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Chapter 9 Learning Objectives

Chapter 9 Learning Objectives. Population Mean: s Unknown. Population Proportion. Tests About a Population Mean: s Unknown. Tests About a Population Mean: s Unknown. Test Statistic. This test statistic has a t distribution with n - 1 degrees of freedom.

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Chapter 9 Learning Objectives

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  1. Chapter 9Learning Objectives • Population Mean: s Unknown • Population Proportion

  2. Tests About a Population Mean:s Unknown

  3. Tests About a Population Mean:s Unknown • Test Statistic This test statistic has a t distribution with n - 1 degrees of freedom.

  4. Tests About a Population Mean:s Unknown • Rejection Rule: Critical Value Approach H1:  Reject H0 if t>t H1:  Reject H0 if t< -t H1:  Reject H0 if t< - t or t>t • Rejection Rule: p -Value Approach Reject H0 if p –value <a

  5. p -Values and the t Distribution • The format of the t distribution table provided in most • statistics textbooks does not have sufficient detail • to determine the exact p-value for a hypothesis test. • However, we can still use the t distribution table to identify a range for the p-value. • An advantage of computer software packages is that the computer output will provide the p-value for the t distribution.

  6. Example: Highway Patrol • One-Tailed Test About a Population Mean: s Unknown A State Highway Patrol periodically samples vehicle speeds at various locations on a particular roadway. The sample speed is used to test the null hypothesis H0: m< 65 The locations where H0 is rejected are deemed the best locations for radar traps.

  7. Example: Highway Patrol At Location F, a sample of 64 vehicles shows a mean speed of 66.2 mph with a standard deviation of 4.2 mph. Use a = .05 to test the above null hypothesis.

  8. One-Tailed Test About a Population Mean:s Unknown 1. Determine the hypotheses. H0: < 65 Ha: m > 65 a = .05 2. Specify the level of significance. 3. Compute the value of the test statistic.

  9. One-Tailed Test About a Population Mean:s Unknown • Using the Critical Value Approach 4. Determine the Critical value and rejection rule. For a = .05 and d.f. = 64 – 1 = 63, t.05 = 1.669 5. Compare Test Statistic with the Critical Value Because 2.286 > 1.669, we reject H0. We are at least 95% confident that the mean speed of vehicles at Location F is greater than 65 mph. Location F is a good candidate for a radar trap.

  10. One-Tailed Test About a Population Mean:s Unknown • Using the p–Value Approach 4. Compute the p –value. For t = 2.286, the p–value lies between .025 (where t = 1.998) and .01 (where t = 2.387). .01 < p–value < .025 5. Determine whether to reject H0. Since any p–value between 0.01 and 0.025 is less than a = .05. Thus, we reject H0. That is, we are at least 95% confident that the mean speed of vehicles at Location F is greater than 65 mph.

  11. Hypotheses Testing of the Population Proportion • In general, a hypothesis test about the value of a population • proportion p takes one of the following three forms (where p0 • is the hypothesized value of the population proportion). One-tailed (lower tail) One-tailed (upper tail) Two-tailed

  12. Tests About a Population Proportion • The Test Statistic where: assuming np> 5 and n(1 – p) > 5

  13. Tests About a Population Proportion • When using the Critical Value Approach H1 : pp Reject H0 if z>z Reject H0 if z< -z H1: p p H1pp Reject H0 if z< -z or z>z • When Using the p –Value Approach Reject H0 if p –value <a

  14. Two-Tailed Test About a Population Proportion • Example: During a Christmas and New Year’s week, the National Safety Council estimates that 500 people would be killed and 25,000 Would be injured on the nation’s roads. The NSC claims that 50% of the accidents would be caused by drunk driving.

  15. Two-Tailed Test About a Population Proportion A sample of 120 accidents showed that 67 were caused by drunk driving. Use these data to test the NSC’s claim with a = .05.

  16. Two-Tailed Test About a Population Proportion 1. Determine the hypotheses. 2. Specify the level of significance. a = .05 3. Compute the value of the test statistic.

  17. Two-Tailed Test About a Population Proportion • Using the Critical Value Approach 4. Determine the critical value For a/2 = .05/2 = .025, z.025 = 1.96. Since the alternative is non-directional, the rejection region would be the area below -1.96 and above 1.96 5. Compare the Test Statistic with the Critical Value As 1.278 falls between -1.96 and 1.96 (the acceptance region), we cannot reject H0.

  18. Two-Tailed Test About a Population Proportion • Using the p-Value Approach 4. Compute the p -value. For z = 1.28, cumulative probability = .8997 p–value = 2(1 - .8997) = .2006 5. Compare the p-value with the significance level As p–value = .2006 > a = .05, we fail to reject H0.

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