Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.1d

Chapter 6, Problem 6.1d

In a survey of 2096 U.S. adults, 1740 think football teams of all levels should require players who suffer a head injury to take a set amount of time off from playing to recover. (Adapted from The Harris Poll)
d. Find the minimum sample size needed to estimate the population proportion at the 99% confidence level to ensure that the estimate is accurate within 3% of the population proportion.

Verified step by step guidance

Step 1: Identify the key components of the problem. The confidence level is 99%, the margin of error (E) is 3% or 0.03, and the sample proportion (p̂) is calculated as the number of favorable responses divided by the total sample size: p̂ = 1740 / 2096.

Step 2: Recall the formula for the minimum sample size needed to estimate a population proportion: n = (Z² * p̂ * (1 - p̂)) / E². Here, Z is the critical value corresponding to the 99% confidence level.

Step 3: Determine the critical value (Z) for a 99% confidence level. Using a Z-table or standard normal distribution, the Z-value for 99% confidence is approximately 2.576.

Step 4: Substitute the values into the formula. Use p̂ from Step 1, Z = 2.576, and E = 0.03. The formula becomes: n = (2.576² * p̂ * (1 - p̂)) / 0.03².

Step 5: Simplify the expression to calculate the minimum sample size. Ensure that the result is rounded 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 context, it represents the percentage of U.S. adults who believe that football players should take time off after a head injury. Understanding this concept is crucial for estimating how representative a sample is of the larger population.

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 survey or experiment are reliable. It takes into account the desired confidence level, margin of error, and the estimated population proportion. In this case, it helps to find the minimum number of respondents needed to accurately estimate the population proportion within a specified margin of error.

Confidence Level

The confidence level indicates the degree of certainty that the population parameter lies within the confidence interval calculated from the sample data. A 99% confidence level means that if the same survey were conducted multiple times, 99% of the calculated intervals would contain the true population proportion. This high level of confidence is important for making informed decisions based on survey results.

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Introduction to Confidence Intervals

Related Practice

Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

c. Increase in the population standard deviation

Textbook Question

Finite Population Correction Factor In Exercises 57 and 58, use the information below.

In this section, you studied the construction of a confidence interval to estimate a population mean. In each case, the underlying assumption was that the sample size n was small in comparison to the population size N. When n ≥ 0.05N however, the formula that determines the standard error of the mean needs to be adjusted, as shown below.

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Recall from the Section 5.4 exercises that the expression sqrt[(N-n)/(n-1)] is called a finite population correction factor. The margin of error is

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Use the finite population correction factor to construct each confidence interval for the population mean.

c. c = 0.95, xbar = 40.3, σ = 0.5, N = 300, n = 68.

Textbook Question

Constructing a Confidence Interval In Exercises 25–28, use the data set to (c) construct a 99% confidence interval for the population mean. Assume the population is normally distributed.

Homework The weekly time spent (in hours) on homework for 18 randomly selected high school students