1 Introduction to Hypothesis Testing

2 What is a Hypothesis? A hypothesis is a claim 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  = $42 Example: The proportion of adults in this city with cell phones is p =.68

3 Null Hypothesis, H 0 States the assumption (numerical) to be tested 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 Is always about a population parameter, not about a sample statistic Types of hypotheses

4 Begin with the assumption that the null hypothesis is true Begin with the assumption that the null hypothesis is true  Similar to the notion of innocent until proven guilty May or may not be rejected May or may not be rejected (continued) Types of hypotheses Null Hypothesis, H 0

5 Alternative Hypothesis, H A Is the opposite of the null hypothesis 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 :  < 3 ) May or may not be accepted May or may not be accepted Is generally the hypothesis that is believed (or needs to be supported) by the researcher Is generally the hypothesis that is believed (or needs to be supported) by the researcher Types of hypotheses

Population Claim: the population mean age is 50. (Null Hypothesis: REJECT Suppose the sample mean age is 20: x = 20 Sample Null Hypothesis 20 likely if  = 50? Is Hypothesis Testing Process If not likely, Now select a random sample H 0 :  = 50 ) x

7 Level of Significance,  Defines unlikely values of sample statistic if null hypothesis is true Defines unlikely values of sample statistic if null hypothesis is true  Defines rejection region of the sampling distribution Is designated by , (level of significance) Is designated by , (level of significance)  Typical values are.01,.05, or.10 Is selected by the researcher at the beginning Is selected by the researcher at the beginning Provides the critical value(s) of the test Provides the critical value(s) of the test

8 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 Represents critical value Lower tail test Level of significance =  0 0  /2  Upper tail test Two tailed test Rejection region is shaded

9 Reject H 0 Do not reject H 0 The cutoff value,, is called a critical value The cutoff value,, is called a critical value Lower Tail Tests  -z α xαxαxαxα -zα-zα-zα-zα 0 μ H 0 : μ ≥ 3 H A : μ < 3 Types of test

10 Reject H 0 Do not reject H 0 The cutoff value, The cutoff value, or, is called a critical value or, is called a critical value Upper Tail Tests  zαzαzαzα xαxαxαxα zαzα xαxα 0 μ H 0 : μ ≤ 3 H A : μ > 3 Types of test

11 Do not reject H 0 Reject H 0 There are two cutoff values (critical values): There are two cutoff values (critical values): or or Two Tailed Tests  /2 -z α/2 x α/2 ± z α/2 x α/2 0 μ0μ0 H 0 : μ = 3 H A : μ  3 z α/2 x α/2 LowerUpper LowerUpper  /2 Types of test

12 Errors in Making Decisions Type I Error 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 Called level of significance of the test Set by researcher in advance Set by researcher in advance

13 Type II Error Type II Error  Fail to reject a false null hypothesis The probability of Type II Error is β (continued) Errors in Making Decisions

14 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 False H 0 True H 0 True Key:Outcome(Probability) No Error ( 1 - β )

15 Critical Value Approach to Testing Convert sample statistic (e.g.: ) to test statistic ( Z or t statistic ) Convert sample statistic (e.g.: ) to test statistic ( Z or t statistic ) Determine the critical value(s) for a specified level of significance  from a table or computer Determine the critical value(s) for a specified level of significance  from a table or computer If the test statistic falls in the rejection region, reject H 0 ; otherwise do not reject H 0 If the test statistic falls in the rejection region, reject H 0 ; otherwise do not reject H 0

16 Convert sample statistic ( ) to a test statistic Convert sample statistic ( ) to a test statistic ( Z or t statistic ) ( Z or t statistic ) x  Known Large Samples  Unknown Hypothesis Tests for  Small Samples Critical Value Approach to Testing

17  Known Large Samples  Unknown Hypothesis Tests for μ Small Samples The test statistic is: Calculating the Test Statistic

18  Known Large Samples  Unknown Hypothesis Tests for  Small Samples The test statistic is: But is sometimes approximated using a z: (continued) Calculating the Test Statistic

19  Known Large Samples  Unknown Hypothesis Tests for  Small Samples The test statistic is: (The population must be approximately normal) Calculating the Test Statistic (continued)

20 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

21 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. H 0 : μ  3 H A : μ < 3 (This is a lower tail test) 2. Suppose that  =.05 is chosen for this test

22 Suppose a sample is taken with the following results: n = 100, x = 2.84 n = 100, x = 2.84 Then the test statistic is: Hypothesis Testing Example But z tab = Since -2.0 < , we reject the null hypothesis that the mean number of TVs in US homes is at least 3

23 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) 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 Form hypothesis test:

24 Suppose a sample is taken with the following results: n = 64, x = 53.1 (  =10 was assumed known)  Then the test statistic is: (continued) Upper Tail z Test for Mean (  Known) But z tab = 1.28 Do not reject H 0 since 0.88 ≤ 1.28 i.e.: there is not sufficient evidence that the mean bill is over $52

25 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 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 s = $ Test at the  = 0.05 level.  = 0.05 level. (Assume the population distribution is normal) H 0 : μ  = 168 H A : μ  168

26   = 0.05  n = 25   is unknown, so use a t statistic  Critical Value: t 24 = ± 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

27 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 p

28 Proportions Sample proportion in the success category is denoted by p  When both np and n(1-p) are at least 5, p can be approximated by a normal distribution with mean and standard deviation  (continued)

29 The sampling distribution of p is normal, so the test statistic is a z value: Hypothesis Tests for Proportions np  5 and n(1-p)  5 Hypothesis Tests for p np < 5 or n(1-p) < 5 Not discussed in this chapter

30 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 p = (500)(.08) = 40 n(1-p) = (500)(.92) = 460

31 Z Test for Proportion: Solution  =.05 n = 500, p =.05 Reject H 0 at  =.05 H 0 : p =.08 H A : p .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

32 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 =

33 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 

34 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)

35 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)

36 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)

37 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

38 Using PHStat Options

39 Sample PHStat Output Input Output

40 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...

41 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)