The mean and standard deviation of a random sample of n measurements are equal t
ID: 3264235 • Letter: T
Question
The mean and standard deviation of a random sample of n measurements are equal to 34.5 and 3.9. respectively. a. Find a 99% confidence interval for mu if n = 49. b. Find a 99% confidence interval for mu if n = 196. c. Find the widths of the confidence intervals found in parts a and b. What is the effect on the width of a confidence interval of quadrupling the sample size while holding the confidence coefficient fixed? a. The 99% confidence interval for mu if n = 49 is approximately (, ). (Round to three decimal places as needed.) b. The 99% confidence interval mu if n = 196 is approximately (, ). (Round to three decimal places as needed.) c. Choose the correct answer below. A. Quadrupling the sample size while holding the confidence coefficient fixed decreases the width of the confidence interval by a factor of 4. B. Quadrupling the sample size while holding the confidence coefficient fixed does not affect the width of the confidence interval. C. Quadrupling the sample size while holding the confidence coefficient fixed increases the width of the confidence interval by a factor of 4. D. Quadrupling the sample size while holding the confidence coefficient fixed increases the width of the confidence interval by a factor of 2. E. Quadrupling the sample size while holding the confidence coefficient fixed decreases the width of the confidence interval by a factor of 2.Explanation / Answer
Solution:
Given xbar = 34.5, s = 3.9
a) Find a 99% confidence interval for µ if n = 49
Given a=0.01, |Z(0.005)|=2.58 (check standard normal table)
So 99% CI is
Confidence interval = xbar ± Z*s/n
=> 34.5 ± 2.58*3.9/7
=> ( 33.06257, 35.93743)
b) Find a 99% confidence interval for µ if n = 196
So 99% CI is = xbar ± Z*s/n
=> 34.5 ± 2.58*3.9/sqrt(196)
=> ( 33.78129, 35.21874)
c) Find the widths of the confidence intervals found in parts a and b. what is the effect on the width of a confidence interval of quadrupling the sample size while holding the confidence coefficient fixed?
The widths for part a is 35.93743-33.06257 = 2.87486
The widths for part b is 35.21874-33.78129 = 1.43745
A. Quadrupling the sample size while holding the confidence fixed decreases the width of the confidence interval by a factor of 4.
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