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 6.1.5
In Exercises 5–8, find the critical value zc necessary to construct a confidence interval at the level of confidence c.
c = 0.80
Verified step by step guidance1
Identify the level of confidence, c, which is given as 0.80. This represents the proportion of the area under the standard normal curve that corresponds to the confidence interval.
Calculate the area in the tails of the standard normal distribution. Since the confidence level is 0.80, the remaining area in the tails is 1 - 0.80 = 0.20. Divide this equally between the two tails, so each tail has an area of 0.10.
Determine the cumulative area to the left of the critical value zc. Since the area in the left tail is 0.10, the cumulative area to the left of zc is 0.10 + 0.80 = 0.90.
Use a standard normal distribution table (z-table) or a statistical calculator to find the z-score that corresponds to a cumulative area of 0.90. This z-score is the critical value zc.
Verify the critical value zc by ensuring that the area between -zc and zc under the standard normal curve equals the confidence level, c = 0.80.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Value (zc)
The critical value, denoted as zc, is a point on the standard normal distribution that corresponds to a specified level of confidence. It defines the boundaries of the confidence interval, indicating how far from the mean we can expect the true population parameter to lie with a certain level of certainty. For example, a confidence level of 80% means that 80% of the distribution lies within zc standard deviations from the mean.
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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. It is calculated using the sample mean, the critical value, and the standard error. For instance, an 80% confidence interval suggests that if we were to take many samples, approximately 80% of the calculated intervals would contain the true parameter.
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Standard Normal Distribution
The standard normal distribution is a probability distribution that has a mean of 0 and a standard deviation of 1. It is used to determine the critical values for various confidence levels. By converting a normal distribution to a standard normal distribution, we can easily find zc values using z-tables or statistical software, which helps in constructing confidence intervals.
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Related Practice
Textbook Question
Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.
In a survey of 2094 U.S. adults who have used an online dating app, 57% said their personal experience with online dating was positive. The survey’s margin of error is ±3.6%. (Source: Pew Research Center)
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 = 0.99, n = 30
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
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.
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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?