Understanding Basic Statistics Fourth Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Nine Hypothesis Testing

Copyright © Houghton Mifflin Company. All rights reserved.9 | 2 Hypothesis testing is used to make decisions concerning the value of a parameter.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 3 Null Hypothesis: H 0 a working hypothesis about the population parameter in question

Copyright © Houghton Mifflin Company. All rights reserved.9 | 4 The value specified in the null hypothesis is often: a historical value a claim a production specification

Copyright © Houghton Mifflin Company. All rights reserved.9 | 5 Alternate Hypothesis: H 1 any hypothesis that differs from the null hypothesis

Copyright © Houghton Mifflin Company. All rights reserved.9 | 6 An alternate hypothesis is constructed in such a way that it is the one to be accepted when the null hypothesis must be rejected.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 7 A manufacturer claims that their light bulbs burn for an average of 1000 hours. We have reason to believe that the bulbs do not last that long. Determine the null and alternate hypotheses.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 8 A manufacturer claims that their light bulbs burn for an average of 1000 hours.... The null hypothesis (the claim) is that the true average life is 1000 hours. H 0 :  = 1000

Copyright © Houghton Mifflin Company. All rights reserved.9 | 9 … A manufacturer claims that their light bulbs burn for an average of 1000 hours. We have reason to believe that the bulbs do not last that long.... If we reject the manufacturer’s claim, we must accept the alternate hypothesis that the light bulbs do not last as long as 1000 hours. H 1 :  < 1000

Copyright © Houghton Mifflin Company. All rights reserved.9 | 10 Types of Statistical Tests Left-tailed: H 1 states that the parameter is less than the value claimed in H 0. Right-tailed: H 1 states that the parameter is greater than the value claimed in H 0. Two-tailed: H 1 states that the parameter is different from (  ) the value claimed in H 0.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 11 Given the Null Hypothesis H 0 :  = k If you believe that  is less than k, Use the left-tailed test: H 1 :  < k

Copyright © Houghton Mifflin Company. All rights reserved.9 | 12 If you believe that  is more than k, Use the right-tailed test: H 1 :  > k Given the Null Hypothesis H 0 :  = k

Copyright © Houghton Mifflin Company. All rights reserved.9 | 13 If you believe that  is different from k, Use the two-tailed test: H 1 :   k Given the Null Hypothesis H 0 :  = k

Copyright © Houghton Mifflin Company. All rights reserved.9 | 14 General Procedure for Hypothesis Testing Formulate the null and alternate hypotheses. Take a simple random sample. Compute a test statistic corresponding to the parameter in H 0. Assess the compatibility of the test statistic with H 0.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 15 Hypothesis Testing about the Mean of a Normal Distribution with a Known Standard Deviation 

Copyright © Houghton Mifflin Company. All rights reserved.9 | 16 P-value of a Statistical Test Assuming H 0 is true, the probability that the test statistic (computed from sample data) will take on values as extreme as or more than the observed test statistic is called the P-value of the test The smaller the P-value computed from sample data, the stronger the evidence against H 0.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 17 P-values for Testing a Mean Using the Standard Normal Distribution

Copyright © Houghton Mifflin Company. All rights reserved.9 | 18 P-value for a Left-tailed Test P-value = probability of getting a test statistic less than

Copyright © Houghton Mifflin Company. All rights reserved.9 | 19 P-value = probability of getting a test statistic greater than P-value for a Right-tailed Test

Copyright © Houghton Mifflin Company. All rights reserved.9 | 20 P-value = probability of getting a test statistic lower than or higher than P-value for a Two-tailed Test

Copyright © Houghton Mifflin Company. All rights reserved.9 | 21 Types of Errors in Hypothesis Testing Type I Type II

Copyright © Houghton Mifflin Company. All rights reserved.9 | 22 Type I Error rejecting a null hypothesis which is, in fact, true

Copyright © Houghton Mifflin Company. All rights reserved.9 | 23 Type II Error not rejecting a null hypothesis which is, in fact, false

Copyright © Houghton Mifflin Company. All rights reserved.9 | 24 Type I and Type II Errors

Copyright © Houghton Mifflin Company. All rights reserved.9 | 25 Level of Significance, Alpha (  ) the probability of rejecting a true hypothesis Alpha is the probability of a type I error

Copyright © Houghton Mifflin Company. All rights reserved.9 | 26 Type II Error Beta = β = probability of a type II error (failing to reject a false hypothesis) In hypothesis testing α and β values should be chosen as small as possible. Usually α is chosen first.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 27 Power of the Test = 1 – β The probability of rejecting H 0 when it is in fact false = 1 – . The power of the test increases as the level of significance (  ) increases. Using a larger value of alpha increases the power of the test but also increases the probability of rejecting a true hypothesis.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 28 Probabilities Associated with a Statistical Test

Copyright © Houghton Mifflin Company. All rights reserved.9 | 29 Hypotheses and Types of Errors A fast food restaurant indicated that the average age of its job applicants is fifteen years. We suspect that the true age is lower than 15. We wish to test the claim with a level of significance of  = Determine the Null and Alternate hypotheses and describe Type I and Type II errors.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 30 … average age of its job applicants is fifteen years. We suspect that the true age is lower than 15. H 0 :  = 15 H 1 :  < 15

Copyright © Houghton Mifflin Company. All rights reserved.9 | 31 H 0 :  = 15 H 1 :  < 15  = 0.01 A type I error would occur if we rejected the claim that the mean age was 15, when in fact the mean age was 15 (or higher). The probability of committing such an error is as much as 1%.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 32 A type II error would occur if we failed to reject the claim that the mean age was 15, when in fact the mean age was lower than 15. The probability of committing such an error is called beta. H 0 :  = 15 H 1 :  < 15  = 0.01

Copyright © Houghton Mifflin Company. All rights reserved.9 | 33 Concluding a Hypothesis Test Using the P-value and Level of Significance α If P-value < α reject the null hypothesis and say that the data are statistically significant at the level α. If P-value > α, do not reject the null hypothesis.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 34 Basic Components of a Statistical Test Null hypothesis, alternate hypothesis and level of significance Test statistic and sampling distribution P-value Test conclusion Interpretation of the test results

Copyright © Houghton Mifflin Company. All rights reserved.9 | 35 Null Hypothesis, Alternate Hypothesis and Level of Significance If the sample data evidence against H 0 is strong enough, we reject H 0 and adopt H 1. The level of significance, α, is the probability of rejecting H 0 when it is in fact true.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 36 Test Statistic and Sampling Distribution Mathematical tools to measure compatibility of sample data and the null hypothesis

Copyright © Houghton Mifflin Company. All rights reserved.9 | 37 P-value The probability of obtaining a test statistic from the sampling distribution that is as extreme as or more extreme than the sample test statistic computed from the data under the assumption that H 0 is true

Copyright © Houghton Mifflin Company. All rights reserved.9 | 38 Test Conclusion If P-value < α reject the null hypothesis and say that the data are statistically significant at the level α. If P-value > α, do not reject the null hypothesis.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 39 Interpretation of Test Results Give a simple explanation of conclusion in the context of the application.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 40 Reject or... When the sample evidence is not strong enough to justify rejection of the null hypothesis, we fail to reject the null hypothesis. Use of the term “accept the null hypothesis” should be avoided. When the null hypothesis cannot be rejected, a confidence interval is frequently used to give a range of possible values for the parameter.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 41 Fail to Reject H 0 There is not enough evidence to reject H 0. The null hypothesis is retained but not proved.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 42 Reject H 0 There is enough evidence to reject H 0. Choose the alternate hypothesis with the understanding that it has not been proven.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 43 Testing the Mean  When  is Known Let x be the appropriate random variable. Obtain a simple random sample (of size n) of x values and compute the sample mean x. State the null and alternate hypotheses and set the level of significance α. If x has a normal distribution, any sample size will work. If we cannot assume a normal distribution, use n > 30.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 44 Testing the Mean  When  is Known Use the test statistic:

Copyright © Houghton Mifflin Company. All rights reserved.9 | 45 Testing the Mean  When  is Known Use the standard normal distribution and the type of test (one-tailed or two- tailed) to find the P-value corresponding to the test statistic. If the P-value α, then do not reject H 0. State your conclusion.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 46 Your college claims that the mean age of its students is 28 years. You wish to check the validity of this statistic with a level of significance of  = Assume the standard deviation is 4.3 years. A random sample of 49 students has a mean age of 26 years. Testing the Mean  When  is Known: Example

Copyright © Houghton Mifflin Company. All rights reserved.9 | 47 H 0 :  = 28 H 1 :   28 Perform a ________-tailed test. two Level of significance = α = 0.05 Hypothesis Test Example

Copyright © Houghton Mifflin Company. All rights reserved.9 | 48 Sample Test Statistic

Copyright © Houghton Mifflin Company. All rights reserved.9 | 49 For a two-tailed test: P-value = 2P(z <  3.26) = 2(0.0006) = Sample Results

Copyright © Houghton Mifflin Company. All rights reserved.9 | 50 P-value and Conclusion P-value = α = Since the P-value < α, we reject the null hypothesis. We conclude that the true average age of students is not 28.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 51 Let x be the appropriate random variable. Obtain a simple random sample (of size n) of x values and compute the sample mean x. State the null and alternate hypotheses and set the level of significance α. If x has a mound shaped symmetric distribution, any sample size will work. If we cannot assume this, use n > 30. Testing the Mean  When  is Unknown

Copyright © Houghton Mifflin Company. All rights reserved.9 | 52 Testing the Mean  When  is Known Use the test statistic:

Copyright © Houghton Mifflin Company. All rights reserved.9 | 53 Testing the Mean  When  is Unknown Use the Student’s t distribution and the type of test (one-tailed or two-tailed) to find (or estimate) the P-value corresponding to the test statistic. If the P-value α, then do not reject H 0. State your conclusion.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 54 Using Table 4 to Estimate P-values Use one-tailed areas as endpoints of the interval containing the P-value for one-tailed tests.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 55 P-value for One-tailed Tests

Copyright © Houghton Mifflin Company. All rights reserved.9 | 56 Use two-tailed areas as endpoints of the interval containing the P-value for one-tailed tests. Using Table 4 to Estimate P-values

Copyright © Houghton Mifflin Company. All rights reserved.9 | 57 P-value for Two-tailed Tests

Copyright © Houghton Mifflin Company. All rights reserved.9 | 58 Testing the Mean  When  is Unknown: Example The Parks Department claims that the mean weight of fish in a lake is 2.1 kg. We believe that the true average weight is lower than 2.1 kg. Assume that the weights are mound-shaped and symmetric and a sample of five fish caught in the lake weighed an average of 1.99 kg with a standard deviation of 0.09 kg.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 59 Determine the P-value when testing the claim that the mean weight of fish caught in a lake is 2.1 kg (against the alternate that the weight is lower). A sample of five fish weighed an average of 1.99 kg with a standard deviation of 0.09 kg.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 60 Test the Claim Using α = 10% Null Hypothesis: H 0 :  = 2.1 kg Alternate Hypothesis: H 1 :  < 2.1 kg α = 0.10

Copyright © Houghton Mifflin Company. All rights reserved.9 | 61 We will complete a left-tailed test with:

Copyright © Houghton Mifflin Company. All rights reserved.9 | 62 The Test Statistic t

Copyright © Houghton Mifflin Company. All rights reserved.9 | 63 Using Table 4 with t =  2.73 and d.f. = 4 Sample t =  2.73

Copyright © Houghton Mifflin Company. All rights reserved.9 | 64 The t value is between two values in the chart.Therefore the P-value will be in a corresponding interval. Sample t =  2.73

Copyright © Houghton Mifflin Company. All rights reserved.9 | 65 Sample t =  2.73 Since we are performing a one-tailed test, we use the “one-tail area” line of the chart.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 66 Sample t =  2.73 Since we are performing a one-tailed test, we use the “one-tail area” line of the chart.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 67 Sample t =  2.73 … < P-value < 0.050

Copyright © Houghton Mifflin Company. All rights reserved.9 | < P-value < Since the range of P-values is less than  (10%), we reject the null hypothesis.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 69 Interpret the results: At level of significance 10% we rejected the null hypothesis that the mean weight of fish in the lake was 2.1 kg. Based on our sample data, we conclude that the true mean weight is actually lower than 2.1 kg.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 70 Critical Region (Traditional) Method for Hypothesis Testing An alternate technique to the P-value method Logically equivalent to the P-value method

Copyright © Houghton Mifflin Company. All rights reserved.9 | 71 Critical Region Procedure for Testing  When  is Known Let x be the appropriate random variable. Obtain a simple random sample (of size n) of x values and compute the sample mean x. State the null and alternate hypotheses and set the level of confidence α. If x has a normal distribution, any sample size will work. If we cannot assume a normal distribution, use n > 30.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 72 Critical Region Method for Testing the Mean  When  is Known Use the test statistic:

Copyright © Houghton Mifflin Company. All rights reserved.9 | 73 Critical Region Method for Testing the Mean  When  is Known Using the level of significance α and the alternate hypothesis, show the critical region and critical values on a graph of the sampling distribution. Conclude the test. If the test statistic is in the critical region, then reject H 0. If not, do not reject H 0. State your conclusion.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 74 Most Common Levels of Significance α = 0.05 and α = 0.01

Copyright © Houghton Mifflin Company. All rights reserved.9 | 75 Critical Region(s) The values of x for which we will reject the null hypothesis. The critical values are the boundaries of the critical region(s).

Copyright © Houghton Mifflin Company. All rights reserved.9 | 76 Compare the sample test statistics to the critical value(s) For a left-tailed test: If the sample test statistic is < critical value, reject H 0. If the sample test statistic is > critical value, fail to reject H 0. Concluding Tests Using the Critical Region Method

Copyright © Houghton Mifflin Company. All rights reserved.9 | 77 Critical Region for H 0 :  = k Left-tailed Test

Copyright © Houghton Mifflin Company. All rights reserved.9 | 78 Concluding Tests Using the Critical Region Method Compare the sample test statistics to the critical value(s) For a right-tailed test: If the sample test statistic is > critical value, reject H 0. If the sample test statistic is < critical value, fail to reject H 0.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 79 Critical Region for H 0 :  = k Right-tailed Test

Copyright © Houghton Mifflin Company. All rights reserved.9 | 80 Compare the sample test statistics to the critical value(s) For a two-tailed test: If the sample test statistic lies beyond the critical values, reject H 0. If the sample test statistic lies between the critical values, fail to reject H 0. Concluding Tests Using the Critical Region Method

Copyright © Houghton Mifflin Company. All rights reserved.9 | 81 Critical Region for H 0 :  = k Two-tailed Test

Copyright © Houghton Mifflin Company. All rights reserved.9 | 82 Critical Values z 0 for α = 0.05 and α = 0.01: Left-tailed Test

Copyright © Houghton Mifflin Company. All rights reserved.9 | 83 Critical Values z 0 for α = 0.05 and α = 0.01: Right-tailed Test

Copyright © Houghton Mifflin Company. All rights reserved.9 | 84 Critical Values z 0 for α = 0.05 and α = 0.01: Two-tailed Test

Copyright © Houghton Mifflin Company. All rights reserved.9 | 85 Your college claims that the mean age of its students is 28 years. You wish to check the validity of this statistic with a level of significance of  = Assume the standard deviation is 4.3 years. A random sample of 49 students has a mean age of 26 years. Testing the Mean  When  is Known: Example

Copyright © Houghton Mifflin Company. All rights reserved.9 | 86 two H 0 :  = 28 H 1 :   28 Perform a ________-tailed test. Level of significance = α = 0.05 Hypothesis Test Example

Copyright © Houghton Mifflin Company. All rights reserved.9 | 87 Sample Test Statistic

Copyright © Houghton Mifflin Company. All rights reserved.9 | 88 Sample Results

Copyright © Houghton Mifflin Company. All rights reserved.9 | 89 Critical Region for a Two-tailed Test with α = 0.05

Copyright © Houghton Mifflin Company. All rights reserved.9 | 90 Our z =  3.26 falls within the critical region. z =  3.26

Copyright © Houghton Mifflin Company. All rights reserved.9 | 91 Since the test statistic is in the critical region we… Reject the Null Hypothesis.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 92 Conclusion We conclude that the true average age of students is not 28.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 93 Tests Involving a Proportion We will test claims that a given percentage of the population fits a certain description.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 94 Let r be the binomial random variable, the number of successes out of n independent trials.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 95 For large samples (np > 5 and nq > 5):

Copyright © Houghton Mifflin Company. All rights reserved.9 | 96 Three Types of Tests of Hypotheses for Tests of Proportions Left-tailed tests Right-tailed tests Two-tailed tests

Copyright © Houghton Mifflin Company. All rights reserved.9 | 97 Left-Tailed Test H 0 : p = k H 1 : p < k

Copyright © Houghton Mifflin Company. All rights reserved.9 | 98 H 0 : p = k H 1 : p > k Right-Tailed Test

Copyright © Houghton Mifflin Company. All rights reserved.9 | 99 H 0 : p = k H 1 : p  k Two-Tailed Test

Copyright © Houghton Mifflin Company. All rights reserved.9 | 100

Copyright © Houghton Mifflin Company. All rights reserved.9 | 101 Testing a Proportion p Consider a binomial experiment with n trials. Let p represent the population probability of success. Let q = 1  p represent the population probability of failure. Let r be a random variable that represents the number of successes out of the n binomial trials.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 102 Testing a Proportion p State the null and alternate hypotheses and set the level of significance α. The number of trials should be sufficiently large so that both np > 5 and nq > 5. (Use p from the null hypothesis.)

Copyright © Houghton Mifflin Company. All rights reserved.9 | 103 Testing a Proportion p

Copyright © Houghton Mifflin Company. All rights reserved.9 | 104 Testing a Proportion p Use the standard normal distribution and the type of test (one-tailed or two- tailed) to find the P-value corresponding to the test statistic. If the P-value α, then do not reject H 0. State your conclusion.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 105 In the past, college officials observed that 40% of students took advantage of early registration. This semester, of 4830 students, 2077 took advantage of early registration. Use a 5% level of significance to test the claim that a higher percentage of students now participates in early registration.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 106 In the past, college officials observed that 40% of students took advantage of early registration.... H 0 :  = 0.40

Copyright © Houghton Mifflin Company. All rights reserved.9 | 107 … test the claim that a higher percentage of students now participates in early registration. H 1 :  > 0.40 Use a right-tailed test.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 108 … This semester, of 4830 students, 2077 took advantage of early registration....

Copyright © Houghton Mifflin Company. All rights reserved.9 | 109 The corresponding z value:

Copyright © Houghton Mifflin Company. All rights reserved.9 | 110 Use Table 3 to Determine the P-value Associated with z = 4.26, The P-value is approximately 0.000

Copyright © Houghton Mifflin Company. All rights reserved.9 | 111 Since α = 0.05 and P < α We reject the null hypothesis.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 112 Conclusion Since we have rejected the hypothesis that 40% of students participate in early registration, we conclude that: A percentage higher than 40% of students now participates in early registration.

Copyright © Houghton Mifflin Company. All rights reserved.9 | 113 H 0 :p = 0.70 H 1 :p  0.70 Use a = 0.01 Suppose that in 120 trials there were 80 successes. Test the hypothesis involving a proportion:

Copyright © Houghton Mifflin Company. All rights reserved.9 | 114 We find that:

Copyright © Houghton Mifflin Company. All rights reserved.9 | 115 Use Table 3 to find the P-value associated with z = –0.72. P(z  –0.72) = Since the test is a two-tailed test, double the area in the left tail to find P. P = 2(0.2358) =

Copyright © Houghton Mifflin Company. All rights reserved.9 | 116 When the P-value is greater than the level of significance, we do not reject the null hypothesis. P =  = 0.01 Do not reject the null hypothesis.