Inferences about Two Populations Practice for the Final Exam Part 1 Inferences about Two population Means. Independent Samples. Construct the indicated confidence interval or build the test for the difference between the two population means. Assume that the assumptions and conditions for inference have been met. 1) Two types of flares are tested for their burning times (in minutes) and sample results are given below. n x s Brand X 35 19.4 1.4 Brand Y 40 15.1 0.8 Construct a 95% confidence interval for the difference between population mean burning times based on the sample data. 2) A grocery store is interested in determining whether or not a difference exists between the shelf life of two different brands of doughnuts. A random sample of 100 boxes of each brand was selected and the shelf life in days was determined for each box. The sample results are given below. Find a 90% confidence interval for A - B, the difference in mean shelf life between brand A and brand B. 3) The table below gives information concerning the gasoline mileage for random samples of trucks of two different types. Find a 95% confidence interval for the difference in the means X - Y. 4) A researcher was interested in comparing the salaries of female and male employees of a particular company. Independent random samples of 8 female employees (sample 1) and 15 male employees (sample 2) yielded the following weekly salaries (in dollars). Female Male 495 722 760 562 556 880 904 520 520 500 1005 1250 743 750 660 1640 518 904 1150 805 480 970 605 mean 705.4 817.1 st.dev 183.9 330.1 n 8 15 Determine a 98% confidence interval for the difference, F - M between the mean weekly salary of all female employees and the mean weekly salary of all male employees. Interpret the given confidence interval. 5) A grocery store is interested in determining whether or not a difference exists between the shelf life of Tasty Choice doughnuts and Sugar Twist doughnuts. A random sample of 100 boxes of each brand was selected and the mean shelf life in days was determined for each brand. A 90% confidence interval for the difference of the means, Tc - St was determined to be ( 1.1, 2.4). A) Based on this sample, we are 90% confident that Sugar Twist doughnuts will last on average between 1.1 and 2.4 days longer than Tasty Choice doughnuts. B) We are 90% confident that a randomly selected box of Tasty Choice doughnuts will have a shelf life that is between 1.1 and 2.4 days longer than a randomly selected box of Sugar Twist doughnuts. C) We know that 90% of Tasty Choice doughnuts last between 1.1 and 2.4 days longer than Sugar Twist doughnuts. D) Based on this sample, we are 90% confident that Tasty Choice doughnuts will last on average between 1.1 and 2.4 days longer than Sugar Twist doughnuts. E) We know that 90% of all random samples done on the population will show that the mean shelf life of Tasty Choice doughnuts is between 1.1 and 2.4 days longer than the mean shelf life of Sugar Twist doughnuts. 6) A researcher was interested in comparing the salaries of female and male employees of a particular company. Independent random samples of female employees (sample 1) and male employees (sample 2) were taken to calculate the mean salary, in dollars per week, for each group. A 90% confidence interval for the difference, 1 - 2 between the mean weekly salary of all female employees and the mean weekly salary of all male employees was determined to be ($-110, $10). A) We are 90% confident that a randomly selected female employee at this company makes between $110 less and $10 more per week than a randomly selected male employee. B) Based on these data, with 90% confidence, female employees at this company average between $110 less and $10 more per week than the male employees. C) Based on these data, with 90% confidence, male employees at this company average between $110 less and $10 more per week than the female employees. D) We know that 90% of female employees at this company make between $110 less and $10 more than the male employees. E) We know that 90% of all random samples done on the employees at this company will show that the average female salary is between $110 less and $10 more per week than the average male salary. 7) A survey was conducted to determine the difference in gasoline mileage for two types of trucks. A random sample was taken for each model of truck, and the mean gasoline mileage, in miles per gallon, was calculated. A 98% confidence interval for the difference in the mean mileage for model A trucks and the mean mileage for model B trucks, A - B was determined to be (2.5, 4.7). A) We know that 98% of model A trucks get mileage that is between 2.5 and 4.7 miles per gallon higher than model B trucks. B) Based on this sample, we are 98% confident that the average mileage for model B trucks is between 2.5 and 4.7 miles per gallon higher than the average mileage for model A trucks. C) We are 98% confident that a randomly selected model A truck will get mileage that is between 2.5 and 4.7 miles per gallon higher than a randomly selected model B truck. D) Based on this sample, we are 98% confident that the average mileage for model A trucks is between 2.5 and 4.7 miles per gallon higher than the average mileage for model B trucks. E) We know that 98% of all random samples done on the population of trucks will show that the average mileage for model A trucks is between 2.5 and 4.7 miles per gallon higher than the average mileage for model B trucks. 8) Two types of flares are tested for their burning times (in minutes) and sample results are given below. n x s Brand X 35 19.4 1.4 Brand Y 40 15.1 0.8 Refer to the sample data to test the claim that the two populations have unequal means. Use a 5% significance level. 9) A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet. Treatment Group n1 = 85 x1 = 189.1 s1 = 38.7 Control Group n2 = 75 x2 = 203.7 s2 = 39.2 Use the sample data to test the claim that the treatment population mean 1 is smaller than the control population mean 2. Test the claim using a significance level of 0.01. 10) A researcher was interested in comparing the amount of time spent watching television by women and by men. Independent random samples of 14 women and 17 men were selected and each person was asked how many hours he or she had watched television during the previous week. The summary statistics are as follows. Do the data provide sufficient evidence to conclude that the mean time for women is less than the mean time for men? Perform a t-test at the 5% significance level. 11) A study was made to determine which cab company gave quicker service. Companies A and B were each called at 50 randomly selected times. The response times were recorded. The results are as follows. Company A Company B Mean response time 7.6 minutes 6.9 minutes Standard deviation 1.4 minutes 1.7 minutes At the 0.02 level of significance, test the claim that the two companies have different mean response times. Answers and Solutions to Part 1 1) (3.75, 4.85) SE= 0.268 Df= 34 t*= 2.032 2) ( -1.03, -0.57) SE= 0.136 Df= 99 t*= 1.660 3) ( -5.03, -3.37) SE= 0.413 Df= 49 t*= 2.009 4) (-$433.1, $209.7) n1 n2 8 15 mean1 mean2 705.4 817.1 s1 s2 183.9 330.1 SE df t* 107.2 7 2.998 5) D 6) B 7) D 8) H0: 1 - 2 = 0 HA: 1 - 2 0 Test statistic t = 16.03, DF=34 Critical value t(34;0.025)= 2.032 P-value 0 Reject the null hypothesis. There is strong evidence that the two types of flares have different mean burn times. Based on these data, it appears that Brand X flares have longer burning times than Brand Y flares. 9) H0: 1- 2 = 0 HA: 1 - 2 < 0 Test statistic t = -2.365, DF = 74 Critical value t* 2.378 0.01 alpha Do not reject the null hypothesis. There is not sufficient evidence to support the claim that the treatment population mean 1 is smaller than the control population mean 2. 10) H0: 1- 2 = 0 HA: 1 - 2 < 0 Test statistic: t = -0.428 DF = 13 Critical value t* = -1.771 P-value > 0.10 Do not reject H0. At the 5% significance level, the data do not provide sufficient evidence to conclude that the mean time for women is less than the mean time for men. 11) H0: 1 - 2 = 0 HA: 1 - 2 0 Test statistic t = 2.248, DF =49 Critical value t 2.403 0.2