7. Sampling Distributions & Confidence Intervals: Mean

When drawing independent random samples from two normal populations, what is the distribution of the difference between the sample means $\mu =x_1-x_2$?

A properly drawn random sample of one thousand individuals is used to estimate a population mean. The sampling error for the sample mean is roughly plus or minus what percent of the population mean (assuming a normal distribution and no prior knowledge of population standard deviation)?

The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)

b. Construct a 90% confidence interval for the population mean. Interpret the results.

Use the standard normal distribution or the t-distribution to construct the indicated confidence interval for the population mean of each data set. Justify your decision. If neither distribution can be used, explain why. Interpret the results.

a. In a random sample of 40 patients, the mean waiting time at a dentist’s office was 20 minutes and the standard deviation was 7.5 minutes. Construct a 95% confidence interval for the population mean.

In the context of confidence intervals for population mean, an interval estimate is used to estimate $\_\_\_\_\_$.

Gas prices are getting more and more expensive. The average gas price, from a random sample of 100 gas stations, was \$3.50. It is assumed that gas prices have a standard deviation of \$0.04. Construct an 80% confidence interval for the true mean gas price in the United States.

Books get more and more expensive every semester, but the distribution of their prices is always normal. 25 randomly selected students in your school spent, on average \$500 with a standard deviation of \$50. Construct a 98% confidence interval for the true spending on books.

You want to purchase one of the new Altima. You randomly select 400 dealerships across the United States and find a mean of \$25,000. Assume a population standard deviation of \$2500. Construct and interpret a 94% confidence interval for the true mean price for the new Nissan Altima.

Find the critical value $t_{\frac{\alpha }{2}}$for an 80% confidence interval given a sample size of 51.

Find the critical value $t_{\frac{\alpha }{2}}$for a 95% confidence interval given a sample size of 6.

For which of the following scenarios can you NOT create a confidence interval using the standard normal or t-distribution?

You ask 16 people in your Statistics class what their grade is. The data appears to be distributed normally. You find a sample mean and sample standard deviation of 60 and 24, respectively. Construct and interpret a 95% confidence interval for the population mean class grade.

You want to take a trip to Paris. You randomly select 225 flights to Europe and find a mean and sample standard deviation of \$1500 and \$900, respectively. Construct and interpret a 95% confidence interval for the true mean price for a trip to Paris.

You want to purchase one of the new Altimas. You randomly select 400 dealerships across the United States and find a mean of \$25,000 and sample standard deviation of \$2500. Construct and interpret a 94% confidence interval for the true mean price for the new Nissan Altima.

Ages of Prisoners The accompanying frequency distribution summarizes sample data consisting of ages of randomly selected inmates in federal prisons (based on data from the Federal Bureau of Prisons). Use the data to construct a 95% confidence interval estimate of the mean age of all inmates in federal prisons.