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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.T.2c
Chapter 6, Problem 6.T.2c

The data set represents the weights (in pounds) of 10 randomly selected black bears from northeast Pennsylvania. Assume the weights are normally distributed. (Source: Pennsylvania Game Commission)
Table displaying weights in pounds of 10 randomly selected black bears from northeast Pennsylvania.
c. Construct a 99% confidence interval for the population standard deviation. Interpret the results.

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Step 1: Calculate the sample variance (s²) using the formula: s² = Σ(xᵢ - x̄)² / (n - 1), where xᵢ represents each data point, x̄ is the sample mean, and n is the sample size. First, compute the sample mean (x̄) by summing all the weights and dividing by the number of data points.
Step 2: Use the Chi-Square distribution to construct the confidence interval for the population variance. The formula for the confidence interval is: ( (n-1)s² / χ²_upper, (n-1)s² / χ²_lower ), where χ²_upper and χ²_lower are the critical values from the Chi-Square distribution table corresponding to the desired confidence level (99%) and degrees of freedom (df = n - 1).
Step 3: Convert the confidence interval for the variance into a confidence interval for the standard deviation by taking the square root of the lower and upper bounds of the variance interval.
Step 4: Interpret the results. The 99% confidence interval for the population standard deviation provides a range within which the true standard deviation of the black bear weights is likely to fall, with 99% certainty.
Step 5: Ensure all calculations are performed accurately, and verify the critical Chi-Square values for the 99% confidence level using a Chi-Square distribution table or statistical software.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In statistics, many natural phenomena, including weights of animals, tend to follow a normal distribution, which is characterized by its bell-shaped curve. Understanding this concept is crucial for constructing confidence intervals and making inferences about population parameters.
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Confidence Interval

A confidence interval is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. The interval is associated with a confidence level, such as 99%, which indicates the probability that the interval will capture the true parameter if the experiment were repeated multiple times. Constructing a confidence interval for the population standard deviation involves using sample data to estimate the variability in the population.
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Chi-Square Distribution

The chi-square 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 standard deviations. When the population is normally distributed, the sample variance follows a chi-square distribution, allowing statisticians to calculate confidence intervals for the population standard deviation using the sample variance and the appropriate chi-square critical values.
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Related Practice
Textbook Question

The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)

b. Construct a 90% confidence interval for the population mean. Interpret the results.

Textbook Question

Use the standard normal distribution or the t-distribution to construct the indicated confidence interval for the population mean of each data set. Justify your decision. If neither distribution can be used, explain why. Interpret the results.

a. In a random sample of 40 patients, the mean waiting time at a dentist’s office was 20 minutes and the standard deviation was 7.5 minutes. Construct a 95% confidence interval for the population mean.

Textbook Question

The data set represents the weights (in pounds) of 10 randomly selected black bears from northeast Pennsylvania. Assume the weights are normally distributed. (Source: Pennsylvania Game Commission)

b. Construct a 95% confidence interval for the population mean. Interpret the results.

Textbook Question

In a survey of 2096 U.S. adults, 1740 think football teams of all levels should require players who suffer a head injury to take a set amount of time off from playing to recover. (Adapted from The Harris Poll)

a. Find the point estimate for the population proportion.

Textbook Question

Since 1935, the Gallup Organization has conducted public opinion polls in the United States and around the world. The table shows the results of Gallup’s World Affairs Poll of 2021, in which 1021 U.S. adults were polled. The remaining percentages not shown in the results are adults who were not sure.

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Find the minimum sample size needed to estimate, with 95% confidence, the population proportion of adults who feel that China’s economic power is a critical or an important economic threat to the United States. Your estimate must be accurate within 2% of the population proportion.

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Textbook Question

The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)

c. Would it be unusual for the population mean to be under 575? Explain.