Textbook Question
Determine the minimum sample size required to be 95% confident that the sample mean waking time is within 10 minutes of the population mean waking time. Use the population standard deviation from Exercise 1.
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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.R.11
In Exercises 9–12, find the critical value tc for the level of confidence c and sample size n.
c = 0.98, n = 15
Verified step by step guidance1
Determine the degrees of freedom (df) for the t-distribution. The formula for degrees of freedom is df = n - 1, where n is the sample size. In this case, df = 15 - 1.
Identify the level of confidence (c). Here, c = 0.98, which means the area in the middle of the t-distribution is 0.98, leaving 0.02 in the two tails combined.
Divide the remaining area (0.02) equally between the two tails to find the area in one tail. This is 0.02 / 2 = 0.01.
Use a t-distribution table or a statistical calculator to find the critical value (tc) that corresponds to the area in one tail (0.01) and the degrees of freedom (df = 14).
Verify the critical value (tc) by ensuring it matches the level of confidence (c = 0.98) and the degrees of freedom (df = 14).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Value
A critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis. In the context of confidence intervals, it represents the value that separates the confidence level from the tail probabilities. For a given confidence level, it helps determine the margin of error in estimating population parameters.
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Critical Values: t-Distribution
t-Distribution
The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It is used when the sample size is small (typically n < 30) and the population standard deviation is unknown. The t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample.
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Degrees of Freedom
Degrees of freedom (df) refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of the t-distribution, degrees of freedom are calculated as n - 1, where n is the sample size. This value is crucial for determining the appropriate critical value from the t-distribution table.
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Related Practice
Textbook Question
[APPLET] The waking times (in minutes past 5:00 A.M.) of 40 people who start work at 8:00 A.M. are shown in the table at the left. Assume the population standard deviation is 45 minutes. Find (a) the point estimate of the population mean μ and (b) the margin of error for a 90% confidence interval.
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Textbook Question
[APPLET] The winning times (in hours) for a sample of 20 randomly selected Boston Marathon Women’s Open Division champions from 1980 to 2019 are shown in the table at the left. Assume the population standard deviation is 0.068 hour. (Source: Boston Athletic Association)
b. Find the margin of error for a 95% confidence level.
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Textbook Question
In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.
c = 0.99, s = 16.5, n = 20, xbar = 25.2
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Textbook Question
In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.
c = 0.90, n = 16
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Textbook Question
In a random sample of 12 senior-level civil engineers, the mean annual earnings were \$133,326 and the standard deviation was \$36,729. Assume the annual earnings are normally distributed and construct a 95% confidence interval for the population mean annual earnings for senior-level civil engineers. Interpret the results. (Adapted from Salary.com)
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