IV A simple random sample of size n is drawn from a populati

IV. A simple random sample of size n is drawn from a population whose population standard deviation, sigma, is known to be 3.8. The sample mean, bar x is determined to be 59.2. A. Construct the 90% confidence interval for mu if the sample size, n, is 45. Construct the 90% confidence interval for mu if the sample size, n, is 55. How does increasing the sample size affect the margin of error, E? Interval __________________________ Effect_______________________________________________ C. Construct the 99% confidence level for mu if the sample size, n, is 45. Compare the results to those obtained in A. How does increasing the level of confidence affect the margin of error, E? Interval _______ Effect________________________________

Solution

(A) Given a=1-0.9=0.1, Z(0.05) = 1.645 (from standard normal table)

So the lower bound is

xbar - Z*s/vn =59.2-1.645*3.8/sqrt(45)=58.26816

So the upper bound is

xbar + Z*s/vn =59.2+1.645*3.8/sqrt(45)=60.13184

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(B)

So the lower bound is

xbar - Z*s/vn =59.2-1.645*3.8/sqrt(55)=58.35712

So the upper bound is

xbar + Z*s/vn =59.2+1.645*3.8/sqrt(45)=60.04288

Effect: The margin of error decreases.

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(C) Given a=1-0.99=0.01, Z(0.005) = 2.58 (from standard normal table)

So the lower bound is

xbar - Z*s/vn =59.2-2.58*3.8/sqrt(45)=57.73851

So the upper bound is

xbar + Z*s/vn =59.2+2.58*3.8/sqrt(45)=60.66149

Effect: The margin of error increases.