Question

A simple random sample of 60 items resulted in a sample mean of 91. 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?

Answer 1

Solution:
Given in the question
No. of samples = 60
Mean = 91
Population standard deviation = 12
95% confidence interval can be calculated as
Mean +/- Zalpha/2 * SD/sqrt(n)
91 +/ 1.96*12/sqrt(60)
91+/- 1.96*12/7.7459666
91 +/- 3.03644
So 95% confidence interval is
87.97 to 94.03 or (87.97,94.03)
And margin of error is 3.03644

If sample is = 120 and mean is same than
Confidence of interval is calculated as follows:
91 +/ 1.96*12/sqrt(120)
91 +/- 1.96*12/10.9544
91 +/- 2.1470
88.85 to 93.15 or (88.85,93.15)

As we can see that margin of error is indirectly proportional to sample size, as sample increase than margin of erro will decrease. So if sample size is 60 than margin of error is 3.036 while when we increase sample size to 120 than we got margin of error is 2.1470 so as no. of sample increase than margin of error decreases.