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.6
In Exercises 5 and 6, use the confidence interval to find the margin of error and the sample mean.
(7.428, 7.562)
Verified step by step guidance1
Identify the given confidence interval, which is (7.428, 7.562). The confidence interval provides the range of plausible values for the population parameter.
To find the margin of error (ME), use the formula: ME = (Upper Limit - Lower Limit) / 2. Substitute the values from the confidence interval into this formula.
To find the sample mean, use the formula: Sample Mean = (Upper Limit + Lower Limit) / 2. Substitute the values from the confidence interval into this formula.
Perform the subtraction and division for the margin of error calculation, ensuring proper order of operations.
Perform the addition and division for the sample mean calculation, ensuring proper order of operations.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Confidence Interval
A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed as an interval (e.g., (7.428, 7.562)) and is associated with a confidence level, typically 95% or 99%, indicating the degree of certainty that the parameter lies within the interval.
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06:33
Introduction to Confidence Intervals
Margin of Error
The margin of error quantifies the uncertainty in a sample estimate. It is calculated as half the width of the confidence interval, representing the maximum expected difference between the sample statistic and the population parameter. In this case, it can be found by subtracting the lower limit of the interval from the upper limit and dividing by two.
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04:08Finding the Minimum Sample Size Needed for a Confidence Interval
Sample Mean
The sample mean is the average of a set of sample observations and serves as a point estimate of the population mean. It can be calculated by taking the midpoint of the confidence interval, which provides a central value around which the interval is constructed. In this example, the sample mean can be found by averaging the two endpoints of the interval.
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05:11Sampling Distribution of Sample Proportion
Related Practice
Textbook Question
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
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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.
In a survey of 912 U.S. adults in Generation Z (born after 1996), 383 said they are at least somewhat likely to consider an electric vehicle for their next vehicle purchase. (Adapted from Pew Research Center)
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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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Textbook Question
In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.
c = 0.99, n = 10
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