1 1 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) Chapter 9 Learning Objectives Population Mean:  Unknown Population Mean:  Unknown Population Proportion Population Proportion

2 2 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) Tests About a Population Mean:  Unknown

3 3 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) n Test Statistic Tests About a Population Mean:  Unknown This test statistic has a t distribution with n - 1 degrees of freedom. with n - 1 degrees of freedom.

4 4 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) n Rejection Rule: p -Value Approach H 1 :   Reject H 0 if t < - t  Reject H 0 if t > t  Reject H 0 if t t  H 1 :   H 1 :   Tests About a Population Mean:  Unknown n Rejection Rule: Critical Value Approach Reject H 0 if p –value < 

5 5 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) p -Values and the t Distribution The format of the t distribution table provided in most The format of the t distribution table provided in most statistics textbooks does not have sufficient detail statistics textbooks does not have sufficient detail to determine the exact p -value for a hypothesis test. to determine the exact p -value for a hypothesis test. However, we can still use the t distribution table to However, we can still use the t distribution table to identify a range for the p -value. identify a range for the p -value. An advantage of computer software packages is that An advantage of computer software packages is that the computer output will provide the p -value for the the computer output will provide the p -value for the t distribution. t distribution.

6 6 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) A State Highway Patrol periodically samples 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 Example: Highway Patrol One-Tailed Test About a Population Mean:  Unknown One-Tailed Test About a Population Mean:  Unknown The locations where H 0 is rejected are deemed The locations where H 0 is rejected are deemed the best locations for radar traps. H 0 :  < 65

7 7 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) 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  =.05 to test the above null hypothesis.

8 8 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) One-Tailed Test About a Population Mean:  Unknown 1. Determine the hypotheses. 2. Specify the level of significance. 3. Compute the value of the test statistic.  =.05 H 0 :  < 65 H a :  > 65

9 9 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) Using the Critical Value Approach Using the Critical Value Approach 5. Compare Test Statistic with the Critical Value 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. Because > 1.669, we reject H 0. One-Tailed Test About a Population Mean:  Unknown For  =.05 and d.f. = 64 – 1 = 63, t.05 = Determine the Critical value and rejection rule.

10 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) One-Tailed Test About a Population Mean:  Unknown Using the p –Value Approach Using the p –Value Approach 5. Determine whether to reject H 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 Since any p –value between 0.01 and is less than  =.05. Thus, we reject H 0. That is, we are at least 95% confident that the mean speed of vehicles at Location F is greater than 65 mph.

11 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse)  0 0 t  = t  = Reject H 0 Do Not Reject H 0 t One-Tailed Test About a Population Mean:  Unknown

12 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) In general, a hypothesis test about the value of a population In general, a hypothesis test about the value of a population proportion p takes one of the following three forms (where p 0 is the hypothesized value of the population proportion). Hypotheses Testing of the Population Proportion One-tailed (lower tail) One-tailed (upper tail) Two-tailed

13 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) n The Test Statistic Tests About a Population Proportion where: assuming np > 5 and n (1 – p ) > 5

14 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) n When Using the p –Value Approach H 1 : p  p  Reject H 0 if z > z  Reject H 0 if z < - z  Reject H 0 if z z  H 1 : p  p  H 1 p  p  Tests About a Population Proportion Reject H 0 if p –value <  n When using the Critical Value Approach

15 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) n Example: During a Christmas and New Year’s week, the 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. Two-Tailed Test About a Population Proportion

16 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) A sample of 120 accidents showed that A sample of 120 accidents showed that 67 were caused by drunk driving. Use these data to test the NSC’s claim with  =.05. Two-Tailed Test About a Population Proportion

17 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) Two-Tailed Test About a Population Proportion 1. Determine the hypotheses. 2. Specify the level of significance. 3. Compute the value of the test statistic.  =.05

18 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) Two-Tailed Test About a Population Proportion Using the Critical Value Approach Using the Critical Value Approach 5. Compare the Test Statistic with the Critical Value For  /2 =.05/2 =.025, z.025 = Since the alternative is non-directional, the rejection region would be the area below and above Determine the critical value As falls between and 1.96 (the acceptance region), we cannot reject H 0.

19 Slide ©2009. Econ-2030-Applied Statistics (Dr. Tadesse) Using the p  Value Approach Using the p  Value Approach 4. Compute the p -value. 5. Compare the p-value with the significance level As p –value =.2006 >  =.05, we fail to reject H 0. Two-Tailed Test About a Population Proportion For z = 1.28, cumulative probability =.8997 p –value = 2(1 .8997) =.2006