Question: The mean and standard deviation of a random sample of n measurements are equal to 33.9 and 3.3, respectively.

a. Find a 95% confidence interval for $\mu$if n = 100.

b. Find a 95% confidence interval for$\mu$ if n = 400.

c. Find the widths of the confidence intervals found in parts a and b. What is the effect on the width of a confidence interval of quadrupling the sample size while holding the confidence coefficient fixed?

As the given confidence interval is 95%, so the significance level will automatically be 5% which means 0.05.

It can be said that $\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.=\frac{0.05}{2}$

=0.025

Now the value of $Z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ can be found from the z table, and the value is 1.96.

The margin of error is calculated below:

The calculation of the confidence intervals of the boundaries are calculated below:

As the given confidence interval is 95%, the significance level will automatically be 5% which means 0.05.

Therefore, it can be said that$\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.=\frac{0.05}{2}$

Now the value of can be found from the z table, and the value is 1.96.

The margin of error is calculated below:

The calculation of the confidence intervals of the boundaries are calculated below:

For part (a), the width of the confidence interval will range between 33.2532 and 34.5468. For part (b), the width of the confidence interval will range between 33.5766 and 34.2234.

If the sample size is quadrupled, the width will not change.The confidence coefficient has already been asked to keep constant, which gets added or subtracted from the respective means to get the value of the widths. With constant coffircient, the mean is constant, so the width will remain constant.