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Chapter 8: Problem 52

Income of Native Americans How large a sample size do we need to estimate the mean annual income of Native Americans in Onondaga County, New York, correct to within $$\$ 1000$$ with probability \(0.99 ?\) No information is available to us about the standard deviation of their annual income. We guess that nearly all of the incomes fall between $$\$ 0$$ and $$\$ 120,000$$ and that this distribution is approximately bell shaped.

Short Answer

Expert verified
The required sample size is 2655.

Step by step solution

01

Understanding the Problem

We need to determine the sample size required to estimate the mean annual income of Native Americans in Onondaga County within a margin of error of $1000 and with a confidence level of 99%. Since the standard deviation is unknown, we will use the range and assume a normal distribution.
02

Approximating Standard Deviation

Given that nearly all incomes range between \(0 and \)120,000, we approximate the range of incomes as $120,000. For a normal distribution, the range is approximately equal to 6 times the standard deviation (6蟽). Thus, \[ 6蟽 鈮� 120,000 \]Solving for 蟽 gives the standard deviation as \[ 蟽 鈮� \frac{120,000}{6} = 20,000 \]
03

Determining the Margin of Error and Z-Score

The margin of error (E) is given as $1000. Since we need a 99% confidence interval, we use the critical value from the normal distribution table for 99% confidence, which is approximately 2.576.
04

Calculating the Sample Size

Use the sample size formula for the margin of error \[ E = \frac{Z蟽}{\sqrt{n}} \]where E is the margin of error, Z is the z-score, 蟽 is the standard deviation, and n is the sample size. Rearrange to solve for n:\[ n = \left(\frac{Z蟽}{E}\right)^2 \]Substituting the values,\[ n = \left(\frac{2.576 \times 20,000}{1,000}\right)^2 \]\[ n 鈮� \left(51.52\right)^2 鈮� 2654.63 \]
05

Rounding the Sample Size

Since the sample size must be a whole number, round 2654.63 up to the nearest whole number. The required sample size is 2655.

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Key Concepts

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

Understanding the Confidence Interval
A confidence interval is a range of values that is believed to contain the true population parameter, such as the mean income in this problem, with a certain level of confidence.
A 99% confidence interval suggests that we are 99% sure that the actual mean falls within this interval.
To construct a confidence interval, we need to know the sample mean, estimated standard deviation, and use a z-score that matches the desired confidence level.
  • The z-score for a 99% confidence level is approximately 2.576.
  • The higher the confidence level, the wider the interval becomes, as we need to "capture" the true mean with more certainty.
The confidence interval tells us how much we can trust the sample mean to represent the population mean. It's crucial for making informed decisions and understanding how sample data reflects the population.
Exploring Normal Distribution
In statistics, normal distribution is often referred to as the "bell curve" because of its bell-like shape. It's a continuous probability distribution that is symmetric around the mean, meaning most data points cluster around the center.
In this problem, it's assumed that annual incomes follow this distribution as incomes generally span a broad range with fewer people on the high and low ends.
  • About 68% of values lie within one standard deviation of the mean in a normal distribution.
  • Approximately 95% fall within two standard deviations, and 99.7% lie within three standard deviations.
When estimating parameters like the mean income from a sample, assuming normal distribution helps in determining margins of error and constructing confidence intervals.
Margin of Error: Precision of Estimates
The margin of error is a statistic expressing the amount of random sampling error in a survey's results. It's a critical part of survey results as it gives an understanding of how reliable the survey is.
For this problem, the margin of error was set at $1000.
  • The margin of error indicates the range of values that the sample result might differ from the true population parameter.
  • A smaller margin of error means more precise estimates, which often requires a larger sample size.
Using the relationship between the z-score, standard deviation, and sample size, we can calculate the necessary sample size to maintain a desired margin of error.
In this problem, the calculation was done using the margin of error formula, ensuring the sample mean would be within $1000 of the true mean 99% of the time. The outcome showed that a sample size of 2655 would be needed.

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Most popular questions from this chapter

The instructor will assign the class a theme to study. Download recent results for variables relating to that theme from sda.berkeley.edu/GSS. Find and interpret confidence intervals for relevant parameters. Prepare a two-page report summarizing results.

Why bootstrap? Explain the purpose of using the bootstrap method.

A Gallup poll taken during June 2011 estimated that \(8.8 \%\) of U.S. adults were unemployed. The poll was based on the responses of 30,000 U.S. adults in the workforce. Gallup reported that the margin of error associated with the poll is 卤0.3 percentage points. Explain how they got this result. (Source: www.gallup.com/poll/125639/Gallup-Daily-Workforce aspx.)

Types of estimates An interval estimate for a mean is more informative than a point estimate, because with an interval estimate you can figure out the point estimate, but with the point estimate alone you have no idea how wide the interval estimate is. Explain why this statement is correct, illustrating using the reported \(95 \%\) confidence interval of (4.0,5.6) for the mean number of dates in the previous month based on a sample of women at a particular college.

Driving after drinking In December \(2004,\) a report based on the National Survey on Drug Use and Health estimated that \(20 \%\) of all Americans of ages 16 to 20 drove under the influence of drugs or alcohol in the previous year (AP, December 30,2004 ). A public health unit in Wellington, New Zealand, plans a similar survey for young people of that age in New Zealand. They want a \(95 \%\) confidence interval to have a margin of error of 0.04 . a. Find the necessary sample size if they expect results similar to those in the United States. b. Suppose that in determining the sample size, they use the safe approach that sets \(\hat{p}=0.50\) in the formula for \(n\). Then, how many records need to be sampled? Compare this to the answer in part a. Explain why it is better to make an educated guess about what to expect for \(\hat{p},\) when possible.
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