MATH 3070: Worksheet No.5
NAME:
1.The caffeine content (in milligrams, mg) was examined for a random sample of 100 cups of black coffee. The mean and the standard deviation were 110mg and 7.1mg, respectively. Find the 95% confidence interval for the mean caffeine content for a single cup. How about 98% confidence interval?
2.The gross profit margin of small businesses is estimated in a city. A random sample of 10 small businesses shows that the mean gross profit margin and their standard deviation are 5.2% and 7.5%, respectively.
(a)What are the limitations in using the confidence interval when the sample size is small?
(b)State the necessary assumptions, and construct 95% confidence interval for the mean gross profit margin of all small businesses in the city.
3.A courier company claims that its mean delivery time to any place in a city is less than 3 hours. 50 deliveries were randomly selected, and the mean delivery time and standard deviation are computed as 2.8 hours and 0.6 hours, respectively.
(a)Construct a 95% confidence interval for the mean delivery time. Does the company's claim appear reasonable?
(b)If a 99% confidence interval is used, would you change your answer in (b)?
4. A study was conducted of 90 adult male patients following a new treatment for congestive heart failure, and the increase in exercise capacity (in minutes) over a four-week treatment period. The data yields the mean increase 2.17 minutes, and the sample standard deviation 1.05 minutes. Researchers wants to evaluate whether the new treatment had improved the exercise capacity in comparison to the standard treatment which has produced an average increase of 2 minutes.
(a)Construct the null and alternative hypothesis for the test.
(b)Calculate the test statistic. Using the significance level 0.05, find the critical region for the test.
(c)What conclusions can you draw from this study?
(d)Describe what is the Type II error in the context of this study.
(e)What is the probability of Type II error if the actual value of mean increase for the new treatment is 2.1?
5.To evaluate the success of a one-year remedial math course, the mathematics scores of the students who complete this course will be compared with the statewide average of 525 points. Researchers know from the previous study that the standard deviation is approximately 80, and they have a reason to believe that the actual mean is 550. If the significance level is chosen to be 0.05, what sample size is needed to have a probability of Type II error at most 0.025?
6.To study the effectiveness of a weight-reducing agent, a clinical trial was conducted. 35 overweight males were placed on a fixed diet for two weeks, and each was weighed at the end of this period. Then for the next two weeks, each is given a supply of the weight-reducing agent in addition to the fixed diet. At the end of the second period weights were obtained again. Researchers want to test at the significance level 0.05 that the mean weight reduction using this agent is greater 5 pounds. Provided that the mean weight reduction is 8 pounds, the test is required to have the power at least 90%. From the previous study the researchers know that the standard deviation is approximately 6.8 pounds. Is a sample size of 35 large enough to meet the researchers' requirements for the test? If not, what sample size is needed?