Textbook Question
In Exercises 13–16, find the margin of error for the values of c, σ and n.
c = 0.975, σ = 4.6, n = 100
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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 6.4.7
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 = 0.99, n = 30
Verified step by step guidance1
Determine the degrees of freedom (df) using the formula: df = n - 1, where n is the sample size. For this problem, df = 30 - 1.
Identify the level of confidence (c) and calculate the significance level (α) using the formula: α = 1 - c. For c = 0.99, α = 1 - 0.99.
Divide the significance level (α) into two tails for a two-tailed test. The left tail will have α/2, and the right tail will also have α/2.
Use a chi-square distribution table or statistical software to find the critical values χL² and χR². For χL², find the value corresponding to the cumulative probability of α/2 with df degrees of freedom. For χR², find the value corresponding to the cumulative probability of 1 - α/2 with df degrees of freedom.
Verify the critical values by ensuring they correspond to the correct cumulative probabilities and degrees of freedom in the chi-square distribution table or software.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chi-Square Distribution
The Chi-Square distribution is a statistical distribution that is commonly used in hypothesis testing, particularly in tests of independence and goodness of fit. It is defined by its degrees of freedom, which are determined by the sample size and the number of parameters estimated. The distribution is right-skewed, meaning it has a longer tail on the right side, and it approaches a normal distribution as the degrees of freedom increase.
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Critical Values
Critical values are the threshold points that define the boundaries of the acceptance region in hypothesis testing. They are determined based on the desired level of confidence (c) and the degrees of freedom associated with the test. For a Chi-Square test, critical values are used to decide whether to reject the null hypothesis, with values falling beyond the critical points indicating significant results.
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Level of Confidence
The level of confidence, denoted as 'c', represents the probability that the confidence interval will contain the true parameter value. Common levels of confidence include 90%, 95%, and 99%. A higher level of confidence corresponds to a wider confidence interval, which reflects greater certainty about the parameter estimate but less precision. In this case, a confidence level of 0.99 indicates a 99% certainty that the true value lies within the calculated interval.
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Related Practice
Textbook Question
In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.
(21.61, 30.15)
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Textbook Question
When estimating the population mean, why not construct a 99% confidence interval every time?
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Textbook Question
Finding the Margin of Error In Exercises 33 and 34, use the confidence interval to find the estimated margin of error. Then find the sample mean. Book Prices A store manager reports a confidence interval of (244.07, 280.97) when estimating the mean price (in dollars) for the population of textbooks.
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Textbook Question
In Exercises 5–8, find the critical value zc necessary to construct a confidence interval at the level of confidence c.
c = 0.80
110
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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 = 0.95, n = 20
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