Question

The length of a confidence interval changes when either the confidence level changes or the sample size changes. To illustrate this, imagine a statistician takes a sample from a population with a population standard deviation of 2. The sample mean is calculated to be 10. Construct 3 different confidence intervals:

3a) Assume the sample size is 30 and the confidence level is 95%

3b) Assume the sample size is 30 and the confidence level is 90%

3c) Assume the sample size is 50 and the confidence level is 95%

3d) In general, if the confidence level is decreased, does the length of the confidence interval increase or decrease? Explain your answer. Hint: Refer to your answers to (3a), (3b) and (3c)

3e) In general, if the sample size is increased, does the length of the confidence interval increase or decrease? Explain your answer.

3f) In order to make a more accurate prediction of a population mean, it is better to have a shorter confidence interval, in other words, a narrower range of possible values. Based on your answers to (3d) and (3e), is it better to increase the accuracy of an estimation by decreasing the confidence level or increasing the sample size? Explain your answer.

Answer 1

3a. Here population standard deviation is known so we will use z distribution

z value for 95% CI z value is 1.96 as P(-1.96<z<1.96)=0.95

So Margin of Error is

Hence CI is

b. z value for 90% CI is 1.645 as P(-1.645<z<1.645)=0.90

So Margin of Error is

Hence CI is

c. Now for n=50

So Margin of Error is

Hence CI is

d. Decreasing the confidence level decrease the Margin of error and so confidence interval decreases.

e. Increasing sample size standard error decreases so Margin of error decreases and hence confidence interval decrease.

f. Both can be used to increase the accuracy