A simple random sample of 60 items resulted in a sample mean

A simple random sample of 60 items resulted in a sample mean of 97. The population standard deviation is 12. a. Compute the 95% confidence interval for the population mean (to 1 decimal). ( , ) b. Assume that the same sample mean was obtained from a sample of 120 items. Provide a 95% confidence interval for the population mean (to 2 decimals). ( , ) c. What is the effect of a larger sample size on the margin of error?

Solution

(a)

n = 60

x-bar = 80

s = 15

% = 95

Standard Error, SE = /n = 15 /60 = 1.936491673

z- score = 1.959963985

Width of the confidence interval = z * SE = 1.95996398454005 * 1.93649167310371 = 3.795453936

Lower Limit of the confidence interval = x-bar - width = 80 - 3.79545393564498 = 76.20454606

Upper Limit of the confidence interval = x-bar + width = 80 + 3.79545393564498 = 83.79545394

The confidence interval is [76.20, 83.80]

(b)

n = 120

x-bar = 80

s = 15

% = 95

Standard Error, SE = /n = 15 /120 = 1.369306394

z- score = 1.959963985

Width of the confidence interval = z * SE = 1.95996398454005 * 1.36930639376292 = 2.683791216

Lower Limit of the confidence interval = x-bar - width = 80 - 2.68379121557574 = 77.31620878

Upper Limit of the confidence interval = x-bar + width = 80 + 2.68379121557574 = 82.68379122

The confidence interval is [77.32, 82.68]

(c) As the sample size increases, the confidence interval narrows down.