Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.3.32

Chapter 6, Problem 6.3.32

Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.
In a survey of 880 unmarried U.S. adults who are living with a partner, 73% say love was a major reason why they decided to move in together. The survey’s margin of error is ±4.8%. (Source: Pew Research Center)

Verified step by step guidance

Step 1: Understand the problem. The problem provides a sample proportion (73%), a sample size (880), and a margin of error (±4.8%). The goal is to translate this information into a confidence interval and approximate the confidence level.

Step 2: Write the formula for a confidence interval for a population proportion. The general formula is:

p̂ ± E

, where

p̂

is the sample proportion and

E

is the margin of error.

Step 3: Substitute the given values into the formula. Here,

p̂ = 0.73

(73% as a decimal) and

E = 0.048

(4.8% as a decimal). The confidence interval is:

0.73 ± 0.048

Step 4: Calculate the lower and upper bounds of the confidence interval. The lower bound is

0.73 - 0.048

, and the upper bound is

0.73 + 0.048

. These bounds represent the range of plausible values for the population proportion.

Step 5: Approximate the confidence level. The margin of error (±4.8%) suggests a typical confidence level of 95%, as this is a common level used in surveys. However, the exact confidence level would depend on the critical value (z*) used in the calculation, which is not explicitly provided in the problem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed as an interval estimate, typically calculated as the sample proportion plus or minus the margin of error. For example, if 73% is the sample proportion and the margin of error is ±4.8%, the confidence interval would be from 68.2% to 77.8%.

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate. It indicates the range within which the true population parameter is expected to fall, based on the sample data. In this case, a margin of error of ±4.8% means that the true percentage of unmarried U.S. adults who believe love was a major reason for cohabitation could vary by this amount from the reported 73%.

Level of Confidence

The level of confidence reflects the degree of certainty that the confidence interval contains the true population parameter. Common levels of confidence are 90%, 95%, and 99%, with higher levels indicating greater certainty but wider intervals. The level of confidence can be approximated based on the sample size and the margin of error, often using standard normal distribution values.

Recommended video:

06:33

Introduction to Confidence Intervals

Related Practice

Textbook Question

In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.

c = 0.90, s^2 = 35, n = 18

Textbook Question

In Exercises 21–24, construct the indicated confidence interval for the population mean μ.

c = 0.80, xbar = 20.6, σ = 4.7, n = 100.

views

Textbook Question

"Finding p^ and q^ In Exercises 3–6, let p be the population proportion for the situation. Find point estimates of p and q.

Tax Fraud In a survey of 1040 U.S. adults, 62 have had someone impersonate them to try to claim tax refunds. (Adapted from Pew Research Center)"

Textbook Question

Does a population have to be normally distributed to use the chi-square distribution?

Textbook Question

Constructing a Confidence Interval In Exercises 17–20, you are given the sample mean and the sample standard deviation. Assume the population is normally distributed and use the t-distribution to find the margin of error and construct a 95% confidence interval for the population mean. Interpret the results.

Commute Time In a random sample of eight people, the mean commute time to work was 35.5 minutes and the standard deviation was 7.2 minute

Textbook Question

Constructing Confidence Intervals In Exercises 25 and 26, use the figure, which shows the results of a survey in which 1051 adults from France, 1042 adults from Germany, 1003 adults from the United Kingdom, and 1000 adults from the United States were asked whether national identity is strongly tied to birthplace. (Source: Pew Research Center)

National Identity Construct a 99% confidence interval for the population proportion of adults who say national identity is strongly tied to birthplace for each country listed.