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Chapter 11: Problem 13

In Problems 13-16, construct a confidence interval for \(p_{1}-p_{2}\) at the given level of confidence. \(x_{1}=368, n_{1}=541, x_{2}=421, n_{2}=593,90 \%\) confidence

Short Answer

Expert verified
(-0.137, 0.017)

Step by step solution

01

- Calculate Sample Proportions

Calculate the sample proportions for both groups using the formula \( \hat{p} = \frac{x}{n} \). For group 1: \( \hat{p}_{1} = \frac{368}{541} \). For group 2: \( \hat{p}_{2} = \frac{421}{593} \).
02

- Compute the Difference in Sample Proportions

Subtract the proportion of group 2 from the proportion of group 1 to get the difference: \( \hat{p}_{1} - \hat{p}_{2} \).
03

- Calculate the Standard Error of Difference

Use the formula for the standard error of the difference between two sample proportions: \[ SE = \sqrt{ \frac{ \hat{p}_{1}(1 - \hat{p}_{1}) }{ n_{1} } + \frac{ \hat{p}_{2}(1 - \hat{p}_{2}) }{ n_{2} } } \].
04

- Find the Critical Value

For a 90% confidence level, determine the critical value (\( z^{*} \)) from the standard normal distribution table, which is approximately 1.645.
05

- Compute the Margin of Error

Calculate the margin of error (ME) using the formula: \( ME = z^{*} \times SE \).
06

- Construct the Confidence Interval

Determine the confidence interval for \( \hat{p}_{1} - \hat{p}_{2} \). It is given by:\[ ( \hat{p}_{1} - \hat{p}_{2} ) - ME < p_{1} - p_{2} < ( \hat{p}_{1} - \hat{p}_{2} ) + ME \].

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Key Concepts

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

Sample Proportions
To start with, sample proportions represent the fraction of individuals in a sample that exhibit a particular characteristic. This is calculated using the formula: \( \hat{p} = \frac{x}{n} \) where \( x \) is the number of individuals with the characteristic and \( n \) is the total number of individuals in the sample. In our example:
  • Group 1 has 368 individuals out of 541 showing the characteristic, making the sample proportion \( \hat{p}_{1} = \frac{368}{541} \).
  • Group 2 has 421 individuals out of 593, with a sample proportion \( \hat{p}_{2} = \frac{421}{593} \).
These proportions provide a basis for comparing the two groups.
Standard Error
The standard error (SE) is essential for understanding the variability in sample proportions. It provides a measure of the expected deviation of the sample proportion difference from the true population proportion difference. Calculating the SE involves the formula:\[ SE = \sqrt{ \frac{\hat{p}_{1}(1 - \hat{p}_{1}) }{ n_{1} } + \frac{\hat{p}_{2}(1 - \hat{p}_{2}) }{ n_{2} } } \]
  • For Group 1: substitute \( \hat{p}_{1} \) and \( n_{1} \) values.
  • For Group 2: substitute \( \hat{p}_{2} \) and \( n_{2} \) values.
This formula helps account for the sample size and the variability observed in each group.
Margin of Error
Once you have the standard error, you can determine the margin of error (ME). The margin of error quantifies the uncertainty inherent in our sample estimate, reflecting how much the sample proportions might vary from the actual population proportions. It's calculated as:\( ME = z^{*} \times SE \)The \( z^{*} \) value comes from a standard normal distribution based on the desired confidence level. Here, for a 90% confidence level, \( z^{*} \) is approximately 1.645. Thus, the margin of error incorporates both the standard error and the confidence level, showing how much we expect our sample results to fluctuate.
Critical Value
The critical value is a key component in statistical inference, as it marks the cutoff for our confidence level. It's derived from the standard normal distribution (often denoted as \( z^{*} \)). For a 90% confidence interval, the critical value to use is 1.645. This means that 90% of the area under the standard normal distribution curve falls between -1.645 and 1.645. This value, when multiplied by the standard error, helps construct the confidence interval, ensuring we are 90% confident that the interval contains the true population difference.
Confidence Interval
Finally, the confidence interval provides a range where the true difference between population proportions likely lies. To calculate the confidence interval for the difference in proportions \( p_{1} - p_{2} \), we use:\[ ( \hat{p}_{1} - \hat{p}_{2} ) - ME < p_{1} - p_{2} < ( \hat{p}_{1} - \hat{p}_{2} ) + ME \]
  • Start with the difference in sample proportions: \( \hat{p}_{1} - \hat{p}_{2} \).
  • Subtract the margin of error to find the lower bound.
  • Add the margin of error to find the upper bound.
This interval provides a range based on our sample data and statistical calculations, telling us the plausible values for the population difference.

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Most popular questions from this chapter

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