"Simulation 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 at least 2000 bootstrap samples. For each sample, find the mean.


d. Determine an estimate of the standard error of the mean from the bootstrap means found in part (c). Compare this result to the theoretical standard error of the mean, sigma divided by square root of n. Compare this result to the estimate of the standard error of the mean based on the sample data, s divided by square root of n.


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

"A second method for finding a bootstrap confidence interval is called the Bootstrap t-Method. This method requires estimating the standard error of the estimate (such as the standard error of the mean) from the bootstrap sample. For any given set of estimates of a parameter, the standard error of the estimate is found by determining the sample standard deviation of the B bootstrap estimates. For example, in Example 1, we found 2000 estimates of the sample mean mpg. The standard deviation of these 2000 sample means is found to be 0.580. The standard deviation of the 16 observations is 2.38, so

s / sqrt(n) = 2.38 / sqrt(16) = 0.595,

is close to the standard error of the estimate found from the bootstrap samples. The estimate of the standard error from Figure 28 in Example 2 is 0.560. We can use an estimate of the standard error (SE_est) along with critical values from Student's t-distribution (Table VII) to construct confidence intervals as follows:

statistic ± t_(alpha/2) * SE_est

For example, in Example 1 we know x̄ = 28.1 and n = 16, so for a 95% confidence interval, t_0.025 = 2.131 using 15 degrees of freedom. Using the standard error estimate from Example 1, we find the lower bound of the confidence interval to be 28.1 – 2.131(0.580) = 26.86 mpg and the upper bound to be 28.1 + 2.131(0.580) = 29.34.

pH of Rain Revisited See Problem 7. Use the Bootstrap t-Method to find a 95% confidence for the mean pH of rainwater in Tucker County, West Virginia."

The procedure for constructing a confidence interval about a mean is ________, which means minor departures from normality do not affect the accuracy of the interval.