91Ó°ÊÓ

What about the sample size \(n\) for confidence intervals for the difference of proportions \(p_{1}-p_{2} ?\) Let us make the following assumptions: equal sample sizes \(n=n_{1}=n_{2}\) and all four quantities \(n_{1} \hat{p}_{1}, n_{1} \hat{q}_{1}, n_{2} \hat{p}_{2},\) and \(n_{2} \hat{q}_{2}\) are greater than \(5 .\) Those readers familiar with algebra can use the procedure outlined in Problem 28 to show that if we have preliminary estimates \(\hat{p}_{1}\) and \(\hat{p}_{2}\) and a given maximal margin of error \(E\) for a specified confidence level \(c,\) then the sample size \(n\) should be at least $$n=\left(\frac{z_{c}}{E}\right)^{2}\left(\hat{p}_{1} \hat{q}_{1}+\hat{p}_{2} \hat{q}_{2}\right)$$ However, if we have no preliminary estimates for \(\hat{p}_{1}\) and \(\hat{p}_{2}\), then the theory similar to that used in this section tells us that the sample size \(n\) should be at least $$n=\frac{1}{2}\left(\frac{z_{c}}{F}\right)^{2}$$ (a) In Problem 17 (Myers-Briggs personality type indicators in common for married couples), suppose we want to be \(99 \%\) confident that our estimate \(\hat{p}_{1}-\hat{p}_{2}\) for the difference \(p_{1}-p_{2}\) has a maximal margin of error \(E=0.04 .\) Use the preliminary estimates \(\hat{p}_{1}=289 / 375\) for the proportion of couples sharing two personality traits and \(\hat{p}_{2}=23 / 571\) for the proportion having no traits in common. How large should the sample size be (assuming equal sample size-i.e., \(n=n_{1}=n_{2}\) )? (b) Suppose that in Problem 17 we have no preliminary estimates for \(\hat{p}_{1}\) and \(\hat{p}_{2}\) and we want to be \(95 \%\) confident that our estimate \(\hat{p}_{1}-\hat{p}_{2}\) for the difference \(p_{1}-p_{2}\) has a maximal margin of error \(E=0.05 .\) How large should the sample size be (assuming equal sample size-i.e., \(n=n_{1}=n_{2}\) )?

Step by step solution

01

Understand Given Values (Part a)

For part (a), we want a 99% confidence interval, so the critical value is computed as \( z_c = 2.576 \). We are provided with preliminary estimates: \( \hat{p}_1 = \frac{289}{375} \) and \( \hat{p}_2 = \frac{23}{571} \). The margin of error (E) is given as 0.04.

02

Calculate Complementary Proportions

Compute the complementary proportions \( \hat{q}_1 \) and \( \hat{q}_2 \) using the formulas: \( \hat{q}_1 = 1 - \hat{p}_1 \) and \( \hat{q}_2 = 1 - \hat{p}_2 \). This gives us \( \hat{q}_1 = 1 - \frac{289}{375} = \frac{86}{375} \) and \( \hat{q}_2 = 1 - \frac{23}{571} = \frac{548}{571} \).

03

Compute Required Sample Size (Part a)

Using the formula for the sample size: \[n=\left(\frac{z_{c}}{E}\right)^{2}\left(\hat{p}_{1} \hat{q}_{1}+\hat{p}_{2} \hat{q}_{2}\right)\]Substitute the known values to compute: \[n = \left(\frac{2.576}{0.04}\right)^{2} \left(\frac{289}{375} \times \frac{86}{375} + \frac{23}{571} \times \frac{548}{571}\right)\] Evaluate and simplify to find \( n \approx 924 \).

04

Understand Given Values (Part b)

For part (b), we want a 95% confidence interval, so the critical value is \( z_c = 1.96 \). No preliminary estimates for proportions are given. The margin of error (E) is provided as 0.05.

05

Compute Required Sample Size Without Estimates (Part b)

Without preliminary estimates, the formula simplifies to \[n=\frac{1}{2}\left(\frac{z_{c}}{E}\right)^{2}\]Substitute the values:\[n = \frac{1}{2}\left(\frac{1.96}{0.05}\right)^{2}\] Simplify and calculate to get \( n \approx 384 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Sample Size Determination

Determining the right sample size is a critical step in designing any survey or experiment where conclusions about a population are drawn from a subset of that population. It affects the reliability of the study's results. When dealing with confidence intervals for the difference in proportions, special considerations must be taken to ensure accuracy.

In the context of comparing proportions, the sample size is crucial, especially if you have preliminary estimates of the proportions you're investigating. These estimates help calculate the variance that contributes to the sample size calculation. This involves the Z-score for the desired confidence level, which reflects how much confidence you have in the interval's accuracy, and a known margin of error.

Suppose you have some prior knowledge in terms of preliminary estimates of the proportions, \(\hat{p}_1\) and \(\hat{p}_2\), the following formula can be used to calculate the sample size (n):

• Given: \(n = \left(\frac{z_c}{E}\right)^2\left(\hat{p}_1 \hat{q}_1 + \hat{p}_2 \hat{q}_2\right)\)
• Here, \(\hat{q}_1 = 1 - \hat{p}_1\) and \(\hat{q}_2 = 1 - \hat{p}_2\).
• This formula accounts for both the proportion itself and its complement (one minus the proportion), thus providing a necessary variance measure.

Proportions Difference

The difference between two proportions is a valuable measure in statistics, particularly when assessing the effect or impact in comparative studies. This difference can reveal significant insights, such as differences between test and control groups in an experiment or variations between demographic segments.

For instance, if you're comparing the proportion of two past surveys, the variables \(\hat{p}_1\) and \(\hat{p}_2\) denote these proportions. The ultimate goal is to reliably estimate the difference \(p_1 - p_2\) between their actual population proportions, using the available sample data.

Whenever you're calculating this difference, initial estimates (denoted as \(\hat{p}_1\) and \(\hat{p}_2\)) can serve as predictors for how large your sample size needs to be, especially in ensuring that the confidence interval for this difference does not overlap zero (indicating significance). If initial estimates are unavailable or unreliable, different strategies for sample size determination may be applied, like increasing manageable sample size or revising preliminary estimates cautiously.

Margin of Error

The margin of error is a fundamental component when calculating confidence intervals. This quantity represents the range in which we can expect the true population parameter to fall, given our sample estimate. A smaller margin of error indicates a more precise estimate.

In our context, if you aim to detect a small difference between proportions, the margin of error must be small, which will naturally increase the sample size required. This is because a smaller margin means we need more data to be more certain about where the true value falls.

Margin of error, denoted as \(E\), is closely linked to the confidence level expressed as a Z-score. For example, for a 95% confidence interval, a Z-score might be 1.96. It shows how much standard deviation away from the mean we consider to be reliable. The formula to insert it into a sample size determination involves squaring this Z-score relative to the desired precision (or \(E\)).

• For instance, without initial estimates: \(n = \frac{1}{2}\left(\frac{z_c}{E}\right)^{2}\)
• This formula simplifies the calculation when preliminary estimates are not present, emphasizing caution since it doubles the effort to secure a reliable survey without starting data.