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Chapter 9: Problem 41

In a Gallup poll, \(64 \%\) of the people polled answered yes to the following question: "Are you in favor of the death penalty for a person convicted of murder?" The margin of error in the poll was \(3 \%,\) and the estimate was made with \(95 \%\) confidence. At least how many people were surveyed?

Short Answer

Expert verified
983 people were surveyed.

Step by step solution

01

Understand the Margin of Error and Confidence Level

The margin of error given is 3% which can be written as 0.03. The confidence level is 95%, which corresponds to a Z-score of 1.96.
02

Use the Formula for Sample Size

The formula to find the minimum sample size (n) is given by \[ n = \frac{Z^2 \times p \times (1-p)}{E^2} \]where:- \( Z \) is the Z-score,- \( p \) is the estimated proportion (0.64 in this case),- \( E \) is the margin of error.
03

Substitute the Values into the Formula

Substitute \( Z = 1.96 \), \( p = 0.64 \), and \( E = 0.03 \) into the formula:\[ n = \frac{(1.96)^2 \times 0.64 \times (1-0.64)}{(0.03)^2} \]
04

Calculate the Values Inside the Formula

First, calculate the numerator:\[ (1.96)^2 = 3.8416 \]\[ 0.64 \times (1-0.64) = 0.64 \times 0.36 = 0.2304 \]So the numerator becomes:\[ 3.8416 \times 0.2304 = 0.884736 \]Next, calculate the denominator:\[ (0.03)^2 = 0.0009 \]
05

Complete the Division to Find the Sample Size

Divide the numerator by the denominator:\[ n = \frac{0.884736}{0.0009} \ n = 982.96 \]
06

Round Up to the Nearest Whole Number

The sample size must be a whole number, so round up 982.96 to 983.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Margin of Error
The margin of error is a measure of the uncertainty or precision of a survey result. In this case, the margin of error is 3% or 0.03. It indicates that the true proportion of people in favor of the death penalty could vary by this amount in either direction from the sample proportion. The smaller the margin of error, the more precise the survey results are. Calculating it involves the sample size, population size, and variability in the data. Margin of error helps gauge the reliability of the survey's reported results.
Confidence Level
A confidence level represents the probability that the sample results accurately reflect the population parameters. Here, the confidence level is 95%. This means we can be 95% sure the true proportion falls within the margin of error range around the sample proportion. In statistical terms, it is how confident we are that repeated sampling would produce the same result. Higher confidence levels mean a broader margin of error, while a lower confidence level means a narrower margin of error. Common confidence levels include 90%, 95%, and 99%.
Z-score
The Z-score is a statistical measure that describes how many standard deviations an element is from the mean. For a 95% confidence level, the Z-score is 1.96. This score corresponds to the value on the standard normal distribution curve that captures 95% of the data under the curve. The Z-score is critical in calculating the margin of error and the required sample size. It helps translate a confidence interval into a probability metric that is used to gauge the reliability of the surveyed proportion.
Proportion
Proportion in statistics refers to the fraction of the sample that exhibits a certain characteristic. In this example, the proportion is 0.64 or 64%, indicating that 64% of the surveyed individuals are in favor of the death penalty. Proportion is often represented as 'p' in statistical formulas. It is used in conjunction with other statistical measures like margin of error and Z-score to perform more accurate and reliable sample size calculations. Understanding proportion is crucial for interpreting survey results.
Statistical Formula
The statistical formula for calculating the sample size is crucial for any survey research. The formula used here is \[ n = \frac{Z^2 \times p \times (1-p)}{E^2} \] where n is the required sample size. Substituting the variables: \[ n = \frac{(1.96)^2 \times 0.64 \times (1-0.64)}{(0.03)^2} \] This formula incorporates the Z-score, proportion, and margin of error to compute the minimum sample size needed to achieve a 95% confidence level. With calculated components: \[ n = \frac{0.884736}{0.0009} = 982.96 \] Finally, rounding to the nearest whole number gives the sample size of 983. This formula ensures that the survey results are statistically significant and reliable.

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Most popular questions from this chapter

A Gallup poll of 547 adult Americans employed full or part time asked, "Generally speaking, would you say your commute to work is - very stressful, somewhat stressful, not that stressful, or not stressful at all?" Gallup reported that \(24 \%\) of American workers said that their commute was "very" or "somewhat" stressful. The margin of error was 4 percentage points with \(95 \%\) confidence. Explain what this means.

A survey of 2306 adult Americans aged 18 and older conducted by Harris Interactive found that 417 have donated blood in the past two years. (a) Obtain a point estimate for the population proportion of adult Americans aged 18 and older who have donated blood in the past two years. (b) Verify that the requirements for constructing a confidence interval about \(p\) are satisfied. (c) Construct a \(90 \%\) confidence interval for the population proportion of adult Americans who have donated blood in the past two years. (d) Interpret the interval.

A researcher wishes to estimate the proportion of households that have broadband Internet access. What size sample should be obtained if she wishes the estimate to be within 0.03 with \(99 \%\) confidence if (a) she uses a 2009 estimate of 0.635 obtained from the National Telecommunications and Information Administration? (b) she does not use any prior estimates?

Construct the appropriate confidence interval. A simple random sample of size \(n=785\) adults was asked if they follow college football. Of the 785 surveyed, 275 responded that they did follow college football. Construct a \(95 \%\) confidence interval for the population proportion of adults who follow college football.

A simple random sample of size \(n<30\) for \(a\) quantitative variable has been obtained. Using the normal probability plot, the correlation between the variable and expected z-score, and the boxplot, judge whether a t-interval should be constructed. $$ n=13 ; \text { Correlation }=0.966 $$
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