Some latest study shows that college students are increasingly skimping on their sleep. A researcher from a small community college wants to estimate the mean weekdays sleep time of its students. He takes a random sample of 84 students and records their weekday mean sleep time as 6.70 hours. Assume that the population standard deviation is fairly stable at 1.40 hours. (a) Construct a 95% confidence interval for the population mean weekday sleep time of all students of this college. Use the following table of z; for most common a values. a 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.1 za 2.575 2.33 2.05 1.88 1.555 1.28 Z; 2.81 2.575 2.33 2.17 1.88 1.645 2 Round the margin of error and the confidence interval endpoints to 2 decimal places. 2 Round the margin of error and the confidence interval endpoints to 2 decimal places. Calculate the margin of error E = C] What is the 95% confidence interval? (b) If the researcher wants to be 98% confident that the sample mean is within 0.20 hours (or 12 min) of the true population mean weekday sleep time, how many students should he include in the sample? 2 Round your answer (:1) UP to the next whole number. The required sample size n isl Helmet Telecom Ltd. claims that, on average, their new design for a wireless router has increased the speed to at least 1.80 Gbps, a big improvement over 1.50 Gbps maximum of the previous version. An independent testing laboratory in Etobicoke found that a random sample of 20 of the newest version routers delivered speed of 1.72 Gbps, on average, with a standard deviation of 0.11 Gbps. Test the manufacturer's claim using a 0.005 significance level. Assume that the speeds are approximately normally distributed as the histogram of the sample values is roughly symmetric and bell-shaped, so t- distribution is applicable for hypothesis testing. Round to 3 decimal places where appropriate. (a) State the null and alternative hypotheses; identify which hypothesis is the claim and what is the type of the test. Ho_-l l H1_-l l Which one is the claim? 0 H0 0 H1 The test is _ For parts (b), (c) use the correct sign for the critical t-value and test statistic. (b) What is the critical t-value? l l (c) What is the test statistic?' l (d) Is the null hypothesis rejected? Is the alternative hypothesis supported? 0 Fail to reject H0 (claim) and fail to support H1 0 Fail to reject H0 and support H1 (claim) 0 Reject H0 (claim) and support H1 0 Reject H0 and fail to support H1 (claim) (e) Select the correct statement. 0 At 0.005 significance level, there is sufficient sample evidence to warrant rejection of the claim that, on average, the new wireless router has increased the speed to at least 1.80 Gbps. OWe are sure that, on average, the speed is at most 1.50 Gbps. 0 We prove that the average speed is still 1.50 Gbps. 0 At 0.005 significance level, there is not sufficient evidence to warrant rejection of the claim that, on average, the new wireless router has increased the speed to at least 1.80 Gbps. 0 None of the above