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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 6 - Confidence IntervalsProblem 6.4.3
Chapter 6, Problem 6.4.3

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

Verified step by step guidance
1
Determine the degrees of freedom (df) using the formula: df = n - 1, where n is the sample size. For this problem, df = 8 - 1.
Identify the level of confidence (c) and calculate the significance level (α) using the formula: α = 1 - c. For c = 0.90, α = 1 - 0.90.
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 have α/2.
Use a chi-square distribution table or statistical software to find the critical values χL² and χR². For the left critical value (χL²), use df and the cumulative probability of α/2. For the right critical value (χR²), use df and the cumulative probability of 1 - α/2.
Verify the critical values by ensuring they correspond to the correct cumulative probabilities in the chi-square distribution table or software output.

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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 used primarily in hypothesis testing and confidence interval estimation for categorical data. It is defined by its degrees of freedom, which are determined by the sample size and the number of categories. The distribution is positively skewed, meaning it has a longer tail on the right side, and is used to assess how observed data compares to expected data.
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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 help to determine whether to reject the null hypothesis by comparing the test statistic to these values.
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Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of the Chi-Square distribution, degrees of freedom are typically calculated as the number of categories minus one or as the sample size minus one. They play a crucial role in determining the shape of the Chi-Square distribution and the corresponding critical values.
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Related Practice
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

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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 Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.

(3.144, 3.176)

91
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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

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

70
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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

92
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