Textbook Question
Determine the minimum sample size required to be 95% confident that the sample mean waking time is within 10 minutes of the population mean waking time. Use the population standard deviation from Exercise 1.
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.5
In Exercises 5 and 6, use the confidence interval to find the margin of error and the sample mean.
(20.75, 24.10)
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Step 1: Understand the problem. The confidence interval is given as (20.75, 24.10). The goal is to find the margin of error and the sample mean.
Step 2: Recall the formula for the margin of error. The margin of error (E) is half the width of the confidence interval. Mathematically, this can be expressed as: .
Step 3: Recall the formula for the sample mean. The sample mean is the midpoint of the confidence interval. Mathematically, this can be expressed as: .
Step 4: Substitute the given values into the formulas. For the margin of error, substitute 24.10 as the upper limit and 20.75 as the lower limit into the formula for E. Similarly, substitute these values into the formula for the sample mean.
Step 5: Simplify the expressions to calculate the margin of error and the sample mean. This will give you the final results for both values.
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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., (20.75, 24.10)) and is associated with a confidence level, typically 95% or 99%, indicating the degree of certainty that the interval contains the parameter.
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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 this case, it can be found by subtracting the lower limit from the upper limit of the interval 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 9–12, find the critical value tc for the level of confidence c and sample size n.
c = 0.98, n = 15
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 = 0.99, s = 16.5, n = 20, xbar = 25.2
Textbook Question
In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.
c = 0.90, n = 16
Textbook Question
In a random sample of 12 senior-level civil engineers, the mean annual earnings were \$133,326 and the standard deviation was \$36,729. Assume the annual earnings are normally distributed and construct a 95% confidence interval for the population mean annual earnings for senior-level civil engineers. Interpret the results. (Adapted from Salary.com)
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.
b. Find the minimum sample size needed, using a prior study that found that 32% of U.S. adults have taken or planned to take a winter vacation in a recent year. (Source: Rasmussen Reports)