Skip to main content
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.R.30
Chapter 6, Problem 6.R.30

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.
c = 0.99, n = 10

Verified step by step guidance
1
Step 1: Understand the problem. The goal is to find the critical values for a given confidence level (c = 0.99) and sample size (n = 10). Critical values are used in hypothesis testing and confidence intervals to determine the range of values that are likely to contain the population parameter.
Step 2: Determine the degrees of freedom (df). For a t-distribution, the degrees of freedom are calculated as df = n - 1. Since n = 10, calculate df = 10 - 1.
Step 3: Identify the confidence level and the corresponding significance level (α). The confidence level is c = 0.99, so the significance level is α = 1 - c = 1 - 0.99.
Step 4: Divide the significance level (α) by 2 to find the area in each tail of the t-distribution. This is because the t-distribution is symmetric, and the critical values are located at the tails. Calculate α/2.
Step 5: Use a t-distribution table or statistical software to find the critical t-values for the given degrees of freedom (df) and the area in each tail (α/2). These critical values will correspond to the boundaries of the confidence interval.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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.
Recommended video:
05:50
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 0.99, 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.
Recommended video:
06:33
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 10 may limit the precision of the confidence interval, affecting the critical values derived from it.
Recommended video:
05:11
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?

84
views
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

97
views
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

73
views
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)

61
views
Textbook Question

In Exercises 19–22, let p be the population proportion for the situation. (a) Find point estimates of p and q, (b) construct 90% and 95% confidence intervals for p, and (c) interpret the results of part (b) and compare the widths of the confidence intervals.

In a survey of 912 U.S. adults in Generation Z (born after 1996), 383 said they are at least somewhat likely to consider an electric vehicle for their next vehicle purchase. (Adapted from Pew Research Center)

89
views
Textbook Question

You wish to estimate, with 95% confidence, the population proportion of U.S. adults who have taken or planned to take a winter vacation in a recent year. Your estimate must be accurate within 5% of the population proportion.

a. No preliminary estimate is available. Find the minimum sample size needed.

119
views