Question 1(a) Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the...

Question 1(a)
Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean is computed as 6.5 hours.
(i) Determine the 99% confidence interval estimate of the population mean. (2 marks)
(ii) Determine the 95% confidence interval estimate of the population mean, changing the sample size to 300. (2 marks)
(iii) Comment on how the confidence interval and changes to sample size will impact on the interval estimates. (3 marks)
Question 1(b)
The amount of time spent by Australian adults playing sports per day is normally distributed, with a mean of 4 hours and standard deviation of 1.25 hours. Use this information to answer the following question(s).
(i) Find the probability that a randomly selected Australian adult plays sport for more than 5 hours per day. ( 1mark)
(ii) Find the probability that if four Australian adults are randomly selected, their average number of hours spent playing sport is more than 5 hours per day. ( 1mark)
(iii) Find the probability that if four Australian adults are randomly selected, all four play sport for more than 5 hours per day (1 mark)
Question 2(a)
At present, many universities in Australia are adopting the practice of having lecture recordings automatically available to students. A university lecturer is trying to investigate whether having lecture recordings available to students has significantly decreased the proportion of students passing her course. When lecture recordings were not provided to students, the proportion of students that passed her course was 80%. The lecturer takes a random sample of 25 students, when lecture recordings are offered to students, and finds that 11 students have passed the course. Is there significant evidence to support this university lecturer’s claim? Use α = 0.01
(5 marks)
Question 2(b)
A drug company is interested in the effectiveness of a new sleeping pill. A random sample of 50 people try the new sleeping pill and the number of additional hours of sleep (compared with the nights without any sleeping pill), X, are recorded. The sample mean of additional hours of sleep is 2.2 hours and the sample standard deviation of X is 3 hours.
Test the claim that the new drug increases the number of hours of sleep at least by 2 hours on average at the 5% level of significance.