Textbook Question
When all other quantities remain the same, how does the indicated change affect the minimum sample size requirement? Explain.
b. Increase in the error tolerance
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 6.3.20b
Alcohol-Impaired Driving You wish to estimate, with 95% confidence, the population proportion of motor vehicle fatalities that were caused by alcohol-impaired driving. Your estimate must be accurate within 5% of the population proportion.
b. Find the minimum sample size needed, using a prior study that found that 28% of motor vehicle fatalities were caused by alcohol-impaired driving. (Source: National Highway Traffic Safety Administration)
Verified step by step guidance1
Step 1: Identify the formula for determining the minimum sample size for estimating a population proportion. The formula is: n = (Z^2 * p * (1 - p)) / E^2, where n is the sample size, Z is the z-score corresponding to the desired confidence level, p is the estimated population proportion, and E is the margin of error.
Step 2: Determine the values for the variables in the formula. For a 95% confidence level, the z-score (Z) is approximately 1.96. The estimated population proportion (p) is 0.28, based on the prior study. The margin of error (E) is 0.05, as the estimate must be accurate within 5%.
Step 3: Substitute the values into the formula. Replace Z with 1.96, p with 0.28, and E with 0.05 in the equation: n = (1.96^2 * 0.28 * (1 - 0.28)) / 0.05^2.
Step 4: Simplify the numerator of the formula. Calculate 1.96^2, then multiply it by 0.28 and by (1 - 0.28), which is 0.72.
Step 5: Simplify the denominator of the formula. Calculate 0.05^2, then divide the simplified numerator by this value to find 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.
Population Proportion
The population proportion refers to the fraction of a population that exhibits a certain characteristic, in this case, the proportion of motor vehicle fatalities caused by alcohol-impaired driving. It is denoted as 'p' and is crucial for estimating the sample size needed to achieve a desired level of confidence in statistical studies.
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Sample Size Calculation
Sample size calculation is a statistical method used to determine the number of observations or replicates needed to ensure that the results of a study are statistically valid. In this context, it involves using the desired margin of error (5%) and the confidence level (95%) to calculate how many samples are necessary to accurately estimate the population proportion.
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Confidence Interval
A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For this question, a 95% confidence interval means that if the study were repeated multiple times, 95% of the calculated intervals would contain the true population proportion of alcohol-impaired driving fatalities.
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Related Practice
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