Use the expression in the accompanying discussion of sample size to find the size of each sample if you want to estimate the difference between proportions of men and women who own smartphones. Assume that you want 95% confidence that your error is no more than 0.035. The sample should includemen and women.(Type whole numbers.) **Sample Size Calculation for Estimating Difference in Population Proportions**To estimate the difference between two population proportions with a margin of error $ E $ and a confidence level of $ 1 - \alpha $, use the following formula:\[E = z_{\alpha/2} \sqrt{\frac{p_1 q_1}{n_1} + \frac{p_2 q_2}{n_2}}\]Where:- $ z_{\alpha/2} $ is the z-value corresponding to the desired confidence level.- $ p_1 $ and $ p_2 $ are the estimated proportions for each population.- $ q_1 = 1 - p_1 $ and $ q_2 = 1 - p_2 $.- $ n_1 $ and $ n_2 $ are the sample sizes for each population.To simplify calculations when the sample sizes are the same ($ n_1 = n_2 = n $), and $ p_1 $, $ q_1 $, $ p_2 $, and $ q_2 $ are unknown, assume these proportions as 0.5 for maximum variability. This simplifies the expression for $ n $:\[n = \frac{z_{\alpha/2}^2}{2E^2}\]This formula helps determine the sample size required to achieve the desired accuracy in estimating the difference between two proportions.

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Answered: want 95% confidence that your error is…

want 95% confidence that your error is no more than 0.035. The sample should include men and women. (Type whole numbers.)

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

Use the expression in the accompanying discussion of sample size to find the size of each sample if you want to estimate the difference between proportions of men and women who own smartphones. Assume that you want 95% confidence that your error is no more than 0.035.

The sample should include

men and women.

(Type whole numbers.)

**Sample Size Calculation for Estimating Difference in Population Proportions**

To estimate the difference between two population proportions with a margin of error $ E $ and a confidence level of $ 1 - \alpha $, use the following formula:

\[
E = z_{\alpha/2} \sqrt{\frac{p_1 q_1}{n_1} + \frac{p_2 q_2}{n_2}}
\]

Where:
- $ z_{\alpha/2} $ is the z-value corresponding to the desired confidence level.
- $ p_1 $ and $ p_2 $ are the estimated proportions for each population.
- $ q_1 = 1 - p_1 $ and $ q_2 = 1 - p_2 $.
- $ n_1 $ and $ n_2 $ are the sample sizes for each population.

To simplify calculations when the sample sizes are the same ($ n_1 = n_2 = n $), and $ p_1 $, $ q_1 $, $ p_2 $, and $ q_2 $ are unknown, assume these proportions as 0.5 for maximum variability. This simplifies the expression for $ n $:

\[
n = \frac{z_{\alpha/2}^2}{2E^2}
\]

This formula helps determine the sample size required to achieve the desired accuracy in estimating the difference between two proportions.

Definition Definition Number of subjects or observations included in a study. A large sample size typically provides more reliable results and better representation of the population. As sample size and width of confidence interval are inversely related, if the sample size is increased, the width of the confidence interval decreases.

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