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Magazine Chapter 11: Problem 7
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
The 90% confidence interval for \( p_{1} - p_{2} \) is (-0.076, 0.016).
Step by step solution
01
Identify the sample proportions
Calculate the sample proportions for each group.\( \hat{p}_{1} = \frac{x_{1}}{n_{1}} = \frac{368}{541} \approx 0.68 \) and \( \hat{p}_{2} = \frac{x_{2}}{n_{2}} = \frac{421}{593} \approx 0.71 \)
02
Calculate the pooled sample proportion
The pooled sample proportion is calculated using \( \hat{p} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}} = \frac{368 + 421}{541 + 593} = \frac{789}{1134} \approx 0.696 \)
03
Determine the standard error
The standard error (SE) for the difference between two proportions is given by: \[SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_{1}} + \frac{1}{n_{2}} \right)} = \sqrt{0.696(1 - 0.696) \left( \frac{1}{541} + \frac{1}{593} \right)} \approx 0.028 \]
04
Find the critical value
For a 90% confidence interval, the critical value (z*) is approximately 1.645. This value corresponds to z-scores that leave 5% in each tail of the normal distribution.
05
Calculate the margin of error
The margin of error (ME) is given by \( ME = z* \cdot SE = 1.645 \cdot 0.028 \approx 0.046 \).
06
Construct the confidence interval
The confidence interval for \( p_{1} - p_{2} \) is \( (\hat{p}_{1} - \hat{p}_{2}) \pm ME = (0.68 - 0.71) \pm 0.046 \). Accordingly, the interval is \( -0.03 \pm 0.046 \) or (-0.076, 0.016).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sample Proportions
Sample proportions give us a sense of the percentage of success from the samples. To find a sample proportion, you need the number of successes divided by the total sample size. In this exercise, the proportions are:
- For group 1: \( \frac{368}{541} \approx 0.68 \)
- For group 2: \( \frac{421}{593} \approx 0.71 \)
Pooled Sample Proportion
When comparing two proportions, we often use a pooled sample proportion to get a more accurate overall proportion. We combine the successes and the sample sizes from both groups to calculate it.
This is done using the formula:\br>\[ \hat{p} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}} \]
For the given exercise, this looks like:
\[ \hat{p} = \frac{368 + 421}{541 + 593} \approx 0.696 \]
The pooled proportion is crucial in the subsequent steps when determining the standard error.
This is done using the formula:\br>\[ \hat{p} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}} \]
For the given exercise, this looks like:
\[ \hat{p} = \frac{368 + 421}{541 + 593} \approx 0.696 \]
The pooled proportion is crucial in the subsequent steps when determining the standard error.
Standard Error Calculation
The standard error measures the variability of the sampling distribution of the difference between two proportions. To calculate it, we use the formula:
\[ SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_{1}} + \frac{1}{n_{2}} \right) } \]
For our example, the calculation is:
\[ SE = \sqrt{0.696(1 - 0.696) \left( \frac{1}{541} + \frac{1}{593} \right) } \approx 0.028 \]
Understanding this helps us to quantify the extent of the expected fluctuation in sample proportions.
\[ SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_{1}} + \frac{1}{n_{2}} \right) } \]
For our example, the calculation is:
\[ SE = \sqrt{0.696(1 - 0.696) \left( \frac{1}{541} + \frac{1}{593} \right) } \approx 0.028 \]
Understanding this helps us to quantify the extent of the expected fluctuation in sample proportions.
Critical Value
The critical value helps in determining the range within which we expect the true parameter to lie, given a certain level of confidence. For a 90% confidence interval, we use the Z-score that corresponds to the desired level. Typically, this value divides the distribution such that 5% of the data are in each tail.
This means we use a critical value of approximately 1.645 for a two-tailed test.
This means we use a critical value of approximately 1.645 for a two-tailed test.
Margin of Error
The margin of error indicates the range of the confidence interval around the estimated difference between proportions. It is calculated by multiplying the critical value and the standard error:
\[ ME = z* \cdot SE \]
In our example, this would be:
\[ ME = 1.645 \cdot 0.028 \approx 0.046 \]
This provides the 'buffer zone' around our estimated proportion difference, helping us to create the confidence interval.
\[ ME = z* \cdot SE \]
In our example, this would be:
\[ ME = 1.645 \cdot 0.028 \approx 0.046 \]
This provides the 'buffer zone' around our estimated proportion difference, helping us to create the confidence interval.
Difference Between Two Proportions
Ultimately, we are interested in the difference between the two sample proportions. Specifically, we want to construct a confidence interval around this difference. Using the sample proportions from earlier:
\[ (0.68 - 0.71) \pm 0.046 \]
This results in a confidence interval of:
\[ -0.03 \pm 0.046 \]
Or, more simply put, the interval is (-0.076, 0.016). This range gives us a plausible set of differences between the two population proportions with 90% confidence.
\[ (0.68 - 0.71) \pm 0.046 \]
This results in a confidence interval of:
\[ -0.03 \pm 0.046 \]
Or, more simply put, the interval is (-0.076, 0.016). This range gives us a plausible set of differences between the two population proportions with 90% confidence.
