Textbook Question
In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.
c = 0.90, s^2 = 35, n = 18
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 6.2.12
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
Verified step by step guidance1
Identify the given values: The confidence level (c) is 0.99, the sample mean (x̄) is 24.7, the sample standard deviation (s) is 4.6, and the sample size (n) is 50.
Determine the degrees of freedom (df) for the t-distribution. The formula is df = n - 1. Substitute n = 50 into the formula to calculate df.
Find the critical t-value (t*) corresponding to the confidence level (c = 0.99) and the degrees of freedom (df). Use a t-distribution table or statistical software to find this value.
Calculate the margin of error (ME) using the formula: ME = t* × (s / √n). Substitute the values of t*, s = 4.6, and n = 50 into the formula.
Construct the confidence interval for the population mean (μ) using the formula: CI = x̄ ± ME. Substitute x̄ = 24.7 and the calculated ME into the formula 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.
Confidence Interval
A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter (like the mean) with a specified level of confidence. For example, a 99% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 99% of those intervals would contain the true population mean.
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Introduction to Confidence Intervals
t-Distribution
The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but has heavier tails. It is used instead of the normal distribution when the sample size is small (typically n < 30) or when the population standard deviation is unknown, making it particularly useful for constructing confidence intervals for the mean.
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Sample Mean and Standard Deviation
The sample mean (x̄) is the average of a set of sample data points, providing an estimate of the population mean (μ). The sample standard deviation (s) measures the dispersion of the sample data around the mean. In constructing a confidence interval, both the sample mean and standard deviation are crucial for determining the range of values that likely contains the population mean.
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08:45Calculating Standard Deviation
Related Practice
Textbook Question
In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.
(3.144, 3.176)
Textbook Question
In Exercise 37, does it seem likely that the population mean could be greater than \$70? Explain.
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
Textbook Question
In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.
c = 0.98, s^2 = 278.1, n =41
Textbook Question
In Exercises 21–24, construct the indicated confidence interval for the population mean μ.
c = 0.80, xbar = 20.6, σ = 4.7, n = 100.
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