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Chapter 11: Problem 16

Construct a confidence interval for \(p_{1}-p_{2}\) at the given level of confidence. \(x_{1}=804, n_{1}=874, x_{2}=892, n_{2}=954,95 \%\) confidence

Short Answer

Expert verified
The 95% confidence interval for the difference in proportions \(p_{1} - p_{2}\) is \([-0.0399, 0.0079]\).

Step by step solution

01

- Find the Sample Proportions

Calculate the sample proportions for each sample using the formula \(\hat{p}_{i} = \frac{x_{i}}{n_{i}}\) for i=1 and 2. \(\hat{p}_{1} = \frac{804}{874} = 0.919\) and \(\hat{p}_{2} = \frac{892}{954} = 0.935\).
02

- Determine the Point Estimate

Find the point estimate for the difference between the two sample proportions: \(\hat{p}_{1} - \hat{p}_{2} = 0.919 - 0.935 = -0.016\).
03

- Calculate the Standard Error

Use the standard error formula for the difference between two proportions: \(SE = \sqrt{\left(\frac{\hat{p}_1 (1 - \hat{p}_1)}{n_1}\right) + \left(\frac{\hat{p}_2 (1 - \hat{p}_2)}{n_2}\right)}\). First, compute each part: \(\frac{0.919 \times (1 - 0.919)}{874} = 0.000087\) and \(\frac{0.935 \times (1 - 0.935)}{954} = 0.000063\). Then, find the sum and take the square root: \(SE = \sqrt{0.000087 + 0.000063} = \sqrt{0.00015} = 0.0122\).
04

- Find the Critical Value

For a 95% confidence interval and large samples, the critical value (z*) is approximately 1.96.
05

- Calculate the Margin of Error

Multiply the standard error by the critical value: \(MOE = 1.96 \times 0.0122 = 0.0239\).
06

- Construct the Confidence Interval

Add and subtract the margin of error from the point estimate to get the interval: \([-0.016 - 0.0239, -0.016 + 0.0239] = [-0.0399, 0.0079]\).

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Key Concepts

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

sample proportions
Sample proportions help us understand how a characteristic is represented in a given sample. They are calculated using the formula \(\hat{p}_i = \frac{x_i}{n_i}\), where \(\hat{p}_i\) is the sample proportion, \(\(x_i)\) is the number of successes, and \(\ n_i\) is the total number of observations in the sample.

In our case, we calculated the sample proportions for two samples:
  • \(\(\hat{p}_1 = \frac{804}{874} = 0.919\)\)
  • \(\hat{p}_2 = \frac{892}{954} = 0.935\)\).
These proportions give us a snapshot of the observed positive outcomes in each sample.
point estimate
A point estimate is a single value used to estimate a population parameter. It's essentially our 'best guess' given the sample data.

In this exercise, we used the point estimate to find the difference between the two sample proportions: \(\hat{p}_1 - \hat{p}_2 = 0.919 - 0.935 = -0.016\).

This means our best estimate for the difference in proportions between the two samples is -0.016. It indicates that sample 2 has a slightly higher proportion compared to sample 1.
standard error
The standard error measures the variability of a sample statistic. It's a crucial component for constructing confidence intervals.

For the difference between two proportions, the standard error can be computed using the formula: \(\ SE = \sqrt{\left( \frac{\hat{p}_1 (1 - \hat{p}_1)}{n_1} \right) + \left( \frac{\hat{p}_2 (1 - \hat{p}_2)}{n_2} \right)}\).

In our example, we calculated:
  • \(\frac{0.919 \times (1 - 0.919)}{874} = 0.000087\)
  • \(\frac{0.935 \times (1 - 0.935)}{954} = 0.000063\).
Adding these, we got \(\ SE = \sqrt{0.00015} = 0.0122\). The smaller the SE, the more precise our estimate.
margin of error
The margin of error (MOE) represents how much the sample estimate might vary from the true population parameter. To calculate it, we multiply the standard error by the critical value.

In this case, for a 95% confidence interval, our critical value (z*) is about 1.96.

Thus, we calculate the MOE as follows: \(\text{MOE} = 1.96 \times 0.0122 = 0.0239\). This tells us that the true difference between the population proportions is expected to be within 0.0239 units of our point estimate.
critical value
The critical value represents the cutoff points in the distribution that determine the confidence level, often denoted by \(z^* \) for the z-distribution.

For a 95% confidence interval, the critical value is around 1.96. This is derived from the standard normal distribution table.

The selection of the critical value hinges on the desired confidence level. Higher confidence levels will have larger critical values.

In our calculations, we used the critical value of 1.96 to determine the margin of error, ensuring that we can state, with 95% confidence, that the true difference lies within our calculated range.

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Most popular questions from this chapter

A random sample of size \(n=41\) results in a sample mean of 125.3 and a sample standard deviation of \(8.5 .\) An independent sample of size \(n=50\) results in a sample mean of 130.8 and sample standard deviation of \(7.3 .\) Does this constitute sufficient evidence to conclude that the population means differ at the \(\alpha=0.01\) level of significance?

Assume that the populations are normally distributed. Test the given hypothesis. \(\sigma_{1}>\sigma_{2}\) at the \(\alpha=0.01\) level of significance $$ \begin{array}{ccc} & \text { Sample 1 } & \text { Sample 2 } \\ \hline n & 26 & 19 \\ \hline s & 9.9 & 6.4 \\ \hline \end{array} $$

Priming Two Dutch researchers conducted a study in which two groups of students were asked to answer 42 questions from Trivial Pursuit. The students in group 1 were asked to spend 5 minutes thinking about what it would mean to be a professor, while the students in group 2 were asked to think about soccer hooligans. The 200 students in group 1 had a mean score of 23.4 with a standard deviation of 4.1 , while the 200 students in group 2 had a mean score of 17.9 with a standard deviation of \(3.9 .\) (a) Determine the \(95 \%\) confidence interval for the difference in scores, \(\mu_{1}-\mu_{2} .\) Interpret the interval. (b) What does this say about priming?

For each study, explain which statistical procedure (estimating a single proportion; estimating a single mean; hypothesis test for a single proportion; hypothesis test for a single mean; hypothesis test or estimation of two proportions, hypothesis test or estimation of two means, dependent or independent) would most likely be used for the research objective given. Assume all model requirements for conducting the appropriate procedure have been satisfied. While exercising by climbing stairs, is it better to take one stair, or two stairs, at a time? Researchers identified 30 volunteers who were asked to climb stairs for two different 15-minute intervals taking both one stair and two stairs at a time. Whether the volunteer did one stair or two stairs first was determined randomly. The goal of the research was to determine if energy expenditure for each exercise routine was different.

Perform the appropriate hypothesis test. A random sample of \(n_{1}=120\) individuals results in \(x_{1}=43\) successes. An independent sample of \(n_{2}=130\) individuals results in \(x_{2}=56\) successes. Does this represent sufficient evidence to conclude that \(p_{1} \neq p_{2}\) at the \(\alpha=0.01\) level of significance?
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