ST107 Exercise 8 In this exercise you will practise interval estimation by constructing confidence intervals and considering sample size determination. Question 1 is a confidence interval for a single mean with known, although since we sample more than 10% of the population size we should apply the finite population correction factor (see the hint). Question 2 concerns a confidence interval for the difference between two population proportions. Finally, Question 3 requires the calculation of a confidence interval for the difference between two population means (with variances unknown and equal). 1. A factory has 2,000 workers. A simple random sample of 250 of these had weekly salaries with a sample mean of 500 and a population standard deviation of 25. (a) Calculate a 98% confidence interval for the mean weekly salary of all workers in the factory. Hint: Use the finite population correction factor, which means the relevant standard error is: r n\u0011 2 \u0010 1 . n N (b) How many more workers should be sampled if it is required that the estimate must be within 2 of the true average (again, with 98% confidence)? Hint: This means a tolerance of 2 - equivalent to a confidence interval width of 4. 2. A wholesaler of electronic goods conducts a survey of 'old' and of 'young' age groups among the public, with the hope of obtaining insight into current and prospective sales prospects for two different brands of media players (namely the jPod and the zPod brands). (a) In an initial pilot study, 77 out of 100 old people preferred the jPod brand, while 36 out of 50 young people preferred the jPod brand. Compute a 99% confidence interval for the difference between the true proportions for the two groups in favour of the jPod brand. (b) For the full study, it is decided to take new samples of both age groups so that the new sample sizes are the same for both age groups, and also so that the final estimator of the difference in proportions is (with 99% confidence) within 0.03 of the true difference. How many more people, i.e. in addition to the sizes of the two pilot study groups, would be needed for each of the new samples of old respondents and young respondents? 1 3. A survey is conducted on time spent recording (in hours per track) for 26 music industry recording studios, classified as successful or unsuccessful according to their recent chart performances. The relevant statistics resulting from this study are: Successful studios Unsuccessful studios Sample size 12 14 Sample mean 7.1 5.6 Sample standard deviation 1.5 1.2 (a) Compute a 95% confidence interval for the difference between population mean recording times between successful and unsuccessful studios. (b) On the basis of this confidence interval, do you consider this to be sufficient evidence of a true difference in mean recording times between the different types of studios? Justify your answer. 2