Question

I need someone to answer this question for me please its for Statistical Modeling and Data Analysis. Thanks.

1.      (65 points) Waiting at an ER.

A hospital administrator hoping to improve wait times decides to estimate the average emergency room waiting time at her hospital. She collects a simple random sample of 64 patients and determines the time (in minutes) between when they checked in to the ER until they were first seen by a doctor. She repeats this 1000 times, and build a distribution of sample means (called sampling distribution) of time. Then they calculate the sample average wait time is 137.5 minutes with the standard deviation 39 minutes.

(a)   Would you expect the sampling distribution follows the normal distribution? Explaining your reasoning. (10 points)

(b)   Calculate the variability of the sampling distribution and state the appropriate term used to refer to this value. (5 points)

Hint: The variability of the sampling distribution is measured by the sampling error of mean. Sampling error is estimated by , where s is the standard deviation of sample.

(c)   Find the 95% level confidence interval. Interpret this interval in context of the data. (10 points)


(d)   Suppose this administrator think 90% confidence level would be more appropriate for this interval. Will this new interval be smaller or larger than the 95% confidence interval? (5 points)

(e)   A local newspaper claims that the average waiting time at this ER is 2.15 hours. Is this claim supported by the 95% confidence interval? Explain your reasoning. (10 points)

(f)    Using a significant level of 0.05, does the sample average wait time provide strong an evidence that the wait time is different from 2.15 hours? (10 points)

(g)   Would the conclusion of the hypothesis test change if the significance level was changed to 0.1? (10 points)

(h)   The higher significant level is; the higher probability is to reject the null hypothesis when it is true. Determine if the statement is true or false. (5 points)

Answer 1

a) Yes sampling distribution follows normal distribution because as long as the data isn't too heavily skewed, we have more than 30 observations, and the sample is independent, the point estimates of any distribution will be normally distributed according to the central limit theorem

b) Standard error= sample standard deviation/sqrt(n)

= 39/sqrt(64)= 39/8= 4.875

c) 95% confidence interval is

is 95% confidence interval.

Interpretation: We are 95% confident that the average waiting time of all patients at this hospital's ER is between 128 and 147 minutes.

d) M = 137.5
Z critical = 1.64
sM = √(39^2/64) = 4.88

μ = M ± Z(sM)
μ = 137.5 ± 1.64*4.88
μ = 137.5 ± 8.019

90% CI [129.481, 145.519].

You can be 90% confident that the population mean (μ) falls between 129 and 146.

this new interval be smaller than the 95% confidence interval.

e) Since 95% confidence interval contain 2.15 hours=129minutes. Therefore this claim supported by the 95 % confidence interval.

f) NULL HYPOTHESIS H0:

ALTERNATIVE HYPOTHESIS Ha:

Z=

Z=

Z criticalvalue for two tailed test is 1.96

Conclusion: Since critical value ofZ is greater than calculated value of Z we therefore fail toreject null hypothesis at 0.05 level of significance.

g) For alpha=0.1 the critical Z value for two tailed test will be 1.64 .

In this case the critical value of Z is smaller than calculated value of Z hence we reject NULL HYPOTHESIS H0 at 0.1 level of significance.

h) The higher significant level is; the higher probability is to reject the null hypothesis when it is true. This statement is TRUE.