7. Sampling Distributions & Confidence Intervals: Mean

In Exercise 28, the population mean weekly time spent on homework by students is 7.8 hours. Does the t-value fall between -t0.99 and t0.99?

In Exercises 51 and 52, a population and sample size are given. (a) Find the mean and standard deviation of the population.

The goals scored in a season by the four starting defenders on a soccer team are 1, 2, 0, and 3. Use a sample size of 2.

[APPLET] The winning times (in hours) for a sample of 20 randomly selected Boston Marathon Women’s Open Division champions from 1980 to 2019 are shown in the table at the left. Assume the population standard deviation is 0.068 hour. (Source: Boston Athletic Association)

b. Find the margin of error for a 95% confidence level.

In Exercises 15 and 16, find the t-value for the given values of xbar, μ, s and n.

xbar = 70.3, μ = 64.8, s = 7.1, n = 16

You wish to estimate the mean winning time for Boston Marathon Women’s Open Division champions. The estimate must be within 2 minutes of the population mean. Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Use the population standard deviation from Exercise 1.

Describe how the t-distribution curve changes as the sample size increases.

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

b. Increase in the sample size

In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.

c = 0.99, s = 16.5, n = 20, xbar = 25.2

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

Find each probability.

(A) $P(T>1.5),df=8$

Age of Death-Row InmatesIn 2002, the mean age of an inmate on death row was 40.7 years, according to data from the U.S. Department of Justice. A sociologist wondered whether the mean age of a death-row inmate has changed since then. She randomly selects 32 death-row inmates and finds that their mean age is 38.9, with a standard deviation of 9.6. Construct a 95% confidence interval about the mean age. What does the interval imply?

In Exercises 51 and 52, a population and sample size are given. (b) List all samples (with replacement) of the given size from the population and find the mean of each. (c) Find the mean and standard deviation of the sampling distribution of sample means and compare them with the mean and standard deviation of the population.

The goals scored in a season by the four starting defenders on a soccer team are 1, 2, 0, and 3. Use a sample size of 2.

In Exercises 53 and 54, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.

The population densities in people per square mile in the 50 U.S. states have a mean of 199.6 and a standard deviation of 265.4. Random samples of size 35 are drawn from this population, and the mean of each sample is determined.

Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (b) find the sample standard deviation

[APPLET] Earnings The annual earnings (in dollars) of 32 randomly selected intermediate level life insurance underwriters (Adapted from Salary.com)

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?

c=0.95, x̅=3.46, s=1.63, n=16