Textbook Question
(a) Construct a 90% confidence interval for the population mean in Exercise 1. Interpret the results. (b) Does it seem likely that the population mean could be within 10% of the sample mean? Explain.
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Ch. 6 - Confidence Intervals
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 6, Problem 6.R.15
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 = 0.98, s = 0.9, n = 12, xbar = 6.8
Verified step by step guidance1
Step 1: Identify the formula for the margin of error (E) when using the t-distribution. The formula is E = t * (s / sqrt(n)), where t is the critical value from the t-distribution table, s is the sample standard deviation, and n is the sample size.
Step 2: Determine the degrees of freedom (df) for the t-distribution. The formula for degrees of freedom is df = n - 1. In this case, n = 12, so df = 12 - 1 = 11.
Step 3: Look up the critical value (t) for a confidence level of c = 0.98 and df = 11 in the t-distribution table. This value corresponds to the two-tailed critical value for a 98% confidence level.
Step 4: Substitute the values of t, s, and n into the margin of error formula E = t * (s / sqrt(n)). Use the given values: s = 0.9 and n = 12. Simplify the expression to calculate E.
Step 5: Construct the confidence interval using the formula: Confidence Interval = x̄ ± E, where x̄ is the sample mean. Substitute x̄ = 6.8 and the calculated margin of error (E) to find the lower and upper bounds of the confidence interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Margin of Error
The margin of error quantifies the uncertainty in a sample estimate. It is calculated using the critical value from the t-distribution, the sample standard deviation, and the square root of the sample size. A larger margin of error indicates less precision in the estimate, while a smaller margin suggests greater reliability.
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Finding the Minimum Sample Size Needed for a Confidence Interval
Confidence Interval
A confidence interval provides a range of values within which we expect the true population parameter to lie, with a specified level of confidence. It is constructed by adding and subtracting the margin of error from the sample mean. For a confidence level of 0.98, we use the t-distribution to find the appropriate critical value based on the sample size.
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t-Distribution
The t-distribution is a probability distribution used when estimating population parameters when the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which accounts for the increased variability in smaller samples. As the sample size increases, the t-distribution approaches the normal distribution.
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Related Practice
Textbook Question
In a random sample of 36 top-rated roller coasters, the average height is 165 feet and the standard deviation is 67 feet. Construct a 90% confidence interval for μ. Interpret the results. (Source: POP World Media, LLC)
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Textbook Question
In Exercises 5 and 6, use the confidence interval to find the margin of error and the sample mean.
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Textbook Question
Determine the minimum sample size required to be 99% confident that the sample mean driving distance to work is within 2 miles of the population mean driving distance to work. Use the population standard deviation from Exercise 2.
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Textbook Question
In Exercises 19–22, let p be the population proportion for the situation. (a) Find point estimates of p and q, (b) construct 90% and 95% confidence intervals for p, and (c) interpret the results of part (b) and compare the widths of the confidence intervals.
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Textbook Question
You wish to estimate, with 95% confidence, the population proportion of U.S. adults who have taken or planned to take a winter vacation in a recent year. Your estimate must be accurate within 5% of the population proportion.
a. No preliminary estimate is available. Find the minimum sample size needed.
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