Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.3.17a

Chapter 6, Problem 6.3.17a

Congress You wish to estimate, with 95% confidence, the population proportion of likely U.S. voters who think Congress is doing a good or excellent job. Your estimate must be accurate within 4% of the population proportion.
a. No preliminary estimate is available. Find the minimum sample size needed.

Verified step by step guidance

Determine the formula for the minimum sample size needed to estimate a population proportion with a given margin of error. The formula is:

n = (z² * p * (1 - p)) / E²

, where

n

is the sample size,

z

is the z-score corresponding to the confidence level,

p

is the estimated population proportion, and

E

is the margin of error.

Since no preliminary estimate of the population proportion is available, use

p = 0.5

. This value maximizes the product

p * (1 - p)

, ensuring the sample size is large enough.

Identify the z-score for a 95% confidence level. For a 95% confidence level, the z-score is approximately

1.96

Substitute the values into the formula:

n = (1.96² * 0.5 * (1 - 0.5)) / 0.04²

. Here,

E = 0.04

represents the margin of error (4%).

Simplify the expression step by step 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.

Population Proportion

The population proportion refers to the fraction of a population that possesses a certain characteristic, in this case, U.S. voters who believe Congress is performing well. It is denoted by 'p' and is crucial for estimating how representative a sample will be in reflecting the views of the entire 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 sample accurately reflects the population. For estimating proportions, the formula incorporates the desired confidence level, margin of error, and the estimated population proportion to derive the minimum sample size required.

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, such as 95%. It provides a measure of uncertainty around the estimate and is essential for understanding the precision of the sample proportion in relation to the true population proportion.

Recommended video:

06:33

Introduction to Confidence Intervals

Related Practice

Textbook Question

Constructing a Confidence Interval In Exercises 25–28, use the data set to (a) find the sample mean. Assume the population is normally distributed.

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Textbook Question

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance σ^2. Interpret the results.

[APPLET] Earnings The annual earnings (in thousands of dollars) of 21 randomly selected level 1 computer hardware engineers are listed. Use a 99% level of confidence. (Adapted from Salary.com)

Textbook Question

Car Batteries The reserve capacities (in hours) of 18 randomly selected automotive batteries have a sample standard deviation of 0.25 hour. Use an 80% level of confidence.

Textbook Question

Soccer Balls A soccer ball manufacturer wants to estimate the mean circumference of soccer balls within 0.15 inch.

a. Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Assume the population standard deviation is 0.5 inch

Textbook Question

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Textbook Question

Paint Can Volumes A paint manufacturer uses a machine to fill gallon cans with paint (see figure). The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.5 ounce. Assume the population of volumes is normally distributed.

a. Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 0.75 ounce.