Question

In New York State, savings banks are permitted to sell a form of life insurance (SBLI). The approval process consists of underwriting, which includes a review of the application, a medical information bureau check, possible requests for additional medical information and medical exams, and a policy compilation stage in which the policy pages are generated and sent to the bank for delivery. The ability to deliver approved policies to customers in a timely manner is critical to the profitability of this service. In the past, the actual mean processing time was 45 days. During a period of one month, a sample of approved policies is selected, and the total processing time, in days, is recorded:

73, 19, 16, 64, 28, 28, 31, 90, 60, 56, 31, 56, 22, 18, 45, 48, 17, 17, 17, 91, 92, 63, 50, 51, 69, 16, and 17.

a). At 5% level of significance, is there sufficient evidence to conclude that the actual mean processing time differs from the past? Your conclusion must be in terms of the P-Value as well as setting up a Rejection Region. You must show all necessary work.

b). Which statistical distribution should be applied in this situation and why? Explain carefully.
c). What type of error is possible and describe this error in terms of the problem.
d). Based on a 95% confidence level, estimate the actual mean processing time.

e). Carefully interpret the interval estimation.
f). Explain carefully whether or not there is sufficient evidence to conclude that the actual mean processing time differs from the past using the estimation in part (d).
g). Do you expect the conclusions of parts (a) and (d) to be the same? Should they be the same? Explain carefully.

Answer 1

a)
H0: mu = 45
Ha: mu not equals to 45

from the given data we have
xbar = 43.8889
s = 25.2835
n = 27

Test statistic,
t = (43.8889 - 45)/(25.2835/sqrt(27))
t = -0.2283

p-value = 0.8212

Rejection region: -2.0555 and 2.0555

Fail to reject the H0

b)
t - distribution is used as population std. dev. is unknown and sample size is less than 30

c)
type II error is possible

d)
t-value = 2.0555
CI = (43.8889 - 2.0555*25.2835/sqrt(27), 43.8889 + 2.0555*25.2835/sqrt(27))
=(33.8872, 53.8906)

As 45 does lie in the above CI, we fail to reject H0.
there are not sufficient evidence

conclusion in a and d are same