?Resampling Douglas Fir Trees Use the data from Problem 15 on the diameter of Douglas fir trees. (a) Construct and interpret a 95% confidence

Find the critical t-value for constructing a confidence interval for a population mean at the given level of confidence for the given sample size, n. (a) 99% confidence; n = 18 (b) 90% confidence; n = 27

Find the critical values and required to construct a confidence interval for s for the given level of confidence and sample size. (a) 95% confidence; n = 22 (b) 99% confidence; n = 12

IQ Scores Many of the examples and exercises in the text have dealt with IQ scores. We now know that IQ scores based on the Stanford-Binet IQ test are approximately normally distributed with a mean of 100 and standard deviation 15. If you were to obtain 100 different simple random samples of size 20 from the population of all adult humans and determine 95% confidence intervals for each of them, how many of the intervals would you expect to include 100? What causes a particular interval to not include 100?

What does the 95% represent in a 95% confidence interval?

For what proportion of samples will a 90% confidence interval for a population mean not capture the true population mean?

The area under the t-distribution with 18 degrees of freedom to the right of t = 1.56 is 0.0681. What is the area under the t-distribution with 18 degrees of freedom to the left of t = -1.56? Why?

Which is larger, the area under the t-distribution with 10 degrees of freedom to the right of t = 2.32 or the area under the standard normal distribution to the right of z = 2.32? Why?

State the properties of Student’s t-distribution.

A simple random sample of size n is drawn from a population. The sample mean, x, is 54.8 and the sample standard deviation is 10.5. (a) Construct the 90% confidence interval for the population mean if the sample size, n, is 30. (b) Construct the 90% confidence interval for the population mean if the sample size, n, is 51. How does increasing the sample size affect the width of the interval? (c) Construct the 99% confidence interval for the population mean if the sample size, n, is 30. Compare the results to those obtained in part (a). How does increasing the level of confidence affect the confidence interval?

A simple random sample of size n is drawn from a population that is known to be normally distributed. The sample mean, x, is determined to be 104.3 and the sample standard deviation, s, is determined to be 15.9. (a) Construct the 90% confidence interval for the population mean if the sample size, n, is 15. (b) Construct the 90% confidence interval for the population mean if the sample size, n, is 25. How does increasing the sample size affect the width of the interval? (c) Construct the 95% confidence interval for the population mean if the sample size, n, is 15. Compare the results to those obtained in part (a). How does increasing the level of confidence affect the confidence interval? (d) Construct the 90% confidence interval for the population standard deviation if the sample size, n, is 15.

Long Life? In a survey of 35 adult Americans, it was found that the mean age (in years) that people would like to live to is 87.9 with a standard deviation of 15.5. An analysis of the raw data indicates the distribution is skewed left. (a) Explain why a large sample size is necessary to construct a confidence interval for the mean age that people would like to live. (b) Construct and interpret a 95% confidence interval for the Mean. (c) How many adult Americans would need to be surveyed to estimate the mean age that people would like to live to within 2 years with 95% confidence?

E-mail The General Social Survey asked: “How many e-mails do you send in a day?” The results of 928 respondents indicate that the mean number of e-mails sent in a day is 10.4, with a standard deviation of 28.5. (a) Given the fact that 1 standard deviation to the left of the mean results in a negative number of e-mails being sent, what shape would you expect the distribution of e-mails sent to have? (b) Construct and interpret a 90% confidence interval for the mean number of e-mails sent per day.

Caffeinated Sports Drinks Researchers conducted an experiment to determine the effectiveness of a commercial caffeinated carbohydrate–electrolyte sports drink compared with a placebo. Sixteen highly trained cyclists each completed two trials of prolonged cycling in a warm environment, one while receiving the sports drink and another while receiving a placebo. For a given trial, one beverage treatment was administered throughout a 2-hour variable-intensity cycling bout followed by a 15-minute performance ride. Total work (in kilojoules) performed during the final 15 minutes was used to measure performance. The beverage order for individual subjects was randomly assigned with a period of at least five days separating the trials. Assume that the researchers verified the normality of the population of total work performed for each treatment. Source: Kirk J. Cureton, Gordon L. Warren, et al., “Caffeinated Sports Drink: Ergogenic Effects and Possible Mechanisms.” International Journal of Sport Nutrition and Exercise Metabolism 17(1):35–55, 2007 (a) Why do you think the sample size was small 1n = 162 for this experiment? (b) For the sports-drink treatment, the mean total work performed during the performance ride for the n = 16 riders was 218 kilojoules, with standard deviation 31 kilojoules. Construct and interpret a 95% confidence interval for the population mean total work performed. (c) Is it possible for the population mean total work performed for the sports-drink treatment to be less than 198 kilojoules? Do you think this is likely? (d) For the placebo treatment, the mean total work performed during the performance ride for the n = 16 riders was 178 kilojoules, with standard deviation 31 kilojoules. Construct and interpret a 95% confidence interval for the population mean total work performed. (e) Is it possible for the population mean total work performed for the placebo treatment to be more than 198 kilojoules? Do you think this is likely? (f) The researchers concluded that the caffeinated carbohydrate–electrolyte sports drink substantially enhanced physical performance during prolonged exercise compared with the placebo. Do your findings in parts (b) and (d) support the researchers’ conclusion? Explain.

Family Size A random sample of 60 married couples who have been married 7 years was asked the number of children they have. The results of the survey are as follows: (a) What is the shape of the distribution of the sample mean? Why? (b) Compute a 95% confidence interval for the mean number of children of all couples who have been married 7 years. Interpret this interval. (c) Compute a 99% confidence interval for the mean number of children of all couples who have been married 7 years. Interpret this interval.

Diameter of Douglas Fir Trees The diameter of the Douglas fir tree is measured at a height of 1.37 meters. The following data represent the diameter in centimeters of a random sample of 12 Douglas firs in the western Washington Cascades. Source: L. Winter. “Live Tree and Tree-Ring Records to Reconstruct the Structural Development of an Old-Growth Douglas Fir/Western Hemlock Stand in the Western Washington Cascades.” Corvallis, OR: Forest Science Data Bank, 2005. (a) Obtain a point estimate for the mean and standard deviation diameter of a Douglas fir tree in the western Washington Cascades. (b) Because the sample size is small, we must verify that the data come from a population that is normally distributed and that the data do not contain any outliers. The figures show the normal probability plot and boxplot. The correlation between the tree diameters and expected z-scores is 0.982. Are the conditions for constructing a confidence interval for the population mean diameter satisfied? (c) Construct a 95% confidence interval for the mean diameter of a Douglas fir tree in the western Washington Cascades. (d) Construct a 95% confidence interval for the standard deviation diameter of a Douglas fir tree in the western Washington Cascades.

Hypertension In a random sample of 678 adult males 20 to 34 years of age, it was determined that 58 of them have hypertension (high blood pressure). Source: The Centers for Disease Control. (a) Obtain a point estimate for the proportion of adult males 20 to 34 years of age who have hypertension. (b) Construct a 95% confidence interval for the proportion of adult males 20 to 34 years of age who have hypertension. Interpret the confidence interval. (c) You wish to conduct your own study to determine the proportion of adult males 20 to 34 years old who have hypertension. What sample size would be needed for the estimate to be within 3 percentage points with 95% confidence if you use the point estimate obtained in part (a)? (d) You wish to conduct your own study to determine the proportion of adult males 20 to 34 years old who have hypertension. What sample size would be needed for the estimate to be within 3 percentage points with 95% confidence if you don’t have a prior estimate?

Resampling Douglas Fir Trees Use the data from Problem 15 on the diameter of Douglas fir trees. (a) Construct and interpret a 95% confidence interval for the mean diameter of a Douglas fir tree using a bootstrap sample with 1000 resamples. Compare the bootstrap confidence interval to the t-interval found in Problem 15(c). (b) Draw a histogram of the 1000 means. What is the shape of the histogram? Why does the result not surprise you? (c) Construct and interpret a 95% confidence interval for the median diameter of a Douglas fir tree using a bootstrap sample with 1000 resamples.

Find the critical t-value for constructing a confidence interval about a population mean at the given level of confidence for the given sample size, n. (a) 96% confidence; n = 26 (b) 98% confidence; n = 18

Find the critical values and required to construct a confidence interval about s for the given level of confidence and sample size. (a) 98% confidence; n = 16 (b) 90% confidence; n = 30

Determine the point estimate of the population mean and margin of error if the confidence interval has lower bound: 125.8 and upper bound: 152.6.

A question on the General Social Survey was this: “How many family members do you know that are in prison?” The results of 499 respondents indicate that the mean number of family members in jail is 1.22, with a standard deviation of 0.59. (a) What shape would you expect the distribution of this variable to have? Why? (b) Construct and interpret a 99% confidence interval for the mean number of family members in jail.

A random sample of 50 recent college graduates results in a mean time to graduate of 4.58 years, with a standard deviation of 1.10 years. Source: Based on data from The Toolbox Revisited by Clifford Adelman, U.S. Department of Education (a) Compute and interpret a 90% confidence interval for time to graduate with a bachelor’s degree. (b) Does this evidence contradict the widely held belief that it takes 4 years to complete a bachelor’s degree? Why?

The campus at Joliet Junior College has a lake. A student used a Secchi disk to measure the clarity of the lake’s water by lowering the disk into the water and measuring the distance below the water surface at which the disk is no longer visible. The following measurements (in inches) were taken on the lake at various points in time over the course of a year. (a) Use the data to compute a point estimate for the population mean and population standard deviation. (b) Because the sample size is small, we must verify that the data are normally distributed and do not contain any outliers. The figures in the next column show the normal probability plot and boxplot. The correlation between depth and expected z-scores is 0.960. Are the conditions for constructing a confidence interval about satisfied? (c) Construct a 95% confidence interval for the mean Secchi disk measurement. Interpret this interval. (d) Construct a 99% confidence interval for the mean Secchi disk measurement. Interpret this interval. (e) Construct a 95% confidence interval for the population standard deviation Secchi disk measurement. Interpret this interval.

From a random sample of 1201 Americans, it was discovered that 1139 of them lived in neighborhoods with acceptable levels of carbon monoxide. Source: Environmental Protection Agency. (a) Obtain a point estimate for the proportion of Americans who live in neighborhoods with acceptable levels of carbon monoxide. (b) Construct a 99% confidence interval for the proportion of Americans who live in neighborhoods with acceptable levels of carbon monoxide. (c) You wish to conduct your own study to determine the proportion of Americans who live in neighborhoods with acceptable levels of carbon monoxide. What sample size would be needed for the estimate to be within 1.5 percentage points with 90% confidence if you use the estimate obtained in part (a)? (d) You wish to conduct your own study to determine the proportion of Americans who live in neighborhoods with acceptable levels of carbon monoxide. What sample size would be needed for the estimate to be within 1.5 percentage points with 90% confidence if you do not have a prior estimate?

Wimbledon Match Lengths A tennis enthusiast wants to estimate the mean length of men’s singles matches held during the Wimbledon tennis tournament. From the Wimbledon history archives, he randomly selects 40 matches played during the tournament since the year 1968 (when professional players were first allowed to participate). The lengths of the 40 selected matches, in minutes, follow: (a) Obtain a point estimate of the population mean length of men’s singles matches during Wimbledon. (b) A frequency histogram of the data is shown next. Explain why a large sample is necessary to construct a confidence interval for the population mean length of men’s singles matches during Wimbledon. (c) Construct and interpret a 99% confidence interval for the population mean length of men’s singles matches during Wimbledon. (d) Construct and interpret a 95% confidence interval for the population mean length of men’s singles matches during Wimbledon. (e) What effect does increasing the level of confidence have on the interval? (f) Do the confidence intervals computed in parts (c) and (d) represent an estimate for the population mean length of men’s singles matches during all professional tennis tournaments? Why?

The given data represent the wait time (in seconds) for a random sample of 14 individuals before connecting with a customer service agent. (a) A boxplot of the data is shown next. The quality control manager of the customer service department wants to estimate the mean wait time. Explain why you would not recommend the construction of a t-interval to estimate the mean. (b) Explain how the bootstrapping method can be used to obtain a 95% confidence interval for the mean using 1000 resamples. (c) Obtain a 95% confidence interval for the mean wait time using 1000 resamples.