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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 6.4.10
Chapter 6, Problem 6.4.10

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 = 0.99, s^2 = 0.64, n = 7

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Step 1: Understand the problem. You are tasked with constructing confidence intervals for both the population variance and the population standard deviation. The sample variance (s²) is given as 0.64, the sample size (n) is 7, and the confidence level (c) is 0.99. The population is assumed to be normally distributed.
Step 2: Recall the formula for the confidence interval of the population variance. The chi-square distribution is used for this purpose. The confidence interval for the population variance is given by: \( \left( \frac{(n-1)s^2}{\chi^2_{\alpha/2}}, \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}} \right) \), where \( \alpha = 1 - c \), \( \chi^2_{\alpha/2} \) and \( \chi^2_{1-\alpha/2} \) are the critical values of the chi-square distribution.
Step 3: Calculate \( \alpha \) and find the critical values. Since \( c = 0.99 \), \( \alpha = 1 - 0.99 = 0.01 \). The degrees of freedom (df) for the chi-square distribution are \( n-1 \), which is \( 7-1 = 6 \). Use a chi-square table or software to find \( \chi^2_{\alpha/2} \) and \( \chi^2_{1-\alpha/2} \) for \( \alpha/2 = 0.005 \) and \( 1-\alpha/2 = 0.995 \).
Step 4: Plug the values into the formula for the confidence interval of the population variance. Substitute \( n-1 = 6 \), \( s^2 = 0.64 \), and the critical values \( \chi^2_{\alpha/2} \) and \( \chi^2_{1-\alpha/2} \) into the formula to compute the lower and upper bounds of the confidence interval for the population variance.
Step 5: To find the confidence interval for the population standard deviation, take the square root of the lower and upper bounds of the confidence interval for the population variance. This will give you the confidence interval for the population standard deviation.

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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 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 parameter.
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Chi-Squared Distribution

The Chi-squared distribution is a statistical distribution that is used to estimate the variance of a population based on sample data. It is particularly important when constructing confidence intervals for population variance and standard deviation, as the test statistic follows a Chi-squared distribution when the population is normally distributed.
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Sample Variance and Standard Deviation

Sample variance (s²) is a measure of how much the values in a sample differ from the sample mean, while the standard deviation (s) is the square root of the variance. These statistics are crucial for estimating the population variance and standard deviation, especially when using sample data to infer characteristics about the larger population.
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Related Practice
Textbook Question

Tennis Ball Manufacturing A company manufactures tennis balls. When the balls are dropped onto a concrete surface from a height of 100 inches, the company wants the mean bounce height to be 55.5 inches. This average is maintained by periodically testing random samples of 25 tennis balls. If the t-value falls between and , then the company will be satisfied that it is manufacturing acceptable tennis balls. For a random sample, the mean bounce height of the sample is 56.0 inches and the standard deviation is 0.25 inch. Assume the bounce heights are approximately normally distributed. Is the company making acceptable tennis balls? Explain.

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Textbook Question

Graphical Analysis In Exercises 9–12, use the values on the number line to find the sampling error.

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Textbook Question

Which statistic is the best unbiased estimator for μ?

a. s

b. xbar

c. the median

d. the mode

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Textbook Question

In Exercises 7 and 8, find the margin of error for the values of c, s, and n.

c = 0.99, s = 3, n = 6

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Textbook Question

In Exercises 7–10, the statement represents a claim. Write its complement and state which is Ho and which is Ha.


μ<33

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Textbook Question

In Exercises 21–24, construct the indicated confidence interval for the population mean μ.

c = 0.95, xbar = 31.39, σ = 0.80, n = 82.

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