Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.4.14a

Chapter 6, Problem 6.4.14a

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance σ^2. Interpret the results.
Drug Concentration The times (in minutes) for the drug concentration to peak when the drug epinephrine is injected into 15 randomly selected patients are listed. Use a 90% level of confidence.

Verified step by step guidance

Step 1: Calculate the sample variance (s^2). First, compute the sample mean (x̄) by summing all the data points and dividing by the number of observations (n = 15). Then, use the formula for variance: s^2 = (Σ(xi - x̄)^2) / (n - 1), where xi represents each data point.

Step 2: Identify the chi-square critical values for the given confidence level (90%) and degrees of freedom (df = n - 1 = 14). Use a chi-square distribution table or calculator to find the values for χ^2_lower and χ^2_upper.

Step 3: Construct the confidence interval for the population variance (σ^2) using the formula: [(n - 1) * s^2 / χ^2_upper, (n - 1) * s^2 / χ^2_lower]. This formula uses the sample variance and the chi-square critical values.

Step 4: To interpret the results, note that the confidence interval provides a range of plausible values for the population variance. A 90% confidence level means that if we were to repeat this process many times, 90% of the intervals would contain the true population variance.

Step 5: If needed, you can also calculate the confidence interval for the population standard deviation (σ) by taking the square root of the lower and upper bounds of the variance 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 true population parameter with a specified level of confidence. For example, a 90% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 90% of those intervals would contain the true population parameter.

Population Variance (σ²)

Population variance is a measure of the dispersion of a set of values in a population. It quantifies how much the values in the population differ from the population mean. In the context of confidence intervals, estimating the population variance is crucial for determining the width of the interval, as it affects the standard error of the mean.

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. Many statistical methods, including the construction of confidence intervals, assume that the underlying population is normally distributed, especially when sample sizes are small.

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Related Practice

Textbook Question

Constructing a Confidence Interval In Exercises 25–28, use the data set to (a) find the sample mean. Assume the population is normally distributed.

Homework The weekly time spent (in hours) on homework for 18 randomly selected high school students

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Textbook Question

Finite Population Correction Factor In Exercises 57 and 58, use the information below.

In this section, you studied the construction of a confidence interval to estimate a population mean. In each case, the underlying assumption was that the sample size n was small in comparison to the population size N. When n ≥ 0.05N however, the formula that determines the standard error of the mean needs to be adjusted, as shown below.

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Recall from the Section 5.4 exercises that the expression sqrt[(N-n)/(n-1)] is called a finite population correction factor. The margin of error is

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Use the finite population correction factor to construct each confidence interval for the population mean.

a. c = 0.99, xbar = 8.6, σ = 4.9, N = 200, n = 25.

197

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Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

a. Increase in the level of confidence

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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

b. What was the greatest value you obtained for p^?

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Textbook Question

Use technology to find a 95% confidence interval for the population proportion of adults who

a. view foreign trade as an economic opportunity.

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