Question

A simple random sample of 60 items resulted in a sample mean of 80. The population standard deviation is σ=15. 

a. Compute the 95% confidence interval for the population mean. Round your answers to one decimal place 

b. Assume that the same sample mean was obtained from a sample of 120 items. Provide a 95% confidence interval for the population mean. Round your answers to two decimal places 

c. What is the effect of a larger sample size on the interval estimate? 

Answer 1

(a)

n = 60     

x-bar = 80     

s = 15     

% = 95     

Standard Error, SE = σ/√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 [76.2, 83.8]

(b)

n = 120     

x-bar = 80     

s = 15     

% = 95     

Standard Error, SE = σ/√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 n increases, the confidence interval narrows down. Larger sample provides a smaller margin of error.