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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.RE.28
Chapter 6, Problem 6.RE.28

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.
c = 0.98, n = 25

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Step 1: Understand the problem. The critical values are associated with the level of confidence (c) and the sample size (n). The level of confidence c = 0.98 indicates that we are looking for the critical values corresponding to a 98% confidence interval.
Step 2: Determine the degrees of freedom (df). For a t-distribution, the degrees of freedom are calculated as df = n - 1, where n is the sample size. In this case, n = 25, so df = 25 - 1 = 24.
Step 3: Identify the tail areas. Since the confidence level is 98%, the remaining area in the tails is 1 - 0.98 = 0.02. Divide this equally between the two tails, so each tail has an area of 0.01.
Step 4: Use a t-distribution table or statistical software to find the critical t-value for df = 24 and a tail area of 0.01. This critical t-value corresponds to the point where the cumulative probability equals 0.99 (since 1 - 0.01 = 0.99).
Step 5: Interpret the critical values. The critical values for a 98% confidence interval are symmetric around the mean, so they will be ±t, where t is the critical t-value obtained in Step 4.

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Key Concepts

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

Critical Values

Critical values are the points on the scale of the test statistic that define the boundaries for rejecting the null hypothesis. They are determined based on the desired level of confidence and the distribution of the test statistic. For example, in a normal distribution, critical values correspond to specific z-scores that capture the central area of the distribution, reflecting the confidence level.
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Critical Values: t-Distribution

Level of Confidence

The level of confidence, denoted as 'c', represents the probability that the confidence interval will contain the true population parameter. A higher confidence level, such as 0.98, indicates a greater certainty that the interval includes the parameter, but it also results in a wider interval. This concept is crucial for understanding how confident we can be in our estimates based on sample data.
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Introduction to Confidence Intervals

Sample Size

Sample size, denoted as 'n', refers to the number of observations or data points collected in a study. It plays a significant role in statistical analysis, as larger sample sizes generally lead to more reliable estimates and narrower confidence intervals. In this context, a sample size of 25 indicates the number of individuals or items from which data is gathered to estimate the population parameters.
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Related Practice
Textbook Question

The Safe Drinking Water Act, which was passed in 1974, allows the Environmental Protection Agency (EPA) to regulate the levels of contaminants in drinking water. The EPA requires that water utilities give their customers water quality reports annually. These reports include the results of daily water quality monitoring, which is performed to determine whether drinking water is safe for consumption. A water department tests for contaminants at water treatment plants and at customers’ taps. These contaminants include microorganisms, organic chemicals, and inorganic chemicals, such as cyanide. Cyanide’s presence in drinking water is the result of discharges from steel, plastics, and fertilizer factories. For drinking water, the maximum contaminant level of cyanide is 0.2 parts per million. As part of your job for your city’s water department, you are preparing a report that includes an analysis of the results shown in the figure at the right. The figure shows the point estimates for the population mean concentration and the 95% confidence intervals for cyanide over a three-year period. The data are based on random water samples taken by the city’s three water treatment plants.

What can the water department do to decrease the size of the confidence intervals, regardless of the amount of variance in cyanide levels?

Textbook Question

(a) Construct a 90% confidence interval for the population mean in Exercise 1. Interpret the results. (b) Does it seem likely that the population mean could be within 10% of the sample mean? Explain.

Textbook Question

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.

c = 0.95, n = 13

Textbook Question

In Exercises 5 and 6, use the confidence interval to find the margin of error and the sample mean.

(7.428, 7.562)

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Textbook Question

Determine the minimum sample size required to be 99% confident that the sample mean driving distance to work is within 2 miles of the population mean driving distance to work. Use the population standard deviation from Exercise 2.

Textbook Question

The Safe Drinking Water Act, which was passed in 1974, allows the Environmental Protection Agency (EPA) to regulate the levels of contaminants in drinking water. The EPA requires that water utilities give their customers water quality reports annually. These reports include the results of daily water quality monitoring, which is performed to determine whether drinking water is safe for consumption. A water department tests for contaminants at water treatment plants and at customers’ taps. These contaminants include microorganisms, organic chemicals, and inorganic chemicals, such as cyanide. Cyanide’s presence in drinking water is the result of discharges from steel, plastics, and fertilizer factories. For drinking water, the maximum contaminant level of cyanide is 0.2 parts per million. As part of your job for your city’s water department, you are preparing a report that includes an analysis of the results shown in the figure at the right. The figure shows the point estimates for the population mean concentration and the 95% confidence intervals for cyanide over a three-year period. The data are based on random water samples taken by the city’s three water treatment plants.

The confidence interval for Year 2 is much larger than that for the other years. What do you think may have caused this larger confidence level?