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Understanding Hypothesis Testing for Population Means and Proportions

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margie
18 diapositives · 318 Views · Jul 17, 2014

This guide explores hypothesis testing concerning population means when the population standard deviation is unknown, focusing on the t-distribution and critical value approach. It covers rejection rules, the p-value approach, and example applications, such as in traffic speed evaluations by the Highway Patrol. Additionally, it delves into hypothesis testing for population proportions, illustrating the process using the National Safety Council's accident statistics related to drunk driving. Key concepts include test statistics, significance levels, and how to interpret results for effective decision-making.

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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.