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Chapter 9: Problem 39

A Gallup poll conducted May \(20-22,2005\) asked Americans how many books, either hardback or paperback, they read during the previous year. How many subjects are needed to estimate the number of books Americans read the previous year within one book with \(95 \%\) confidence? Initial survey results indicate that \(\sigma=16.6\) books.

Short Answer

Expert verified
The minimum sample size needed is 1059 subjects.

Step by step solution

01

Understand the Problem

Determine the sample size required to estimate the number of books read with a certain confidence level and margin of error, given the standard deviation.
02

Identify Given Values

The standard deviation \(\sigma\) is given as \(16.6\) books. The desired margin of error \(E\) is \(1\) book. The confidence level is \(95\%\), which corresponds to a Z-score of approximately \(1.96\).
03

Use the Sample Size Formula

Use the formula for determining sample size \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \] where \(n\) is the sample size, \(Z\) is the Z-score, \(\sigma\) is the standard deviation, and \(E\) is the margin of error.
04

Substitute the Values

Plug the given values into the formula: \[ n = \left( \frac{1.96 \cdot 16.6}{1} \right)^2 \]
05

Calculate the Sample Size

Perform the calculation: \[ n = \left( 32.536 \right)^2 = 1058.779 \] Round up to the nearest whole number since the sample size must be a whole number, so \(n = 1059\).

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Key Concepts

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

Confidence Interval
A confidence interval is a range of values used to estimate a population parameter. When we say we have a confidence level of 95%, it means we are 95% certain that the population parameter lies within this range. The confidence interval consists of the sample statistic (like the mean) plus and minus the margin of error. This concept helps us understand the degree of uncertainty around our sample estimates, making it an essential tool in statistics. For instance, in the Gallup poll example, the confidence interval helps estimate how many books Americans read in the previous year with high certainty.
Margin of Error
The margin of error is the amount allowed for in case of miscalculation or change of circumstances. It gives a buffer zone around the sample statistic. In the exercise, the desired margin of error is 1 book, meaning we want our estimate to be within 1 book of the actual number people read. The margin of error is directly related to the sample size: a smaller margin of error requires a larger sample size. This buffer ensures that the findings are reliable even if there are small variations in the data.
Standard Deviation
Standard deviation \(\sigma\) measures how spread out numbers are in a dataset. A large standard deviation indicates that data points are far from the mean, while a small one shows they are close. In the problem, the standard deviation of 16.6 books indicates variability in the number of books read. Knowing the standard deviation helps us understand the spread of our data and is critical for determining the necessary sample size using the sample size formula.
Z-Score
The Z-score measures how many standard deviations an element is from the mean. It is used in statistics to denote the number of standard deviations a data point is from the mean of a data set. For a 95% confidence interval, the Z-score is approximately 1.96. This score helps in standardizing data points and is crucial for calculating the sample size. By converting the confidence level to a Z-score, we can better interpret our data and calculate the required sample size.

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Most popular questions from this chapter

A simple random sample of size \(n\) is drawn from a population that is normally distributed with population standard deviation, \(\sigma,\) known to be \(13 .\) The sample mean, \(\bar{x},\) is found to be 108. (a) Compute the \(96 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is 25 (b) Compute the \(96 \%\) confidence interval about \(\mu\) if the sample size, \(n\), is \(10 .\) How does decreasing the sample size affect the margin of error, \(E ?\) (c) Compute the \(88 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(25 .\) Compare the results to those obtained in part (a). How does decreasing the level of confidence affect the size of the margin of error, \(E ?\) (d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? Why? (e) Suppose an analysis of the sample data revealed three outliers greater than the mean. How would this affect the confidence interval?

State the circumstances under which we construct a \(t\) interval. What are the circumstances under which we construct a Z-interval?

Taking It Easy In a Gallup poll conducted December \(11-14,2003,455\) of 1011 randomly selected adults aged 18 and older said they had too little time for relaxing or doing nothing. (a) Verify that the requirements for constructing a confidence interval about \(\hat{p}\) are satisfied. (b) Construct a \(92 \%\) confidence interval for the proportion of adults aged 18 and older who say they have too little time for relaxing or doing nothing. Interpret this interval. (c) Construct a \(96 \%\) confidence interval for the proportion of adults aged 18 and older who say they have too little time for relaxing or doing nothing. Interpret this interval. (d) What is the effect of increasing the level of confidence on the width of the interval?

A student constructs a \(95 \%\) confidence interval for the mean age of students at his college. The lower bound is 21.4 years and the upper bound is 28.8 years. He interprets the interval as, "There is a \(95 \%\) probability that the mean age of a student is between 21.4 years and 28.8 years" What is wrong with this interpretation? What could be done to increase the precision of the interval?

Death Penalty In a Harris Poll conducted in July 2000 , \(64 \%\) of the people polled answered yes to the following question: "Do you believe in capital punishment, that is, the death penalty, or are you opposed to it?" The margin of error in the poll was \(\pm 3 \%\) = 卤0.03 , and the estimate was made with \(95 \%\) confidence. How many people were surveyed?
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