Problem 1.

A survey company asked 100 randomly selected people how much time they spend daily on watching TV. According to the survey response, the average time spent was 4.7 hours and the standard deviation was 1.5 hours. Based on these results, answer the following questions:

(a) Find the 95% confidence interval of the true average time (i.e., the population mean) people spend on watching TV every day.

(b) Suppose we want the length of the 95% confidence interval to be strictly less than 0.4. What is the minimum sample size needed?

Problem 2.

A company that produces soy wax aromatherapy candles claims that their candles last 30 hours on average. To verify their claim, a consumer organization randomly tested 75 candles manufactured by that company.

(a) Suppose the average life time of the tested candles was 30.41 with a standard deviation of 2.32 hours. At a confidence level of 95%, is there a reason to believe that their claim may not be true? Statistically justify your answer.

(b) Now, suppose the average was 29.32 hours with a standard deviation of 2.43 hours. Is there a reason to believe that the actual lifetime would be less than 30 hours at a 99% confidence level? Statistically justify your answer.

**Subject Mathematics Statistics-R Programming**