Textbook Question
Graphical Analysis In Exercises 9–12, use the values on the number line to find the sampling error.
Ch. 6 - Confidence Intervals
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 6, Problem 6.3.12
Constructing Confidence Intervals In Exercises 11 and 12, construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals.
New Year’s Resolutions In a survey of 1790 U.S. adults in a recent year, 816 have a New Year’s resolution related to their health. (Adapted from Finder)
Verified step by step guidance1
Step 1: Identify the sample proportion (p̂) and the sample size (n). The sample proportion is calculated as p̂ = x / n, where x is the number of successes (816) and n is the total sample size (1790).
Step 2: Determine the critical z-scores for the desired confidence levels. For a 90% confidence interval, the critical z-score is approximately 1.645. For a 95% confidence interval, the critical z-score is approximately 1.96.
Step 3: Calculate the standard error (SE) of the sample proportion using the formula SE = sqrt((p̂ * (1 - p̂)) / n).
Step 4: Construct the confidence intervals using the formula: CI = p̂ ± (z * SE), where z is the critical z-score for the desired confidence level. Perform this calculation separately for the 90% and 95% confidence levels.
Step 5: Compare the widths of the confidence intervals. The width of a confidence interval is given by 2 * (z * SE). Note that the 95% confidence interval will be wider than the 90% confidence interval because a higher confidence level requires a larger margin of error.
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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, such as 90% or 95%. It provides an estimate of uncertainty around the sample proportion, indicating how much the sample result might vary if different samples were taken.
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Introduction to Confidence Intervals
Population Proportion
The population proportion is the ratio of members of a population that have a particular characteristic, expressed as a fraction or percentage. In this context, it refers to the proportion of U.S. adults who have a New Year’s resolution related to their health, which is estimated from the sample data.
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05:45Constructing Confidence Intervals for Proportions
Width of Confidence Intervals
The width of a confidence interval reflects the precision of the estimate; narrower intervals indicate more precise estimates of the population parameter. The width is influenced by the sample size and the confidence level chosen; higher confidence levels result in wider intervals, while larger sample sizes yield narrower intervals, allowing for more accurate estimates.
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Related Practice
Textbook Question
In Exercises 5–8, find the critical value zc necessary to construct a confidence interval at the level of confidence c.
c = 0.80
Textbook Question
Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.
c = 0.95, n = 20
Textbook Question
You research prices of cell phones and find that the population mean is \$431.61. In Exercise 19, does the t-value fall between -t0.95 and t0.95?
Textbook Question
Finding p^ and q^ In Exercises 3–6, let p be the population proportion for the situation. Find point estimates of p and q.
Private Internet Browsing In a survey of 4272 U.S. adults, 1025 knew that private browsing mode only prevents someone using the same computer from seeing one’s online activities. (Adapted from Pew Research Center)
Textbook Question
In Exercises 35–40, use the standard normal distribution or the t-distribution to construct a 95% confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results.
The population standard deviation of the weights of the two-year-old males on a pediatrician’s patient list is 2.49 pounds. The mean weight of a sample of 10 of the two–year–old males is 13.68 pounds. Weights are known to be normally distributed.