence and compute the t test statistic 6. Reach a decision and interpret the result

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-45 Example: Two-Tail Test (  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 = $ and s = $ Test at the  = 0.05 level. (Assume the population distribution is normal) H 0 : μ  = 168 H A : μ  168

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-46   = 0.05  n = 25  Critical Values: t 24 = ±  is unknown, so use a t statistic Example Solution: Two-Tail Test Do not reject H 0 : not sufficient evidence that true mean cost is different than $168 Reject H 0  /2=.025 -t α/2 Do not reject H 0 0 t α/2  /2= H 0 : μ  = 168 H A : μ  168

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-47 Hypothesis Tests for Proportions Involves categorical values Two possible outcomes “Success” (possesses a certain characteristic) “Failure” (does not possesses that characteristic) Fraction or proportion of population in the “success” category is denoted by π

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-48 Proportions The sample proportion of successes is denoted by p : When both nπ and n(1- π) are at least 5, p is approximately normally distributed with mean and standard deviation (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-49 The sampling distribution of p is normal, so the test statistic is a z value: Hypothesis Tests for Proportions nπ  5 and n(1-π)  5 Hypothesis Tests for π nπ < 5 or n(1-π) < 5 Not discussed in this chapter

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-50 Example: z Test for Proportion A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the  =.05 significance level. Check: n π = (500)(.08) = 40 n(1-π) = (500)(.92) = 460

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-51 Z Test for Proportion: Solution  =.05 n = 500, p =.05 Reject H 0 at  =.05 H 0 : π =.08 H A : π .08 Critical Values: ± 1.96 Test Statistic: Decision: Conclusion: z 0 Reject There is sufficient evidence to reject the company’s claim of 8% response rate

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-52 Do not reject H 0 Reject H 0  /2 = z = Calculate the p-value and compare to  (For a two sided test the p-value is always two sided) (continued) p-value =.0136: p -Value Solution Reject H 0 since p-value =.0136 <  =.05 z =  /2 =

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-53 Reject H 0 : μ  52 Do not reject H 0 : μ  52 Type II Error Type II error is the probability of failing to reject a false H Suppose we fail to reject H 0 : μ  52 when in fact the true mean is μ = 50 

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-54 Reject H 0 :   52 Do not reject H 0 :   52 Type II Error Suppose we do not reject H 0 :   52 when in fact the true mean is  = This is the true distribution of x if  = 50 This is the range of x where H 0 is not rejected (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-55 Reject H 0 : μ  52 Do not reject H 0 : μ  52 Type II Error Suppose we do not reject H 0 : μ  52 when in fact the true mean is μ = 50  5250 β Here, β = P( x  cutoff ) if μ = 50 (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-56 Reject H 0 : μ  52 Do not reject H 0 : μ  52 Suppose n = 64, σ = 6, and  =.05  5250 So β = P( x  ) if μ = 50 Calculating β (for H 0 : μ  52)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-57 Reject H 0 : μ  52 Do not reject H 0 : μ  52 Suppose n = 64, σ = 6, and  =.05  5250 Calculating β (continued) Probability of type II error: β =.1539

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-58 Hypothesis Tests in PHStat Options

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-59 Sample PHStat Output Input Output

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-60 Chapter Summary Addressed hypothesis testing methodology Performed z Test for the mean (σ known) Discussed p–value approach to hypothesis testing Performed one-tail and two-tail tests...

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 9-61 Chapter Summary Performed t test for the mean (σ unknown) Performed z test for the proportion Discussed Type II error and computed its probability (continued)