compute the probability that the total premiums collected will be altered by more than $500 from the original value

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Question 1 When collecting the premiums, Insurance rounds 60,000 motor insurance premiums to the nearest ten dollars.

(a) Stating any assumptions made, compute the probability that the total premiums collected will be altered by more than $500 from the original value (before rounding). Show all details of your working. (15 marks)

(b) Comment on whether Apex Insurance should be worried that the amount altered with this rounding exceeds $2000? Explain your answer briefly. (5 marks)

Question 2 Three trainee salespersons were being monitored for their level of sales. Salesperson A made a total of 440 sales in 5 days, salesperson B made a total of 630 sales in 7 days, and salesperson C made a total of 690 sales in 10 days. The sum of squares of all 22 daily sales is calculated as: ∑𝑥𝑖 2 = 146,840.

(a) Calculate the various entries in the one-way ANOVA table (p-values may be excluded). Show all details in your workings. (14 marks)

(b) By performing the ANOVA test, comment on whether there is a significant difference among the average daily sales of the 3 salespersons? Justify your answer at a 5% level of significance. (6 marks)

Question 3 The following sample is collected: 1.6, 1.7, 1.7, 1.8, 1.5, 1.9. (a) Apply an appropriate hypothesis test to check if the sample is likely to be from a population with mean 1.5. Show full details of the test conducted. (16 marks)

(b) State the assumptions made in performing the test in Question 3(a). (4 marks)

Question 4 The I.Q.’s (intelligence quotients) of 16 students from the rural regions of a small country showed a mean of 107, with standard deviation of 10, while the I.Q.’s of 14 students from the urban regions showed a mean of 112 with a standard deviation of 8.

(a) Apply an appropriate hypothesis test (using critical region) to check if there is evidence at 5% level of a significant difference between the I.Q.’s of students from the two types if regions. Show full details of the test carried out. (14 marks)

(b) Describe and comment on the changes to the test carried out in

Question 4(a) if

(i) the test is now conducted at a 1% level of significance,

(ii) the interest is now to check if the I.Q.’s of students from the urban regions are higher in general compared to those from the rural regions.

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