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.R.25a

Chapter 6, Problem 6.R.25a

You wish to estimate, with 95% confidence, the population proportion of U.S. adults who have taken or planned to take a winter vacation in a recent year. Your estimate must be accurate within 5% 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 = \frac{z^{2}}{}

, where

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 = \frac{1}{2}

(or 0.5). This value maximizes the product

p (1 - p)

, ensuring the sample size is large enough.

Identify the z-score corresponding to a 95% confidence level. For a 95% confidence level, the z-score is approximately 1.96. This value is derived from the standard normal distribution.

Set the margin of error

E

to 0.05 (5%). This is the desired level of accuracy for the estimate.

Substitute the values into the formula:

n = \frac{{1.96}^{2}}{}

. Simplify the expression to calculate the minimum sample size needed.

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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. adults who have taken or planned to take a winter vacation. It is denoted by 'p' and is crucial for estimating how widespread a behavior or opinion is within a 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 study are statistically valid. In this context, it involves using the desired confidence level and margin of error to find the minimum number of respondents required to accurately estimate the population proportion.

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 reflects the uncertainty associated with estimating the population proportion and is influenced by the sample size and variability in the data.

Recommended video:

06:33

Introduction to Confidence Intervals

Related Practice

Textbook Question

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.

c = 0.95, n = 13

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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 = 0.98, n = 25

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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 = 0.98, s = 0.9, n = 12, xbar = 6.8

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Textbook Question

In Exercises 5 and 6, use the confidence interval to find the margin of error and the sample mean.

(7.428, 7.562)

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Textbook Question

In Exercises 19–22, let p be the population proportion for the situation. (a) Find point estimates of p and q, (b) construct 90% and 95% confidence intervals for p, and (c) interpret the results of part (b) and compare the widths of the confidence intervals.

In a survey of 912 U.S. adults in Generation Z (born after 1996), 383 said they are at least somewhat likely to consider an electric vehicle for their next vehicle purchase. (Adapted from Pew Research Center)

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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 = 0.99, n = 10

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