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Chapter 9: Problem 40

A Gallup poll conducted January \(23,2003-\) February \(10,2003,\) asked American teens (aged 13 to 17 ) how much time they spent each week using the Internet. How many subjects are needed to estimate the time American teens spend on the Internet each week within 0.5 hour with \(95 \%\) confidence? Initial survey results indicate that \(\sigma=6.6\) hours.

Short Answer

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671 subjects

Step by step solution

01

Identify the given values

Determine the given information in the problem: - Confidence Level (in this case, 95%) - Margin of Error (Ei.e., 0.5 hour) - Standard Deviation (蟽in this case, 6.6 hours).
02

Determine the z-score

The z-score for a 95% confidence level is found using standard normal distribution tables or statistical tools. For a 95% confidence level, the z-score is approximately 1.96.
03

Apply the sample size formula

Use the formula for calculating sample size: n = ((z * 蟽) / E)^2 where n is the sample size, z is the z-score, 蟽 is the standard deviation, and E is the margin of error.
04

Substitute the given values into the formula

Replace the values in the formula: n = ((1.96 * 6.6) / 0.5)^2
05

Calculate the required sample size

- Calculate the inside part first: (1.96 * 6.6) / 0.5 = 25.896 - Now square the result: (25.896)^2 = 670.75
06

Round up if needed

Since you cannot survey a fraction of a person, round up to the nearest whole number. Thus, the needed sample size is 671.

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Key Concepts

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

confidence level
The confidence level is a key concept when determining sample size. It reflects how sure you can be about your results. A common confidence level is 95%, which means you can be 95% certain that your estimate falls within the margin of error. The remaining 5% is the chance that your estimate is off. The confidence level relates to the z-score, which is used in the sample size formula. The higher the confidence level, the larger the z-score, which typically means you'll need a larger sample size to maintain the same margin of error.
margin of error
The margin of error (E) indicates the range in which the true population parameter is expected to fall. In simple terms, it tells you how precise your estimate is. A smaller margin of error requires a larger sample size but provides more precise results. For example, if you want to estimate the average time teens spend online within 0.5 hours, your margin of error is 0.5. When inputting values in the sample size formula, the margin of error helps determine how wide or narrow your confidence interval will be.
standard deviation
Standard deviation (蟽) measures the amount of variation or dispersion in a set of values. It shows how much individual data points differ from the mean. A higher standard deviation means data points are more spread out, while a lower standard deviation means they are closer to the mean. Standard deviation is crucial in sample size determination because it helps understand the spread of the data. In the given exercise, the standard deviation is 6.6 hours, indicating that teen internet usage has moderate variability. This value is used in the formula to calculate the necessary sample size to achieve the desired precision in your estimate.

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Most popular questions from this chapter

Suppose the arrival of cars at Burger King's drive-through follows a Poisson process with \(\mu=4\) cars every 10 minutes. (a) Simulate obtaining 30 samples of size \(n=40\) from this population. (b) Construct \(90 \%\) confidence intervals for each of the 30 samples. [Note: \(\sigma=\sqrt{\mu}\) in a Poisson process.] (c) How many of the intervals do you expect to include the population mean? How many actually contain the population mean?

In a random sample of 100 estate tax returns that was audited by the Internal Revenue Service, it was determined that the mean amount of additional tax owed was \(\$ 3137 .\) Assuming the population standard deviation of the additional amount owed is \(\$ 2694,\) construct and interpret a \(90 \%\) confidence interval for the mean additional amount of tax owed for estate tax returns.

Construct the appropriate confidence interval. In a February 2005 Harris Poll, 769 of 1010 randomly selected adults said that they always wear their seat belt. Construct and interpret a \(98 \%\) confidence interval for the proportion of adults who always wear their seat belt.

Credit Card Debt A school administrator is concerned about the amount of credit card debt college students have. She wishes to conduct a poll to estimate the percentage of full-time college students who have credit card debt of \(\$ 2000\) or more. What size sample should be obtained if she wishes the estimate to be within 2.5 percentage points with \(94 \%\) confidence if (a) a pilot study indicates the percentage is \(34 \% ?\) (b) no prior estimates are used?

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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