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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.27
Chapter 6, Problem 6.RE.27

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.
c = 0.95, n = 13

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Step 1: Understand the problem. You are tasked with finding the critical values for a given confidence level (c = 0.95) and sample size (n = 13). Critical values are used in hypothesis testing and confidence intervals to determine the range within which the true population parameter lies.
Step 2: Identify the appropriate distribution. Since the sample size is small (n < 30), and assuming the population standard deviation is unknown, you will use the t-distribution to find the critical values.
Step 3: Determine the degrees of freedom (df). The degrees of freedom for the t-distribution is calculated as: df=n-1. For this problem, df=13-1=12.
Step 4: Use the confidence level to find the critical values. The confidence level c = 0.95 corresponds to a two-tailed test, meaning the area in each tail is 12(1-c). For c = 0.95, the area in each tail is 12(1-0.95)=0.025. The critical values correspond to the t-scores where the cumulative probability equals 0.025 in the lower tail and 0.975 in the upper tail.
Step 5: Look up the t-scores in a t-distribution table or use statistical software. Using the degrees of freedom (df = 12) and the cumulative probabilities (0.025 and 0.975), find the critical values. These values will be symmetric around zero, with one positive and one negative.

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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 (c)

The level of confidence, denoted as 'c', represents the probability that the confidence interval will contain the true population parameter. A common level of confidence is 95%, which implies that if we were to take many samples and construct confidence intervals, approximately 95% of those intervals would contain the true parameter. This level influences the width of the confidence interval.
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Introduction to Confidence Intervals

Sample Size (n)

Sample size, denoted as 'n', refers to the number of observations or data points collected in a study. The sample size affects the precision of the estimates and the width of the confidence intervals; larger sample sizes generally lead to more reliable estimates and narrower intervals. In this case, with n = 13, the sample size is relatively small, which may impact the critical values derived from the statistical distribution.
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Sampling Distribution of Sample Proportion
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 = 0.98, n = 25

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

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)

d. Determine the minimum sample size required to be 95% confident that the sample mean test score is within 10 points of the population mean test score.

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?