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Chapter 7: Problem 32

You plan to conduct a survey to estimate the percentage of adults who have had chickenpox. Find the number of people who must be surveyed if you want to be \(90 \%\) confident that the sample percentage is within two percentage points of the true percentage for the population of all adults. a. Assume that nothing is known about the prevalence of chickenpox. b. Assume that about \(95 \%\) of adults have had chickenpox. c. Does the added knowledge in part (b) have much of an effect on the sample size?

Short Answer

Expert verified
a) Approximately 1692 samples; b) Approximately 322 samples; c) Knowledge significantly reduces the sample size needed.

Step by step solution

01

Understanding the Confidence Interval

The confidence interval provides a range of values within which we expect the true population parameter to lie. Here we want to be 90% confident that the sample percentage is within two percentage points of the true percentage.
02

Identify the Margin of Error (E)

The problem states that the desired margin of error is two percentage points. Thus, we will use E = 0.02 in our formulas.
03

Determine the Z-Score for 90% Confidence

For a 90% confidence level, the Z-score (z) is approximately 1.645.
04

Formula for Sample Size with Unknown Proportion

When the population proportion (p) is unknown, we use 0.5 as an estimate for maximum variability. The formula is: \[ n = \left( \frac{z^2 \, p (1-p)}{E^2} \right) \] Substituting the values, we get: \[ n = \left( \frac{(1.645)^2 \, 0.5 \, (1-0.5)}{0.02^2} \right) \]
05

Calculate the Sample Size for Unknown Proportion

Using the formula: \[ n = \left( \frac{(1.645)^2 \, 0.5 \, 0.5}{0.02^2} \right) = \frac{2.706025 \, 0.25}{0.0004} = \frac{0.67650625}{0.0004} \approx 1691.27 \] So, the sample size needed is approximately 1692.
06

Formula for Sample Size with Known Proportion

When the population proportion (p) is known, the formula is: \[ n = \left( \frac{z^2 \, p (1-p)}{E^2} \right) \] Given that 95% of adults have had chickenpox, we use p = 0.95.
07

Calculate the Sample Size for Known Proportion

Using the formula: \[ n = \left( \frac{(1.645)^2 \, 0.95 \, (1-0.95)}{0.02^2} \right) = \frac{2.706025 \, 0.0475}{0.0004} = \frac{0.1285361875}{0.0004} \approx 321.34 \] So, the sample size needed is approximately 322.
08

Compare the Results

We compare the sample sizes calculated. Without prior knowledge, we need approximately 1692 samples. With prior knowledge, we need only approximately 322 samples. Therefore, the added knowledge in part (b) has a significant effect on reducing the sample size needed.

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Key Concepts

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

Confidence Interval
The confidence interval lets us understand how precise an estimate is from a statistical sample. Imagine you survey a group to estimate the percentage of people who had chickenpox. The confidence interval gives a range in which the true population percentage likely falls. The confidence level (like 90% in this case) tells us how sure we feel about our interval. It's like saying, 'We're 90% confident the true percentage is between X and Y.' This is why a higher confidence level means a broader range. We use the Z-score, margin of error, and sample size to calculate this interval. So, our confidence depends on these numbers.
Margin of Error
The margin of error (E) is crucial in surveys. It shows how much the survey results might differ from the true population value. In simple terms, it's how 'off' our survey might be. If the margin of error is 2%, and our survey finds that 50% have had chickenpox, the actual percentage could be between 48% and 52%. The smaller the margin of error, the closer our estimate is to the true value. However, a smaller margin needs a larger sample size. So when we want our survey to be accurate within 2 percentage points, E = 0.02 in our calculations.
Z-score
The Z-score tells us how far a data point is from the mean, in terms of standard deviations. For confidence intervals, it represents the number of standard deviations needed for a certain confidence level. For a 90% confidence level, the Z-score is approximately 1.645. This number comes from standard statistical tables. We plug this Z-score into our sample size formula to ensure we're catching the correct range of values with our desired confidence. Different confidence levels (like 95% or 99%) will use different Z-scores.
Population Proportion
The population proportion (p) represents the percentage of the entire population that has a specific trait. For example, if 95% of adults have had chickenpox, p = 0.95. This proportion is crucial when calculating sample size. If p is unknown, we maximize variability by assuming p = 0.5. Using this number gives the most cautious estimate, leading to the largest sample size. However, if we know p (like 0.95 for chickenpox), it significantly reduces the needed sample size as there's less uncertainty. Knowing p can make our surveys much more efficient.

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Most popular questions from this chapter

Find the critical value \(z_{\alpha / 2}\) that corresponds to the given confidence level. \(99 \%\)

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