n 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)
You'll need to calculate the mean and standard deviation.
Once you have those values, try a t-test formula since your sample size is small.
Formula:
t = (sample mean - population mean)/(standard deviation divided by the square root of the sample size)
Hypotheses:
Ho: µ = 45 -->null hypothesis
Ha: µ does not equal 45 -->alternate hypothesis
The alternate hypothesis is a two-tailed test because part a) asks if the actual mean differs from the past.
Calculate t-test statistic. To find the P-value, use a t-test table. The P-value is the actual level of the test statistic.
You can draw your conclusions once you determine the P-value. If Ha is accepted (null rejected), there will be sufficient evidence to conclude a difference. If the null is not rejected, you cannot conclude a difference.
I hope this will get you started.
To answer these questions, we need to perform a hypothesis test and calculate a confidence interval.
a) Hypothesis Test:
Null Hypothesis (H0): The actual mean processing time is equal to the past mean processing time (μ = 45)
Alternative Hypothesis (Ha): The actual mean processing time differs from the past mean processing time (μ ≠ 45)
To determine if there is sufficient evidence to support the alternative hypothesis, we can calculate the p-value. The p-value is the probability of observing a sample mean as extreme as the one calculated or more extreme, assuming the null hypothesis is true.
1. Calculate the sample mean (x̄) and sample size (n) from the provided data.
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 = 45.926
2. Calculate the sample standard deviation (s) from the provided data.
s = √[(∑(xi - x̄)²) / (n-1)] = 24.200
3. Calculate the test statistic (t) using the formula:
t = (x̄ - μ) / (s / √n)
t = (45.926 - 45) / (24.200 / √27) = 0.052
4. Determine the degrees of freedom (df) using n-1.
df = 27 - 1 = 26
5. Look up the critical t-value from the t-distribution table using a 5% level of significance (two-tailed test) and degrees of freedom 26.
The critical t-value is approximately ±2.056.
6. Calculate the p-value using the t-distribution.
P-value = probability(|t| > |observed t|)
P-value = 2 * (1 - tcdf(|t|, df))
P-value = 2 * (1 - tcdf(0.052, 26))
By comparing the calculated t-value (0.052) with the critical t-value (±2.056), we can determine if there is sufficient evidence to reject the null hypothesis.
b) Statistical Distribution:
The appropriate statistical distribution to use in this case is the t-distribution because the sample size is relatively small (n<30) and the population standard deviation is unknown.
c) Type of Error:
In hypothesis testing, there are two types of errors: Type I Error (rejecting a true null hypothesis) and Type II Error (failing to reject a false null hypothesis). In this context, a Type I Error would occur if we conclude that the actual mean processing time differs from the past when it actually does not.
d) Confidence Interval:
To estimate the actual mean processing time with a 95% confidence level, we can construct a confidence interval using the t-distribution.
The formula for a confidence interval is:
Confidence Interval = x̄ ± (t * (s / √n))
x̄ = 45.926 (sample mean)
t = critical t-value at a 95% confidence level (from the t-distribution, with 26 degrees of freedom)
s = 24.200 (sample standard deviation)
n = 27 (sample size)
Calculate the confidence interval using the formula.
e) Interpretation of the Confidence Interval:
The confidence interval represents the range of values within which we are fairly confident (with a 95% confidence level) that the true population mean processing time lies. Based on the calculated confidence interval, we can estimate the true mean processing time to be within a certain range.
f) Decision based on Estimation:
If the confidence interval includes the past mean processing time (45), we can conclude there is not sufficient evidence to conclude that the actual mean processing time differs from the past. If the confidence interval does not include the past mean processing time, we can conclude there is sufficient evidence to support that the actual mean processing time differs from the past.