Title: Analysis of the Effectiveness of Titan Insurance Company's New Incentive Payment Scheme
The Titan Insurance Company has recently implemented a new incentive payment scheme for its life policy sales force. The company is interested in evaluating the success or failure of the scheme. Although there are indications that the sales force is selling more policies, the monthly sales vary unpredictably, making it unclear whether the scheme has made a significant difference.
Life insurance companies typically measure a salesperson's monthly output by calculating the total sum assured for the policies sold by that person during the month. For example, if salesperson X sold seven policies with sums assured of £1000, £2500, £3000, £5000, £10000, and £35000, X's output for the month would be £61,500.
Under Titan's new scheme, the sales force receives low regular salaries but is eligible for large bonuses based on their output (i.e., the total sum assured of policies sold by them). Although the scheme is expensive for the company, they expect the sales increases to compensate for the costs. The agreement with the sales force is that if the scheme does not break even for the company within six months, it will be abandoned.
The scheme has been in operation for four months and has stabilized after initial fluctuations during the first two months due to the changeover. To assess the effectiveness of the scheme, Titan has taken a random sample of 30 salespeople and measured their output in the penultimate month before the changeover and in the fourth month after the changeover (choosing months that are not too close to the changeover).
The outputs of the salespeople are presented in Table 1:
Table 1: Output (£000)
Salesperson Old Scheme New Scheme
1. 57 62
2. 103 122
3. 59 54
4. 75 82
5. 84 84
6. 73 86
7. 35 32
8. 110 104
9. 44 38
10. 82 107
11. 67 84
12. 64 85
13. 78 99
14. 53 39
15. 41 34
16. 39 58
17. 80 73
18. 87 53
19. 73 66
20. 65 78
21. 28 41
22. 62 71
23. 49 38
24. 84 95
25. 63 81
26. 77 58
27. 67 75
28. 101 94
29. 91 100
30. 50 68
a. Describe the five percent significance test you would apply to these data to determine whether the new scheme has significantly raised outputs.
b. What conclusion does the test lead to?
c. What reservations do you have about this result?
d. Suppose it has been calculated that in order for Titan to break even, the average output must increase by £5000. If this figure is the alternative hypothesis, what is:
(i) The probability of a type 1 error?
(ii) The probability of a type 2 error?
(iii) The power of the test?
e. Are Type 1 and Type 2 errors the same in this case? Should they be equal? Why or why not? If they are to be equated, suggest a way to do so. (Hint: would a change in sample size work?)