Chapter 6 Homework
Construct the confidence interval for the population mean μ. c=0.90, x=8.5, σ=0.7, and n=50
A 90% confidence interval for μ is (8.34,8.66)
Construct the indicated confidence interval for the population mean μ using the t-distribution. Assume the population is normally distributed. c=0.90, x=12.9, s=3.0, n=7
(10.7,15.1)
Find the critical value zc necessary to form a confidence interval at the level of confidence shown below. c=0.81
1.31
Find the critical value zc necessary to form a confidence interval at the level of confidence shown below. c=0.89
1.60
Find the minimum sample size n needed to estimate μ for the given values of c, σ, and E. c=0.90, σ=5.7, and E=1
Assume that a preliminary sample has at least 30 members. n= 88
In a survey of 3353 adults, 1424 say they have started paying bills online in the last year. Construct a 99% confidence interval for the population proportion. Interpret the results.
A 99% confidence interval for the population proportion is (.403,.447) Interpret results: With 99% confidence, it can be said that the population proportion of adults who say they have started paying bills online in the last year is between the endpoints of the given confidence interval.
You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. If convenient, use technology to construct the confidence intervals. A random sample of 50 home theater systems has a mean price of $144.00. Assume the population standard deviation is $15.20.
Construct a 90% confidence interval for the population mean. The 90% confidence interval is (140.45,147.55) Construct a 95% confidence interval for the population mean. The 95% confidence interval is (139.79,148.21) Interpret the results. With 90% confidence, it can be said that the population mean price lies in the first interval. With 95% confidence, it can be said that the population mean price lies in the second interval. The 95% confidence interval is wider than the 90%.
Use the given confidence interval to find the margin of error and the sample proportion. (0.716,0.742)
E= .013 p=.729
Find the margin of error for the given values of c, σ, and n. c=.90,σ=3.5,n=81
E=.640
You construct a 95% confidence interval for a population mean using a random sample. The confidence interval is 24.9<μ<31.5. Is the probability that μ is in this interval 0.95? Explain.
No. With 95% confidence, the mean is in the interval (24.9,31.5).
In a random sample of 27 people, the mean commute time to work was 32.3 minutes and the standard deviation was 7.2 minutes. Assume the population is normally distributed and use a t-distribution to construct a 99% confidence interval for the population mean μ. What is the margin of error of μ? Interpret the results.
The confidence interval for the population mean μ is (28.5,36.2) The margin of error of μ is 3.8 Interpret the results. With 99% confidence, it can be said that the population mean commute time is between the bounds of the confidence interval.
In a random sample of 21 people, the mean commute time to work was 32.7 minutes and the standard deviation was 7.3 minutes. Assume the population is normally distributed and use a t-distribution to construct a 80% confidence interval for the population mean μ. What is the margin of error of μ? Interpret the results.
The confidence interval for the population mean μ is (30.6,34.8) The margin of error of μ is 2 Interpret the results. With 80% confidence, it can be said that the population mean commute time is between the bounds of the confidence interval.
Use the confidence interval to find the margin of error and the sample mean. (0.308,0.410)
The margin of error is .051 The sample mean is .359
Find the margin of error for the given values of c, s, and n. c=0.99, s=4, n= 24
The margin of error is 2.292
In a survey of 2059 adults in a recent year, 1474 say they have made a New Year's resolution. Construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals.
The 90% confidence interval for the population proportion p is (0.7,0.732) The 95% confidence interval for the population proportion p is (0.697,0.735) With the given confidence, it can be said that the population proportion of adults who say they have made a New Year's resolution is between the endpoints of the given confidence interval. Compare the widths of the confidence intervals. The 95% confidence interval is wider.
T OR F The point estimate for the population proportion of failures is 1−p.
True
When estimating a population mean, are you more likely to be correct when you use a point estimate or an interval estimate? Explain your reasoning.
You are more likely to be correct using an interval estimate because it is unlikely that a point estimate will exactly equal the population mean.
Let p be the population proportion for the following condition. Find the point estimates for p and q. In a survey of 1227 adults from country A, 486 said that they were not confident that the food they eat in country A is safe.
p=.396 q=.604
If all other quantities remain the same, how does the indicated change affect the minimum sample size requirement? (a) Increase in the level of confidence (b) Increase in the error tolerance (c) Increase in the population standard deviation
(a) How does an increase in the level of confidence affect the width of a confidence interval? Choose the correct answer below. An increase in the level of confidence will increase the minimum sample size requirement. (b) How does an increase in the sample size affect the width of a confidence interval? Choose the correct answer below. An increase in the sample size will decrease the minimum sample size requirement. (c) How does an increase in the population standard deviation affect the width of a confidence interval? Choose the correct answer below. An increase in the population standard deviation will increase the minimum sample size requirement.
If all other quantities remain the same, how does the indicated change affect the width of a confidence interval? (a) Increase in the level of confidence (b) Increase in the sample size (c) Increase in the population standard deviation
(a) How does an increase in the level of confidence affect the width of a confidence interval? Choose the correct answer below. An increase in the level of confidence will widen the confidence interval. (b) How does an increase in the sample size affect the width of a confidence interval? Choose the correct answer below. An increase in the sample size will narrow the confidence interval. (c) How does an increase in the population standard deviation affect the width of a confidence interval? Choose the correct answer below. An increase in the population standard deviation will widen the confidence interval.
A researcher wishes to estimate, with 90% confidence, the population proportion of likely U.S. voters who think Congress is doing a good or excellent job. Her estimate must be accurate within 3% of the true proportion. (a) No preliminary estimate is available. Find the minimum sample size needed. (b) Find the minimum sample size needed, using a prior study that found that 28% of the respondents said they think Congress is doing a good or excellent job. (c) Compare the results from parts (a) and (b).
(a) What is the minimum sample size needed assuming that no prior information is available n= 752 (b) What is the minimum sample size needed using a prior study that found that 28% of the respondents said they think Congress is doing a good or excellent job? n= 607 Interpret results: Having an estimate of the population proportion reduces the minimum sample size needed.
For the same sample statistics, which level of confidence would produce the widest confidence interval? Explain your reasoning.
99%, because as the level of confidence increases, zc increases.
