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 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)
b)Which statistical distribution should be applied and why?
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)
I need a complete answer 😁
a) To determine if there is sufficient evidence to conclude that the actual mean processing time differs from the past, we can perform a hypothesis test.
Null Hypothesis (H0): The actual mean processing time is equal to the past mean processing time.
Alternative Hypothesis (Ha): The actual mean processing time is different from the past mean processing time.
We will use a two-tailed t-test at a 5% level of significance to compare the sample mean processing time to the past mean processing time.
To calculate the test statistic, we need to first find the sample mean, sample standard deviation, and the degrees of freedom. The sample mean is simply the average of the recorded processing times.
Sample mean (x̄) = (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 + 17) / 27 = 44.15
Next, we calculate the sample standard deviation (s) using the formula:
s = sqrt((∑(x - x̄)^2) / (n - 1))
where x is each individual processing time.
Using this formula, we find the value of s to be approximately 27.14.
The degrees of freedom (df) is equal to the sample size minus 1, which in this case is 26.
Now, we can calculate the t-value using the formula:
t = (x̄ - μ) / (s / sqrt(n))
where μ is the past mean processing time and n is the sample size.
Assuming the past mean processing time is 45, we find the t-value to be approximately -0.110.
Using a t-table or a statistical software, we can find the p-value associated with this t-value and degrees of freedom. The p-value represents the probability of obtaining a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
If the p-value is less than the significance level of 0.05, we can reject the null hypothesis in favor of the alternative hypothesis. Otherwise, we fail to reject the null hypothesis.
Rejection Region: The rejection region would be in both tails of the t-distribution curve, representing extreme values that are unlikely to occur by chance alone.
b) The statistical distribution that should be applied in this case is the t-distribution. We use the t-distribution because the sample size is relatively small (less than 30) and the population standard deviation is unknown.
c) In this problem, there are two types of errors that can occur.
Type I Error: This occurs when we reject the null hypothesis when it is actually true. In this case, it would mean concluding that the actual mean processing time differs from the past mean processing time when it actually doesn't.
Type II Error: This occurs when we fail to reject the null hypothesis when it is actually false. In this case, it would mean failing to conclude that the actual mean processing time differs from the past mean processing time when it actually does.
d) To estimate the actual mean processing time with a 95% confidence level, we can construct a confidence interval.
The formula for a confidence interval for the mean is:
Confidence Interval = x̄ ± t * (s / sqrt(n))
where x̄ is the sample mean, s is the sample standard deviation, n is the sample size, and t is the critical value from the t-distribution corresponding to the desired confidence level and degrees of freedom (26 in this case).
Using the formula with a 95% confidence level, we have:
Confidence Interval = 44.15 ± 2.056 * (27.14 / sqrt(27))
After computation, we find the confidence interval to be approximately (33.64, 54.66).
e) The 95% confidence interval (33.64, 54.66) means that we are 95% confident that the true population mean processing time falls within this range. This interval estimation suggests that the actual mean processing time is likely to be between 33.64 and 54.66 days.
f) Based on the estimation in part (d), we can interpret that there is not sufficient evidence to conclude that the actual mean processing time differs from the past mean processing time. This is because the confidence interval includes the past mean processing time of 45 days.