Textbook Question
True or False? In Exercises 1 and 2, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
The point estimate for the population proportion of failures is 1-p^
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 6.1.31
In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.
c = 0.80, σ = 4.1, E = 2.
Verified step by step guidance1
Identify the formula for determining the minimum sample size n for estimating a population mean: , where zc is the critical z-value, σ is the population standard deviation, and E is the margin of error.
Determine the critical z-value (zc) for the given confidence level c = 0.80. Use a z-table or statistical software to find the z-value corresponding to the middle 80% of the standard normal distribution.
Substitute the given values into the formula: c = 0.80, σ = 4.1, and E = 2. Replace zc with the critical z-value obtained in the previous step.
Simplify the numerator by squaring the critical z-value (zc) and multiplying it by the square of the population standard deviation (σ²).
Divide the result from the numerator by the square of the margin of error (E²) to calculate the minimum sample size n. Round up to the nearest whole number, as sample size must be an integer.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sample Size Determination
Sample size determination is the process of calculating the number of observations or replicates needed in a statistical study to ensure that the results are reliable and valid. It is crucial for achieving a desired level of confidence and margin of error in estimates. The formula often used involves the population standard deviation, the desired confidence level, and the margin of error.
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06:14
Coefficient of Determination
Confidence Level (c)
The confidence level represents the probability that the sample accurately reflects the population parameter within a specified margin of error. A confidence level of 0.80, for instance, indicates that if the same sampling procedure were repeated multiple times, approximately 80% of the intervals would contain the true population parameter. This concept is essential for understanding the reliability of the estimate.
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Margin of Error (E)
The margin of error is the range within which the true population parameter is expected to fall, given a certain confidence level. It quantifies the uncertainty associated with the sample estimate. A smaller margin of error requires a larger sample size, as it indicates a more precise estimate of the population parameter, which is critical for making informed decisions based on the data.
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Related Practice
Textbook Question
LGBT Identification In a survey of 15,349 U.S. adults, 860 identify as lesbian, gay, bisexual, or transgender. Construct a 95% confidence interval for the population proportion of U.S. adults who identify as lesbian, gay, bisexual, or transgender. (Adapted from Gallup)
Textbook Question
What happens to the shape of the chi-square distribution as the degrees of freedom increase?
Textbook Question
When estimating the population mean, why not construct a 99% confidence interval every time?
Textbook Question
In Exercise 31, the population mean salary is \$67,319. Does the t-value fall between -t0.98 and t0.98? (Source: Salary.com)
Textbook Question
Matching In Exercises 17–20, match the level of confidence c with the appropriate confidence interval. Assume each confidence interval is constructed for the same sample statistics.
c = 0.98