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Chapter 8: Problem 127

Use the following information to answer the next three exercises: According to a Field Poll, 79% of California adults (actual results are 400 out of 506 surveyed) feel that education and our schools is one of the top issues facing California. We wish to construct a 90% confidence interval for the true proportion of California adults who feel that education and the schools is one of the top issues facing California. The error bound is approximately _____. a. 1.581 b. 0.791 c. 0.059 d. 0.030

Short Answer

Expert verified
The error bound is approximately 0.030, so the correct answer is d.

Step by step solution

01

Understanding the Sample Proportion

First, calculate the sample proportion using the given data. The sample proportion \( \hat{p} \) is calculated as \( \hat{p} = \frac{400}{506} \approx 0.79 \).
02

Determine the Z-Score for 90% Confidence Level

For a 90% confidence interval, the Z-score is approximately \( 1.645 \). This value comes from the standard normal distribution table, often used for confidence intervals.
03

Calculate the Standard Error

The standard error (SE) for the proportion is calculated using the formula: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \] Substituting the given values, \( SE = \sqrt{\frac{0.79 \times 0.21}{506}} \approx 0.0181 \).
04

Calculate the Error Bound (E)

Finally, calculate the error bound using the formula: \[ E = Z \times SE \] Here, \( E = 1.645 \times 0.0181 \approx 0.030 \).

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Key Concepts

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

Understanding Sample Proportion
A sample proportion is an estimate of the proportion of a certain outcome in a population, derived from a sample of data. In this exercise, we're concerned about the proportion of California adults who consider education and schools as a top priority, calculated from a sample of 506 individuals.
The formula to determine the sample proportion, denoted as \( \hat{p} \), is:\[\hat{p} = \frac{x}{n}\]where:
  • \( x \) is the number of favorable outcomes (400 adults considering education crucial)
  • \( n \) is the total number of individuals surveyed (506)
Using the data, \( \hat{p} = \frac{400}{506} \approx 0.79 \), meaning approximately 79% of the sample believes education is a key issue. This percentage serves as our sample proportion.
Calculating Error Bound
The error bound, also known as the margin of error, indicates how much the sample proportion might differ from the true population proportion. It's calculated using the confidence level and the standard error.
To find the error bound (E), you use the formula:\[ E = Z \times SE \]where:
  • \( Z \) represents the Z-score corresponding to your confidence interval
  • \( SE \) is the standard error of the sample proportion
In our example, a 90% confidence interval gives us a Z-score of approximately 1.645. We already calculated the standard error as 0.0181. Substituting these values in gives us:\[ E = 1.645 \times 0.0181 \approx 0.030 \]Thus, the error bound tells us the estimate of the population proportion could vary by 0.030 above or below 79%.
Decoding the Z-Score
The Z-score is a statistical measurement that describes the number of standard deviations a data point is from the mean. In the context of confidence intervals, the Z-score is crucial for determining the width of the interval.
For different confidence levels, you select different Z-scores:
  • 90% confidence level: \( Z \approx 1.645 \)
  • 95% confidence level: \( Z \approx 1.96 \)
  • 99% confidence level: \( Z \approx 2.576 \)
In this exercise, we use a Z-score of 1.645 for a 90% confidence interval, meaning we are 90% confident the calculated interval contains the true population proportion. This selection comes from standard normal distribution tables, which provide critical values for different confidence levels.
What is Standard Error?
The standard error (SE) is a measure of the variability or dispersion of a sample statistic over different samples. In simpler terms, it tells us how much the sample proportion might change if we were to repeat the sampling process.
The formula for the standard error of a proportion is:\[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]where:
  • \( \hat{p} \) is the sample proportion (0.79)
  • \( n \) is the sample size (506)
Substitute these values into the formula:\[ SE = \sqrt{\frac{0.79 \times 0.21}{506}} \approx 0.0181 \]The standard error can be seen as the average distance that an observed proportion would be from the true population proportion. Smaller SE indicates more precise estimates, leading to narrower confidence intervals.

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Most popular questions from this chapter

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