1. Two contractors, Bill and Ben have put in the same bid for a landscaping job. You want to decide between them based on their past performance at completing jobs. The following tables are probability distributions showing the amount of time it takes Bill & Ben to finish similar jobs.

(a) How long on average would it take (i) Bill, (ii) Ben to finish the job?

(b) What is the standard deviation of the time it would take each of Bill and Ben to finish the job? Interpret the results.

(c) Who would you choose to do the job?

2. The following dataset contains ages of 10 randomly selected students in a school. Complete (a) to (e) below. (6 marks)
9, 11, 12, 13, 13 , 15, 16, 21, 28, 42

(a) Compute the mean and standard deviation.

(b) Find the median, and compare the mean and median to determine the skewness of the dataset.

(c) Calculate the coefficient of variation.

(d) Locate the first quartile, third quartile

(e) Locate the 80th percentile.

(e) Find out the range and inter-quartile range for the dataset. Suppose the last
observation (42) was incorrectly entered. If the correct age was 80, how would the correction affect the range and the inter-quartile range?

3. Using the Z-table and inverse Z-table, find the following. Illustrate your solutions with appropriate diagrams (i.e. appropriately shaded Z-curves).

(a) P(0 < Z < 2.21)

(b) P(Z < -3) (c) P(Z > -0.5)

(d) P(Z < -2 or > 2) (e) P(Z >0.6 and <2.5)

(f) The Z value such that its left tail is 0.20.

(g) The symmetric Z values which contain 95% of all observations.

4. The clean air legislation in Australia requires that vehicle exhaust emissions do not exceed certain limits for various types of pollutants. In Queensland, the Department of Environment has recently received complaints that some car repair workshops certified vehicles that did not meet the emission standard. The regulator decides to check a random sample of cars that a suspect repair shop has certified as okay. They will revoke the shop's license if they find significant evidence that the shop is issuing certificates for cars that do not meet the standard.

(a) In this context, what is a type I error?

(b) In this context, what is a type II error?

(c) Which type of error would the shop owner consider more serious?

(d) Which type of error might the environmentalists consider more serious?

5. Why does σ pose a problem for sample size calculation for a mean? How can σ be approximated when it is unknown?

6. Why is it better to say "fail to reject H0" instead of "accept H0"?