Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses Statistics

McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 2 Where We’ve Been Calculated point estimators of population parameters Used the sampling distribution of a statistic to assess the reliability of an estimate through a confidence interval

McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 3 Where We’re Going Test a specific value of a population parameter Measure the reliability of the test

8.1: The Elements of a Test of Hypotheses McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 4 Confidence Interval Where on the number line do the data point us? (No prior idea about the value of the parameter.) Hypothesis Test Do the data point us to this particular value? (We have a value in mind from the outset.) µ? µ0?µ0?

8.1: The Elements of a Test of Hypotheses McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 5 Null Hypothesis: H 0 This will be supported unless the data provide evidence that it is false The status quo Alternative Hypothesis: H a This will be supported if the data provide sufficient evidence that it is true The research hypothesis

8.1: The Elements of a Test of Hypotheses If the test statistic has a high probability when H 0 is true, then H 0 is not rejected. If the test statistic has a (very) low probability when H 0 is true, then H 0 is rejected. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 6

8.1: The Elements of a Test of Hypotheses McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 7

8.1: The Elements of a Test of Hypotheses Reality ↓ / Test Result → Do not reject H 0 Reject H 0 H 0 is true Correct! Type I Error: rejecting a true null hypothesis P(Type I error) =  H 0 is false Type II Error: not rejecting a false null hypothesis P(Type II error) =  Correct! McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 8

8.1: The Elements of a Test of Hypotheses McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 9 Note: Null hypotheses are either rejected, or else there is insufficient evidence to reject them. (I.e., we don’t accept null hypotheses.)

8.1: The Elements of a Test of Hypotheses Null hypothesis (H 0 ): A theory about the values of one or more parameters Ex.: H 0 : µ = µ 0 (a specified value for µ) Alternative hypothesis (H a ): Contradicts the null hypothesis Ex.: H 0 : µ ≠ µ 0 Test Statistic: The sample statistic to be used to test the hypothesis Rejection region: The values for the test statistic which lead to rejection of the null hypothesis Assumptions: Clear statements about any assumptions concerning the target population Experiment and calculation of test statistic: The appropriate calculation for the test based on the sample data Conclusion: Reject the null hypothesis (with possible Type I error) or do not reject it (with possible Type II error) McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 10

8.1: The Elements of a Test of Hypotheses Suppose a new interpretation of the rules by soccer referees is expected to increase the number of yellow cards per game. The average number of yellow cards per game had been 4. A sample of 121 matches produced an average of 4.7 yellow cards per game, with a standard deviation of.5 cards. At the 5% significance level, has there been a change in infractions called? McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 11

8.1: The Elements of a Test of Hypotheses McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 12 H 0 : µ = 4 H a : µ ≠ 4 Sample statistic:  = 4.7 "=.05 Assume the sampling distribution is normal. Test statistic: Conclusion: z.05 = Since z* > z.05, reject H 0. (That is, there do seem to be more yellow cards.)

8.2: Large-Sample Test of a Hypothesis about a Population Mean McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 13 H0: µ = µ0 Ha: µ < µ0 Ha: µ ≠ µ0Ha: µ > µ0 The null hypothesis is usually stated as an equality … … even though the alternative hypothesis can be either an equality or an inequality.

8.2: Large-Sample Test of a Hypothesis about a Population Mean McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 14

8.2: Large-Sample Test of a Hypothesis about a Population Mean Lower TailedUpper TailedTwo tailed  =.10z < z > 1.28| z | >  =.05z < z > 1.645| z | > 1.96  =.01z < z > 2.33| z | > McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 15 Rejection Regions for Common Values of 

8.2: Large-Sample Test of a Hypothesis about a Population Mean One-Tailed Test H 0 : µ = µ 0 H a : µ µ 0 Test Statistic: Rejection Region: | z | > z  Two-Tailed Test McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 16 H 0 : µ = µ 0 H a : µ ≠ µ 0 Test Statistic: Rejection Region: | z | > z  /2 Conditions: 1) A random sample is selected from the target population. 2) The sample size n is large.

8.2: Large-Sample Test of a Hypothesis about a Population Mean The Economics of Education Review (Vol. 21, 2002) reported a mean salary for males with postgraduate degrees of $61,340, with an estimated standard error (s  ) equal to $2,185. We wish to test, at the  =.05 level, H 0 : µ = $60,000. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 17

8.2: Large-Sample Test of a Hypothesis about a Population Mean The Economics of Education Review (Vol. 21, 2002) reported a mean salary for males with postgraduate degrees of $61,340, with an estimated standard error (s  ) equal to $2,185. We wish to test, at the  =.05 level, H 0 : µ = $60,000. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 18 H 0 : µ = 60,000 H a : µ ≠ 60,000 Test Statistic: Rejection Region: | z | > z.025 = 1.96 Do not reject H 0

8.3:Observed Significance Levels: p - Values McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 19 Suppose z = P(z > 2.12) = Reject H 0 at the  =.05 levelDo not reject H 0 at the  =.01 level But it’s pretty close, isn’t it?

8.3:Observed Significance Levels: p - Values McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 20 The observed significance level, or p-value, for a test is the probability of observing the results actually observed (z*) assuming the null hypothesis is true. The lower this probability, the less likely H 0 is true.

Let’s go back to the Economics of Education Review report (  = $61,340, s  = $2,185). This time we’ll test H 0 : µ = $65,000. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 21 H 0 : µ = 65,000 H a : µ ≠ 65,000 Test Statistic: p-value: P(   61,340 |H 0 ) = P(|z| > 1.675) = :Observed Significance Levels: p - Values

Reporting test results  Choose the maximum tolerable value of   If the p-value < , reject H 0 If the p-value > , do not reject H 0 McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 22

Some stats packages will only report two-tailed p-values. 8.3:Observed Significance Levels: p - Values McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 23 Converting a Two-Tailed p-Value to a One-Tailed p-Value

Some stats packages will only report two-tailed p-values. 8.3:Observed Significance Levels: p - Values McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 24 Converting a Two-Tailed p-Value to a One-Tailed p-Value

8.4: Small-Sample Test of a Hypothesis about a Population Mean McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 25 If the sample is small and  is unknown, testing hypotheses about µ requires the t-distribution instead of the z-distribution.

One-Tailed Test H 0 : µ = µ 0 H a : µ µ 0 Test Statistic: Rejection Region: | t | > t  Two-Tailed Test McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 26 H 0 : µ = µ 0 H a : µ ≠ µ 0 Test Statistic: Rejection Region: | t | > t  /2 Conditions: 1) A random sample is selected from the target population. 2) The population from which the sample is selected is approximately normal. 3) The value of t  is based on (n – 1) degrees of freedom 8.4: Small-Sample Test of a Hypothesis about a Population Mean

Suppose copiers average 100,000 between paper jams. A salesman claims his are better, and offers to leave 5 units for testing. The average number of copies between jams is 100,987, with a standard deviation of 157. Does his claim seem believable? McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 27

H 0 : µ = 100,000 H a : µ > 100,000 Test Statistic: p-value: P(   100,987|H 0 ) = P(|t df=4 | > 14.06) < : Small-Sample Test of a Hypothesis about a Population Mean Suppose copiers average 100,000 between paper jams. A salesman claims his are better, and offers to leave 5 units for testing. The average number of copies between jams is 100,987, with a standard deviation of 157. Does his claim seem believable? McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 28

H 0 : µ = 100,000 H a : µ > 100,000 Test Statistic: p-value: P(   100,987|H 0 ) = P(|t df=4 | > 14.06) < : Small-Sample Test of a Hypothesis about a Population Mean Suppose copiers average 100,000 between paper jams. A salesman claims his are better, and offers to leave 5 units for testing. The average number of copies between jams is 100,987, with a standard deviation of 157. Does his claim seem believable? McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 29 Reject the null hypothesis based on the very low probability of seeing the observed results if the null were true. So, the claim does seem plausible.

One-Tailed Test H 0 : p = p 0 H a : p p 0 Test Statistic: Rejection Region: | z | > z  Two-Tailed Test McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 30 H 0 : p = p 0 H a : p ≠ p 0 Test Statistic: Rejection Region: | z | > z  /2 Conditions: 1) A random sample is selected from a binomial population. 2) The sample size n is large (i.e., np 0 and nq 0 are both  15). 8.5: Large-Sample Test of a Hypothesis about a Population Proportion p 0 = hypothesized value of p,, and q 0 = 1 - p 0

McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses : Large-Sample Test of a Hypothesis about a Population Proportion Rope designed for use in the theatre must withstand unusual stresses. Assume a brand of 3” three-strand rope is expected to have a breaking strength of 1400 lbs. A vendor receives a shipment of rope and needs to (destructively) test it. The vendor will reject any shipment which cannot pass a 1% defect test (that’s harsh, but so is falling scenery during an aria) sections of rope are tested, with 20 pieces failing the test. At the  =.01 level, should the shipment be rejected?

McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses : Large-Sample Test of a Hypothesis about a Population Proportion The vendor will reject any shipment that cannot pass a 1% defects test sections of rope are tested, with 20 pieces failing the test. At the  =.01 level, should the shipment be rejected? H 0 : p =.01 H a : p >.01 Rejection region: |z| > Test statistic:

McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses : Large-Sample Test of a Hypothesis about a Population Proportion The vendor will reject any shipment that cannot pass a 1% defects test sections of rope are tested, with 20 pieces failing the test. At the  =.01 level, should the shipment be rejected? H 0 : p =.01 H a : p >.01 Rejection region: |z| > Test statistic: There is insufficient evidence to reject the null hypothesis based on the sample results.

8.6: Calculating Type II Error Probabilities: More about  To calculate P(Type II), or , … 1. Calculate the value(s) of  that divide the “do not reject” region from the “reject” region(s). Upper-tailed test: Lower-tailed test: Two-tailed test: McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 34

8.6: Calculating Type II Error Probabilities: More about  To calculate P(Type II), or , … 1. Calculate the value(s) of  that divide the “do not reject” region from the “reject” region(s). 2. Calculate the z-value of  0 assuming the alternative hypothesis mean is the true µ: The probability of getting this z-value is . McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 35

8.6: Calculating Type II Error Probabilities: More about  The power of a test is the probability that the test will correctly lead to the rejection of the null hypothesis for a particular value of µ in the alternative hypothesis. The power of a test is calculated as (1 -  ). McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 36

8.6: Calculating Type II Error Probabilities: More about  McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 37 The Economics of Education Review (Vol. 21, 2002) reported a mean salary for males with postgraduate degrees of $61,340, with an estimated standard error (s  ) equal to $2,185. We wish to test, at the  =.05 level, H 0 : µ = $60,000. H 0 : µ = 60,000 H a : µ ≠ 60,000 Test Statistic: z =.613; z  =.025 = 1.96 We did not reject this null hypothesis earlier, but what if the true mean were $62,000?

8.6: Calculating Type II Error Probabilities: More about  McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 38 The Economics of Education Review (Vol. 21, 2002) reported a mean salary for males with postgraduate degrees of $61,340, with s  equal to $2,185. We did not reject this null hypothesis earlier, but what if the true mean were $62,000? The power of this test is =.6179

8.6: Calculating Type II Error Probabilities: More about  For fixed n and , the value of  decreases and the power increases as the distance between µ 0 and µ a increases. For fixed n, µ 0 and µ a, the value of  increases and the power decreases as the value of  is decreased. For fixed , µ 0 and µ a, the value of  decreases and the power increases as n is increased. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 39

For fixed n and   decreases and (1-  ) increases as |µ0 - µa| increases For fixed n, µ0 and µa  increases and (1 -  ) decreases as  decreases  decreases and (1 -  ) increases as n increases For fixed , µ0 and µa 8.6: Calculating Type II Error Probabilities: More about  McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 40

8.7: Tests of Hypotheses about a Population Variance McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 41 Random Variable X Sample Statistic:  or p-hat Hypothesis Test on µ or pBased on zBased on t Sample Statistic: s 2 Hypothesis Test on  2 or  Based on  2

8.7: Tests of Hypotheses about a Population Variance McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 42 The chi-square distribution is really a family of distributions, depending on the number of degrees of freedom. But, the population must be normally distributed for the hypothesis tests on  2 (or  ) to be reliable!

8.7: Tests of Hypotheses about a Population Variance One-Tailed Test Test statistic: Rejection region: McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 43 Two-Tailed Test Test statistic: Rejection region:

8.7: Tests of Hypotheses about a Population Variance Conditions Required for a Valid Large- Sample Hypothesis Test for  2 1. A random sample is selected from the target population. 2. The population from which the sample is selected is approximately normal. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 44

8.7: Tests of Hypotheses about a Population Variance McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 45 Earlier, we considered the average number of copies between jams for a brand of copiers. The salesman also claims his copiers are more predictable, in that the standard deviation of jams is 125. In the sample of 5 copiers, that sample standard deviation was 157. Does his claim seem believable, at the  =.10 level?

8.7: Tests of Hypotheses about a Population Variance Two-Tailed Test Test statistic: Rejection criterion: McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 46 Earlier, we considered the average number of copies between jams for a brand of copiers. The salesman also claims his copiers are more predictable, in that the standard deviation of jams is 125. In the sample of 5 copiers, that sample standard deviation was 157. Does his claim seem believable, at the  =.10 level?

8.7: Tests of Hypotheses about a Population Variance Two-Tailed Test Test statistic: Rejection criterion: McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses 47 Earlier, we considered the average number of copies between jams for a brand of copiers. The salesman also claims his copiers are more reliable, in that the standard deviation of jams is 125. In the sample of 5 copiers, that sample standard deviation was 157. Does his claim seem believable, at the  =.10 level? Do not reject the null hypothesis.