Question

Aetna, a major health insurance company, would like to estimate average wait time for a patient seeking emergency room services. A random sample of 25 emergency room patients had an average wait time of 222 minutes with a sample standard deviation of 76 minutes. (16 points) a. What is the best point estimate for the population mean? b. What is the 99% confidence interval for the population mean? c. Find a 99% confidence interval for the population mean in the above example when all things remain same except that the sample size has changed to 16. d. Discuss what happens to the width of the confidence interval as the sample size declines and explain why?

Answer 1

a)

Sample mean is best point estimate for the popuation mean.

So,

Best point estimate for the population mean = = 222

b)

df = n-1 = 25 - 1 = 24

t critical value at 0.01 significance level for 24 df = 2.797

99% confidence interval for is

- t * S / sqrt(n) < < + t * S / sqrt(n)

222 - 2.797 * 76 / sqrt(25) < < 222 + 2.797 * 76 / sqrt(25)

179.4856 < < 264.4992

99% CI is ( 179.4856 , 264.4992)

c)

When, n = 16

df = n-1 = 16-1 = 15

t critical value at 0.01 significance level with 15 df = 2.947

99% confidence interval for is

- t * S / sqrt(n) < < + t * S / sqrt(n)

222 - 2.947 * 76 / sqrt(16) < < 222 + 2.947 * 76 / sqrt(16)

166.007 < < 277.993

99% CI is ( 166.007 , 277.993 )

d)

Width of confidence interval for size n = 25 is

264.4992 - 179.4856 = 85.0135

Width of confidence interval for size n = 16 is

277.993 - 166.007 = 111.986

Therefore,

Width of confidence interval increase as sample size decreases.

If sample size reduced, margin of error quantity ( t * S / sqrt(n) ) decrease so width of confidence

interval increases.