IQ scores are known to be approximately normally distributed with mean 100 and standard deviation 15 .

(a) Simulate obtaining a random sample of 12 IQ scores from this population.

(b) Use the data from part

(a) to construct a \(95 \%\) confidence interval for the mean IQ using Student's \(t\)-distribution.

(c) Use the data from part (a) to obtain 1000 bootstrap samples. For each sample, find the mean.

(d) Determine an estimate of the standard error of the mean from the 1000 bootstrap means found in part (c). Compare this result to the theoretical standard error of the mean, \(\frac{\sigma}{\sqrt{n}}\). Compare this result to the estimate of the standard error of the mean based on the sample data, \(\frac{s}{\sqrt{n}}\).

(e) Construct a 95\% confidence interval for the mean IQ using the bootstrap sample from part (c).