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## Statistics

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**Statistics**Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**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**Where We’re Going**• Test a specific value of a population parameter • Measure the reliability of the test McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.1: The Elements of a Test of Hypotheses**Confidence Interval Where on the number line do the data point us? (No prior idea about the value of the parameter.) µ? µ? Hypothesis Test Do the data point us to this particular value? (We have a value in mind from the outset.) µ0? McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.1: The Elements of a Test of Hypotheses**• Null Hypothesis: H0 • This will be supported unless the data provide evidence that it is false • The status quo • Alternative Hypothesis: Ha • This will be supported if the data provide sufficient evidence that it is true • The research hypothesis McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.1: The Elements of a Test of Hypotheses**• If the test statistic has a high probability when H0is true, then H0 is not rejected. • If the test statistic has a (very) low probability when H0is true, then H0 is rejected. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.1: The Elements of a Test of Hypotheses**McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.1: The Elements of a Test of Hypotheses**McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.1: The Elements of a Test of Hypotheses**Note:Null hypotheses are either rejected, or else there is insufficient evidence to reject them. (I.e., we don’t accept null hypotheses.) McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.1: The Elements of a Test of Hypotheses**• Null hypothesis (H0): A theory about the values of one or more parameters • Ex.: H0: µ = µ0 (a specified value for µ) • Alternative hypothesis (Ha): Contradicts the null hypothesis • Ex.: H0: µ ≠ µ0 • 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**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**8.1: The Elements of a Test of Hypotheses**H0: µ = 4 Ha: µ ≠ 4 Sample statistic: = 4.7 α= .05 Assume the sampling distribution is normal. Test statistic: Conclusion: z.025 = 1.96. Since z* > z.025 , reject H0. (That is, there do seem to be more yellow cards.) McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.2: Large-Sample Test of a Hypothesis about a Population**Mean The null hypothesis is usually stated as an equality … … even though the alternative hypothesis can be either an equality or an inequality. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**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**8.2: Large-Sample Test of a Hypothesis about a Population**Mean Rejection Regions for Common Values of α McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.2: Large-Sample Test of a Hypothesis about a Population**Mean One-Tailed Test Two-Tailed Test • H0: µ = µ0 Ha: µ ≠ µ0 Test Statistic: Rejection Region: | z | > zα/2 • H0: µ = µ0 Ha: µ < or > µ0 Test Statistic: Rejection Region: | z | > zα Conditions: 1) A random sample is selected from the target population. 2) The sample size n is large. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**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 α = .05 level, H0: µ = $60,000. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.2: Large-Sample Test of a Hypothesis about a Population**Mean • H0: µ = 60,000 Ha: µ ≠ 60,000 Test Statistic: Rejection Region: | z | > z.025 = 1.96 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 α = .05 level, H0: µ = $60,000. Do not reject H0 McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.3:Observed Significance Levels: p - Values**Suppose z = 2.12. P(z > 2.12) = .0170. Reject H0 at the α= .05 level Do not reject H0 at the α= .01 level But it’s pretty close, isn’t it? McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.3:Observed Significance Levels: p - Values**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 H0 is true. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.3:Observed Significance Levels: p - Values**H0: µ = 65,000 Ha: µ ≠ 65,000 Test Statistic: p-value: P(|z| > 1.675) = .047 * 2 =0.094 Let’s go back to the Economics of Education Review report (= $61,340, s = $2,185). This time we’ll test H0: µ = $65,000. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.3:Observed Significance Levels: p - Values**• Reporting test results • Choose the maximum tolerable value of α • If the p-value≤α, reject H0 If the p-value> α, do not reject H0 McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.4: Small-Sample Test of a Hypothesis about a Population**Mean If the sample is small and is unknown, testing hypotheses about µ requires the t-distribution instead of the z-distribution. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.4: Small-Sample Test of a Hypothesis about a Population**Mean One-Tailed Test Two-Tailed Test • H0: µ = µ0 Ha: µ ≠ µ0 Test Statistic: Rejection Region: | t | > tα/2 • H0: µ = µ0 Ha: µ < or > µ0 Test Statistic: Rejection Region: | t | > tα 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 tα is based on (n – 1) degrees of freedom McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**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**8.4: Small-Sample Test of a Hypothesis about a Population**Mean H0: µ = 100,000 Ha: µ > 100,000 Test Statistic: p-value: P(tdf=4 > 14.06) < .0005 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**8.4: Small-Sample Test of a Hypothesis about a Population**Mean H0: µ = 100,000 Ha: µ > 100,000 Test Statistic: p-value: P(|tdf=4| > 14.06) < .0005 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. 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**8.5: Large-Sample Test of a Hypothesis about a Population**Proportion One-Tailed Test Two-Tailed Test • H0: p = p0 Ha: p ≠ p0 Test Statistic: Rejection Region: | z | > zα/2 • H0: p = p0 Ha: p < or > p0 Test Statistic: Rejection Region: | z | > zα p0 = hypothesized value of p, , and q0 = 1 - p0 Conditions: 1) A random sample is selected from a binomial population. 2) The sample size n is large (i.e., np0 and nq0 are both ≥ 15). McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.5: 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). 1500 sections of rope are tested, with 20 pieces failing the test. At the α = .01 level, should the shipment be rejected? McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.5: Large-Sample Test of a Hypothesis about a Population**Proportion The vendor will reject any shipment that cannot pass a 1% defects test . 1500 sections of rope are tested, with 20 pieces failing the test. At the α = .01 level, should the shipment be rejected? H0: p = .01 Ha: p > .01 Rejection region: |z| > 2.33 Test statistic: McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.5: Large-Sample Test of a Hypothesis about a Population**Proportion There is insufficient evidence to reject the null hypothesis based on the sample results. The vendor will reject any shipment that cannot pass a 1% defects test . 1500 sections of rope are tested, with 20 pieces failing the test. At the α = .01 level, should the shipment be rejected? H0: p = .01 Ha: p > .01 Rejection region: |z| > 2.33 Test statistic: McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.6: Calculating Type II Error Probabilities: More about β**To calculate P(Type II), or β, … 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**8.6: Calculating Type II Error Probabilities: More about β**To calculate P(Type II), or β, … 1. Calculate the value(s) of that divide the “do not reject” region from the “reject” region(s). 2. Calculate the z-value of 0assuming the alternative hypothesis mean is the true µ: The probability of getting this z-value for the acceptance region is β. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.6: Calculating Type II Error Probabilities: More about β**• 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 - β). McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.6: Calculating Type II Error Probabilities: More about β**• H0: µ = 60,000 Ha: µ ≠ 60,000 • Test Statistic: z = .613; • zα/2=.025= 1.96 • We did not reject this null hypothesis earlier, but what if the true mean were $62,000? 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 α = .05 level, H0: µ = $60,000. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.6: Calculating Type II Error Probabilities: More about β**xO,L= 60,000 - 1.96*2,185 = 55,717.4 xO,U= 60,000 +1.96*2,185 = 64,282.6 ZL = (55,717.4 – 62,000)/2,185 = -2.875 ZU = (64,282.6 - 62,000)/2,185 = 1.045 β = P(-2.875≤Z≤1.045) = 0.8511 The power of this test is 1 – β = 1 – 0.8511 = 0.1489 The Economics of Education Review (Vol. 21, 2002) reported a mean salary for males with postgraduate degrees of $61,340, with sequal to $2,185. We did not reject this null hypothesis earlier, but what if the true mean were $62,000? McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.6: Calculating Type II Error Probabilities: More about β**• For fixed n and α, the value of βdecreases and the power increases as the distance between µ0 and µa increases. • For fixed n, µ0 and µa, the value of β increases and the power decreases as the value of α is decreased. • For fixed α, µ0 and µa, the value of β decreases and the power increases as n is increased. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.6: Calculating Type II Error Probabilities: More about β**McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.7: Tests of Hypotheses about a Population Variance**McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.7: Tests of Hypotheses about a Population Variance**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! McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.7: Tests of Hypotheses about a Population Variance**One-Tailed Test Test statistic: Rejection region: Two-Tailed Test Test statistic: Rejection region: McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**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**8.7: Tests of Hypotheses about a Population Variance**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 α = .10 level? McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.7: Tests of Hypotheses about a Population Variance**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 α = .10 level? Two-Tailed Test Test statistic: Rejection criterion: McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses**8.7: Tests of Hypotheses about a Population Variance**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 α = .10 level? Two-Tailed Test Test statistic: Rejection criterion: Do not reject the null hypothesis. McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses