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1.A 95% confidence interval for...

1.A 95% confidence interval for the mean income of shop assistants in a certain city is found to be (£12,000, £15,000). Explain what this means. Would a 99% confidence interval be better than a 95% confidence interval? Justify your answer.

2. A charity believes that when it puts out an appeal for charitable donations the donations it receives will be normally distributed with a mean of £50 and a standard deviation of £6.

(a) Find the probability that the first donation it receives will be less than £40. (b) Find the value x such that 5% of donations are more than £x.

3. A consultant for Dell was investigating computer usage among students at a particular university. 200 undergraduates and 100 postgraduates were chosen and asked if they owned a laptop. It was found that 81 of the undergraduates and 63 of the postgraduates owned a laptop. The consultant calculated that 48% (144 out of 300) of the students interviewed owned a laptop. Explain, with reasons, whether the figure of 48% will be a good estimate of the proportion of all students who own a laptop.

4. For a certain variable, the standard deviation in a large population is equal to 15. How big a sample is needed to be 95% sure that the sample mean is within 2.25 units of the population mean?

5. (a) What conclusions would you draw from a test which is significant at the 1% significance level?

(b) What conclusions would you draw from a test which is significant at the 10% significance level, but not the 5% significance level?

(c) An accounting firm wishes to test the claim that no more than 6% of a large number of transactions contains errors. In order to test this claim, they examine a random sample of 225 transactions and find that exactly 20 of these are in error. What conclusion should the firm draw? Use a 5% significance level.

2

6. A profit-maximising retailer can obtain cameras from the manufacturer at a cost of £50 per camera. The retailer has been selling the cameras at a price of £80, and at this price consumers have been buying 40 cameras per month. The retailer is planning to lower the price to stimulate sales and knows that for each £5 reduction in the price, 10 more cameras will be sold each month. Assuming price is a multiple of £5, what price should the retailer charge and what will the monthly profits be?

7. Explain the purpose of:

(a) modelling (b) sampling.

Discuss any advantages and limitations.

8. A lumber company makes plywood for the furniture industry. One product it makes is 3/4-inch panels. It is very important that the panels conform to specifications. One specification calls for the panels to be made to an average thickness of 0.75 inches. Each hour, 5 panels are selected at random and measured. After 20 hours, a total of 100 panels have been measured. The data are in the file ‘Lumber.xls’.

(a) Formulate the appropriate null and alternative hypotheses relative to the thickness specification.

(b) Based on the sample data, what should the company conclude about the status of its product meeting the thickness specification? Test at a 1% significance level. Discuss your results in a brief ‘executive summary’ to the production manager.

(c) The production manager has looked at the results of your test. He wishes to repeat this sampling process and wants a rule to compare future X ̄ values, i.e. based just on a sample mean, and not conducting a formal hypothesis test, he wishes to decide whether the thickness specification is being met, again using a 1% significance level. Develop a suitable rule.

(d) The manager wishes to know what error in part (b) could have been made and what the potential consequences of the error might be. He also wishes to know how you could be certain (if possible) whether such an error could have been made. Provide this information to the manager.

2. A charity believes that when it puts out an appeal for charitable donations the donations it receives will be normally distributed with a mean of £50 and a standard deviation of £6.

(a) Find the probability that the first donation it receives will be less than £40. (b) Find the value x such that 5% of donations are more than £x.

3. A consultant for Dell was investigating computer usage among students at a particular university. 200 undergraduates and 100 postgraduates were chosen and asked if they owned a laptop. It was found that 81 of the undergraduates and 63 of the postgraduates owned a laptop. The consultant calculated that 48% (144 out of 300) of the students interviewed owned a laptop. Explain, with reasons, whether the figure of 48% will be a good estimate of the proportion of all students who own a laptop.

4. For a certain variable, the standard deviation in a large population is equal to 15. How big a sample is needed to be 95% sure that the sample mean is within 2.25 units of the population mean?

5. (a) What conclusions would you draw from a test which is significant at the 1% significance level?

(b) What conclusions would you draw from a test which is significant at the 10% significance level, but not the 5% significance level?

(c) An accounting firm wishes to test the claim that no more than 6% of a large number of transactions contains errors. In order to test this claim, they examine a random sample of 225 transactions and find that exactly 20 of these are in error. What conclusion should the firm draw? Use a 5% significance level.

2

6. A profit-maximising retailer can obtain cameras from the manufacturer at a cost of £50 per camera. The retailer has been selling the cameras at a price of £80, and at this price consumers have been buying 40 cameras per month. The retailer is planning to lower the price to stimulate sales and knows that for each £5 reduction in the price, 10 more cameras will be sold each month. Assuming price is a multiple of £5, what price should the retailer charge and what will the monthly profits be?

7. Explain the purpose of:

(a) modelling (b) sampling.

Discuss any advantages and limitations.

8. A lumber company makes plywood for the furniture industry. One product it makes is 3/4-inch panels. It is very important that the panels conform to specifications. One specification calls for the panels to be made to an average thickness of 0.75 inches. Each hour, 5 panels are selected at random and measured. After 20 hours, a total of 100 panels have been measured. The data are in the file ‘Lumber.xls’.

(a) Formulate the appropriate null and alternative hypotheses relative to the thickness specification.

(b) Based on the sample data, what should the company conclude about the status of its product meeting the thickness specification? Test at a 1% significance level. Discuss your results in a brief ‘executive summary’ to the production manager.

(c) The production manager has looked at the results of your test. He wishes to repeat this sampling process and wants a rule to compare future X ̄ values, i.e. based just on a sample mean, and not conducting a formal hypothesis test, he wishes to decide whether the thickness specification is being met, again using a 1% significance level. Develop a suitable rule.

(d) The manager wishes to know what error in part (b) could have been made and what the potential consequences of the error might be. He also wishes to know how you could be certain (if possible) whether such an error could have been made. Provide this information to the manager.

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