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Expand Your Knowledge: Sample Size, Difference of Proportions 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 $$ n=\frac{1}{2}\left(\frac{z_{c}}{E}\right)^{2} $$ (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

Calculate preliminary estimates

Given \( \hat{p}_1 = \frac{289}{375} \) and \( \hat{p}_2 = \frac{23}{571} \), first calculate \( \hat{q}_1 \) and \( \hat{q}_2 \). \( \hat{q}_1 = 1 - \hat{p}_1 = 1 - \frac{289}{375} \) and \( \hat{q}_2 = 1 - \hat{p}_2 = 1 - \frac{23}{571} \).

02

Calculate critical value for confidence level

For a 99% confidence level, find the critical value \( z_c \). This corresponds to \( z_{0.005} \) in a z-table, which is approximately 2.576.

03

Calculate sample size with estimates

Using the formula \( n=\left(\frac{z_c}{E}\right)^{2}\left(\hat{p}_1 \hat{q}_1+\hat{p}_2 \hat{q}_2\right) \), substitute the values: \( z_c = 2.576 \), \( E = 0.04 \), \( \hat{p}_1 = \frac{289}{375} \), \( \hat{q}_1 \) from step 1, \( \hat{p}_2 = \frac{23}{571} \), \( \hat{q}_2 \) from step 1. Calculate \( n \).

04

Calculate \( \hat{q}_1 \) and \( \hat{q}_2 \)

Compute \( \hat{q}_1 = 1 - \hat{p}_1 = \frac{86}{375} \) and \( \hat{q}_2 = 1 - \hat{p}_2 = \frac{548}{571} \).

05

Substitute values into formula

Substitute these values into the sample size equation: \( n = \left(\frac{2.576}{0.04}\right)^{2} \left(\frac{289}{375}\cdot\frac{86}{375} + \frac{23}{571}\cdot\frac{548}{571}\right) \) and calculate \( n \).

06

Calculate using formula without estimates

For part (b), use the formula \( n=\frac{1}{2}\left(\frac{z_c}{E}\right)^{2} \) for a 95% confidence level. Here, \( z_c = 1.96 \) for 95% confidence level, \( E = 0.05 \). Substitute these values into the formula.

07

Calculate sample size for part (b)

Calculate \( n = \frac{1}{2} \left( \frac{1.96}{0.05} \right)^2 \). Simplifying gives the required sample size for part (b).

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Key Concepts

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

Sample Size

When planning to conduct statistical analysis, determining the appropriate sample size is crucial. Sample size refers to the number of observations or participants included in a study. It deeply influences the accuracy of the results. In statistics, especially when dealing with confidence intervals and differences of proportions, having a suitable sample size ensures the stability and reliability of the estimations made.

There are various formulas to calculate sample size, depending on the data available:

• If preliminary estimates \(\hat{p}_1\) and \(\hat{p}_2\) exist, use the formula \[ n=\left(\frac{z_c}{E}\right)^{2}\left(\hat{p}_1 \hat{q}_1+\hat{p}_2 \hat{q}_2\right) \] to compute the needed sample size.
• If no preliminary estimates are available, the formula \[ n=\frac{1}{2}\left(\frac{z_c}{E}\right)^{2} \] is recommended, ensuring accurate assessments with limited information.

These formulas are essential to achieve desired precision in research results, offering specific guidance based on available information.

Confidence Intervals

Confidence intervals are a fundamental concept in statistics. They represent a range of values, derived from sample data, within which the true population parameter is expected to lie. This range is constructed with a certain level of confidence, often expressed as a percentage, like 95% or 99%.

Confidence intervals are key because they give us an idea of the uncertainty surrounding our estimate. For example, in checking the difference between two proportions (say, \(p_1\) and \(p_2\)), a confidence interval provides a range in which the actual difference might exist. A 99% confidence interval means you can be 99% sure that the true difference lies within this interval.

• Higher confidence levels (e.g., 99%) will result in wider intervals compared to lower levels (e.g., 95%), reflecting more uncertainty at higher confidence.
• The width of the confidence interval is also influenced by the sample size and the variability of the data.

Understanding confidence intervals empowers researchers to better interpret their results and understand the limitations of their data.

Difference of Proportions

The difference of proportions is a common way to compare two groups and is calculated by taking the difference between two sample proportions, \(\hat{p}_1\) and \(\hat{p}_2\). This measure is particularly useful in experiments or surveys where you want to see if there is a significant disparity between two results.

In statistical analysis, calculating the difference in proportions involves several steps:

• Determine individual sample proportions, \(\hat{p}_1\) and \(\hat{p}_2\).
• Compute the difference \(\hat{p}_1 - \hat{p}_2\).
• Evaluate the significance of this difference, often using a calculated confidence interval to ascertain if the observed difference is statistically significant.

By analyzing differences in proportions, one can draw conclusions about potential differences in populations or behaviors, contributing significantly to data-driven decision-making. Understanding this concept is crucial for anyone involved in conducting comparative research.