Small-Sample Inferences for the Difference Between Two Population Means: Independent Random Samples and
Small-Sample Inferences for the Difference Between Two Means: A Paired-Difference Test
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In a study on the effect of an oral rinse on plaque buildup on teeth, eighteen people whose teeth were thoroughly cleaned and polished were randomly assigned to two groups of nine subjects each. Both groups were assigned to use oral rinses (no brushing) for a 2-week period. Group 1 used a rinse that contained an antiplaque agent. Group 2, the control group, received a similar rinse except that the rinse contained no antiplaque agent. A measure of plaque buildup was recorded at 14 days with means and standard deviations for the two groups shown in the table. Control Group Antiplaque Group Sample Size 9 9 Mean 1.23 0.77 Standard Deviation 0.36 0.36 USE SALT (a) State the null and alternative hypotheses that should be used to test the effectiveness of the antiplaque oral rinse. Ho: (H1 H2) 0 Ho (12) 0 versus Ha: (#1 - 2) 0 Ho (H1 H2) = 0 versus Ha: (1 - 2) 0 (b) Do the data provide sufficient evidence to indicate that the oral antiplaque rinse is effective? Test using a = 0.05. State the test statistic. (Round your answer to three decimal places.) t= State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three decimal places.) t > t 0 State the test statistic. t = State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three places.) t> t 0 Ho (H1 H2) 0 versus Ha: (1 - H) 0 State the test statistic. (Round your answer to three decimal places.) t = State the p-value. p-value 0.200 State the conclusion. Ho is not rejected. There is sufficient evidence to conclude that there is a significant difference in the mean numbers of uncontaminated eggplants for the two disinfectants used. Ho is not rejected. There is insufficient evidence to conclude that there is a significant difference in the mean numbers of uncontaminated eggplants for the two disinfectants used. Ho is rejected. There is insufficient evidence to conclude that there is a significant difference in the mean numbers of uncontaminated eggplants for the two disinfectants used. Ho is rejected. There is sufficient evidence to conclude that there is a significant difference in the mean numbers of uncontaminated eggplants for the two disinfectants used. You may need to use the appropriate appendix table to answer this question. To compare the average swimming times for two swimmers, each swimmer was asked to swim freestyle for a distance of 100 yards at randomly selected times. The swimmers were thoroughly rested between laps and did not race against each other, so that each sample of times was an independent random sample. The times for each of 10 trials are shown for the two swimmers. Swimmer 1 Swimmer 2 59.63 59.75 59.80 59.42 59.49 59.44 59.31 59.64 59.64 59.73 59.77 59.51 59.51 59.64 59.65 59.82 60.00 59.69 59.87 59.50 USE SALT Suppose that swimmer 2 was last year's winner when the two swimmers raced. Does it appear that the average time for swimmer 2 is still faster than the average time for swimmer 1 in the 100-yard freestyle? (Use Swimmer 1 - Swimmer 2. Use a = 0.05.) State the null and alternative hypotheses. Ho (H1 H2) = 0 versus Ha: (#1 - H2) 0 Ho (H1 H2) 0 versus Ha: (#1 - 2) = 0 Ho (H1 H2) 0 Ho (H1 H2)=0 versus Ha: (1 - 2) > 0 Ho (H1 H2) = 0 versus Ha: (#1 - 2) 0.100 Interpret the results. Ho is rejected. There is sufficient evidence to conclude that the average time for swimmer 2 is still faster than that of swimmer 1. Ho is not rejected. There is sufficient evidence to conclude that the average time for swimmer 2 is still faster than that of swimmer 1. Ho is rejected. There is insufficient evidence to conclude that the average time for swimmer 2 is still faster than that of swimmer 1. Ho is not rejected. There is insufficient evidence to conclude that the average time for swimmer 2 is still faster than that of swimmer 1. You may need to use the appropriate appendix table to answer this question. How does Alex Smith, quarterback for the Kansas City Chiefs, compare to Joe Flacco, quarterback for the Baltimore Ravens? The following table shows the number of completed passes for each athlete during the 2017 NFL football season. 25 25 27 Joe Flacco 22 19 + Alex Smith 27 29 23 25 27 29 34 31 20 14 16 26 10 8 19 25 21 27 25 223 23 19 19 28 23 24 24 9 20 2-SampTTest H1 H2 t=0.3250234901 p 0.7474965338 df 29 x1 22.73333333 x2=22 Sx1 4.463609473 Sx2 7.589466384 (a) The TI-84 Plus analysis uses the pooled estimate of 2. Is the assumption of equal variances reasonable? Why or why not? Yes, because the ratio of the larger variance to smaller variance is less than 3. Yes, because the ratio of the larger variance to smaller variance is more than 3. No, because the ratio of the larger variance to smaller variance is not equal to 1. No, because the ratio of the larger variance to smaller variance is less than 3. No, because the ratio of the larger variance to smaller variance is more than 3. (b) Do the data indicate that there is a difference in the average number of completed passes for the two quarterbacks? Test using a = 0.05. (Use for the population mean for Alex Smith and 2 for the population mean for Joe Flacco.) State the null and alternative hypotheses. Ho: (H1 H2) 0 versus Ha: (#1 - 2) = 0 Ho (H1 H2) 0 versus Ha: (#1 - 2) 0 Ho (H1 H2) 0. Ho (H1 H2) = 0 versus Ha: (#1 - 2) 0 State the test statistic. (Round your answer to three decimal places.) t = State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three decimal places.) t> t 0.200 Which of the following could be appropriate conclusions based on your p-value? (Select all that apply.). At the 1% level of significance, Ho is not rejected. At the 1% level of significance, Ho is rejected. At the 5% level of significance, Ho is not rejected. At the 5% level of significance, Ho is rejected. At the 10% level of significance, Ho is not rejected. At the 10% level of significance, Ho is rejected. You may need to use the appropriate appendix table to answer this question. Calculate the observed value of the t statistic for testing the difference between the two population means using paired data. (Round your answer to three decimal places.) d=5.9, sd 253, n = 19, Ha Hd t = USE SALT Approximate the p-value for the test. p-value 0. Ho: d0 versus Ha: d = 0 Ho Hd 0 versus Ha: d 0 Ho Hd 0 versus Ha: Hd #0 State the test statistic. (Round your answer to three decimal places.) t = State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three decimal places.) t> t 0 Ho: d0 versus Ha: "d=0 Ho d = 0 versus Ha: "d0 Ho Hd 0 versus Ha: "d> 0 State the test statistic. (Round your answer to three decimal places.) t = State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three decimal places.) t: > t 0.200 (c) Construct a 99% confidence interval for the difference in the average prices for the two supermarket chains. (Use advertiser "competitor. Round your answers to two decimal places.) $ to $ Interpret this interval. Since 0 does not fall in the confidence interval, there is sufficient evidence to indicate that the means are different. Since 0 falls in the confidence interval, there is sufficient evidence to indicate that the means are different. Since 0 does not fall in the confidence interval, there is insufficient evidence to indicate that the means are different. Since 0 falls in the confidence interval, there is insufficient evidence to indicate that the means are different. You may need to use the appropriate appendix table or technology to answer this question. In an experiment to study an oral rinse designed to prevent plaque buildup, subjects were divided into two groups-one group used rinse with an antiplaque ingredient, and the control group used a rinse containing inactive ingredients. Suppose that the plaque growth on each person's teeth was measured after using the rinse after 4 hours and then again after 8 hours. If you wish to estimate the difference in plaque growth from 4 to 8 hours, should you use a confidence interval based on a paired or an unpaired analysis? Explain. If two measurements are taken on the same person, then the measurements --Select--independent, thus you should use --Select--analysis
