1. An engineering company is considering buying a smaller but older company who have an excellent reputation in the area. The sale will include a substantial amount of stock that the selling company have in storage. The purchasing company believe that much of this stock is obsolete but cannot conduct an audit. They obtain a sample of 380 items and find that 209 of the items are obsolete.

(a) Provide a point estimate of the proportion of selling company's stock which is obsolete.

(b) To give a measure of the uncertainty in this estimate, construct a 90% confidence interval for the true proportion of stock which is obsolete.

(c) Based on the confidence interval you constructed in part (b), are you convinced that the majority of the stock is obsolete? Explain.

(d) Suppose that only 120 items were sampled and that 66 were found to be obsolete. Construct a 90% confidence interval for the true proportion of stock which is obsolete. Does the interval provide convincing evidence that the majority of the stock is obsolete.

(e) Compare the sample proportions and the confidence intervals found in parts (b) and (d). Use these results to discuss the role sample size plays in helping make decisions from sample data.

(f) Assume that the obsolete proportion of the stock is the same is as in parts (b) and (d). What is the minimum number of items that
should be sampled in order to obtain a 95% confidence interval whose width is less than 0.1?

2. The research and development department of Bubble Man Tyres are developing a new tyre. The engineers believe that the mean distance, µ, travelled on its new tyre compound until the tyre is no longer roadworthy is 11,000km. In order to test this, a sample of 50 tyres are tested in a machine to see how many kilometers are travelled until the tread depth less than that required by law. The sample average distance is 10,680km and the sample standard deviation is 1,056km.

(a) Find the rejection region for the following test, using a significance level α = 0.04.

H0 : µ = 11, 000km vs. HA : µ 6= 11, 000km

Calculate the relevant test statistic and state your conclusion.

(b) Calculate a 96% confidence interval for µ.

(c) Compare the interval found in part (b) to the rejection region of part (a) and comment on the relationship between confidence intervals and hypothesis tests.

(d) Calculate the p-value for the test in part (a).

3. An engineer is deciding which type of support beam to use in the construction of a new office building. Support beams made from two different metal alloys, S1 and S2, are to be compared. Each of 40 beams made from S1 and 45 beams made from S2 are tested to see what their maximum load capacity is. The data are in the file PS2 Q4.xlsx which is available on blackboard. Use Minitab to answer the following questions. Include a printout of the session window with your answers.

(a) Provide a 90% confidence interval for the mean maximum load capacity, (µ1), of beams made from S1.

(b) Provide a 99% confidence interval for the mean maximum load capacity, (µ2), of beams made from S2.

(c) Provide a 95% confidence interval for the difference in the mean maximum load capacity, (µ1 - µ2), of beams made from S1 and S2. (Note: There is no option in Minitab to construct a confidence interval for the difference between two large sample means. Use Minitab to calculate the required summary statistics of each sample then construct the interval by hand.)

(d) The company who supply the beams say there is no difference in the mean maximum load capacity of the two types of beams but the engineer thinks that the mean maximum load capacity of beams made from S2 is greater than the mean maximum load capacity of beams made from S1. Test this hypothesis based on the data provided.

State the null and alternative hypotheses and your conclusions. Use α = 0.1. (Note: Use the summary statistics produced in part (c) to perform this test by hand.)