Question

1. The mean arrival rate of flights at O’Hara International Airport in marginal weather is 195 flights per hour with a population standard deviation of 13 flights per hour. To increase arrivals, a new air traffic control procedure is implemented. In the next 30 days of marginal weather, the mean number of arrivals increased to 200. Given this information, please answer the following questions:
A. Compute the standard error of the sampling distribution of mean arrival rates?
B. What assumption must we make about the parent population to conduct statistical inference?
C. Find the 90% confidence interval estimate for the true mean arrival rate at O’Hara Airport in marginal weather.
D. Use the p-value method or critical value method to test the following hypothesis:
H0: μ ≤ 195
H1: μ ˃ 195
E. Interpret your finding in part (D) as it relates to mean arrival rates.

2. Machelon Health Clinic claims that the true population mean wait time for patients is 20 minutes or less. A random sample of 15 patients revealed a mean wait time of 17.77 minutes with a sample standard deviation of 4 minutes. Given this information, please answer the following questions:
A. What assumption must be made about the parent population of patient wait times to perform statistical inference?
B. Find the 95% confidence interval.
C. Perform a hypothesis test to test Machelon’s claim that wait times are less than 20 minutes per patient using the p-value method. (If you cannot do the p-value method, use the critical value method and test at the 10% level of significance).
D. Interpret your findings from part (C).

3. A shareholder’s group claimed that the mean tenure for a chief executive offer (CEO) was exactly nine years. A survey of 185 companies (reported in the Wall Street Journal) found a sample mean tenure of 7.27 years with a sample standard deviation of 6.38 years.
A. Are any assumption necessary to perform statistical inference? If so, what assumption(s) are needed?
B. Compute the standard error.
C. Test the shareholder’s claim at the 5% level of significance using the p-value method. (If you cannot do the p-value method, use the critical value method)
D. Interpret your findings

4. State the Central Limit Theorem. Discuss why it is important as it relates to statistical inference.

Answer 1

Solution:-

1)

A)

SE = S.D / sqrt(n)

n = 30

S.E = 2.3735
B) The required assumptions are:-

The population is normally distributed.

The sample size is sufficiently large.

The sample is selected randomly.
C) 90% confidence interval estimate for the true mean arrival rate at O’Hara Airport in marginal weather is C.I = (191.10 , 198.90).

$C.I = \bar{x}\pm z_{\alpha /2}\ast \frac{\sigma }{\sqrt{n}}$

C.I = 195 + 1.645*2.3735

C.I = 195 + 3.904

C.I = (191.10 , 198.90)

D)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u < 195
Alternative hypothesis: u > 195

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the sample mean is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a one-sample z-test.

Analyze sample data. Using sample data, we compute the standard error (SE), z statistic test statistic (z).

SE = s / sqrt(n)

S.E = 2.3735
z = (x - u) / SE

z = 2.11

where s is the standard deviation of the sample, x is the sample mean, u is the hypothesized population mean, and n is the sample size.

The observed sample mean produced a t statistic test statistic of 2.11

Thus the P-value in this analysis is 0.017.

Interpret results. Since the P-value (0.017) is less than the significance level (0.05), we have to reject the null hypothesis.

E) From the above test we have sufficient evidence in the favor of the claim that the mean number of arrivals has increased.