The mean and standard deviation of a random sample of n measurements are equal to 34.8 and 3.9, respectively. a. Find a 90% confidence interval for mu if nequals81. b. Find a 90% confidence interval for mu if nequals324. 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?

a) The 90% confidence interval for the mean is (34.08, 35.52).

b) The 90% confidence interval for the mean is (34.44, 35.16).

c) Width for n=81: 1.44

Width for n=324: 0.72

d) The width of the interval is reduced by a factor of √4=2.

Step-by-step explanation:

a) We have to calculate a 90% confidence interval for the mean.

The population standard deviation is not known, so we have to estimate it from the sample standard deviation and use a t-students distribution to calculate the critical value.

The sample mean is M=34.8.

The sample size is N=81.

When σ is not known, s divided by the square root of N is used as an estimate of σM:

The degrees of freedom for this sample size are:

The t-value for a 90% confidence interval and 80 degrees of freedom is t=1.66.

The margin of error (MOE) can be calculated as:

Then, the lower and upper bounds of the confidence interval are:

The 90% confidence interval for the mean is (34.08, 35.52).

b) As the sample size has changed, we recalculate the standard error:

The degrees of freedom for this sample size are:

The t-value for a 90% confidence interval and 323 degrees of freedom is t=1.65.

The margin of error (MOE) can be calculated as:

Then, the lower and upper bounds of the confidence interval are:

The 90% confidence interval for the mean is (34.44, 35.16).

c. The widths are:

For n=81

For n=324

d. The effect of quadrupling the sample size, with all the other parameters constant, is that the width of the interval is reduced by a factor of 2.

This is because the standard error, and therefore the margin of error, is reduced by a factor of √4=2, when the sample size is quadrupled.

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