Textbook Question
You research prices of cell phones and find that the population mean is \$431.61. In Exercise 19, does the t-value fall between -t0.95 and t0.95?
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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 6.1.28
In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.
(3.144, 3.176)
Verified step by step guidance1
Identify the given confidence interval, which is (3.144, 3.176). The lower bound is 3.144, and the upper bound is 3.176.
To find the margin of error (E), use the formula: . Substitute the values of the upper and lower bounds into this formula.
To find the sample mean (), use the formula: . Substitute the values of the upper and lower bounds into this formula.
Perform the subtraction and division in the margin of error formula to calculate the margin of error.
Perform the addition and division in the sample mean formula to calculate the sample mean.
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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., (3.144, 3.176)) and is associated with a confidence level, indicating the probability that the interval contains the parameter. Understanding confidence intervals is crucial for estimating population parameters and assessing the precision of sample estimates.
Recommended video:
06:33
Introduction to Confidence Intervals
Margin of Error
The margin of error quantifies the uncertainty associated with 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 the given interval (3.144, 3.176), the margin of error would be 0.016, indicating how much the sample mean could vary from the true mean.
Recommended video:
04:08Finding the Minimum Sample Size Needed for a Confidence Interval
Sample Mean
The sample mean is the average of a set of observations from a sample, serving as a point estimate of the population mean. It is calculated by summing all sample values and dividing by the number of observations. In the context of the confidence interval (3.144, 3.176), the sample mean can be found as the midpoint of the interval, which provides a central value around which the data is distributed.
Recommended video:
05:11Sampling Distribution of Sample Proportion
Related Practice
Textbook Question
Finding p^ and q^ In Exercises 3–6, let p be the population proportion for the situation. Find point estimates of p and q.
Private Internet Browsing In a survey of 4272 U.S. adults, 1025 knew that private browsing mode only prevents someone using the same computer from seeing one’s online activities. (Adapted from Pew Research Center)
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Textbook Question
In Exercises 35–40, use the standard normal distribution or the t-distribution to construct a 95% confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results.
The population standard deviation of the weights of the two-year-old males on a pediatrician’s patient list is 2.49 pounds. The mean weight of a sample of 10 of the two–year–old males is 13.68 pounds. Weights are known to be normally distributed.
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Textbook Question
In Exercise 37, does it seem likely that the population mean could be greater than \$70? Explain.
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Textbook Question
Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.
c = 0.90, n = 8
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Textbook Question
In Exercises 9–12, construct the indicated confidence interval for the population mean μ using the t-distribution. Assume the population is normally distributed.
c = 0.99, xbar = 24.7, s = 4.6, n = 50
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