91Ó°ÊÓ

The sample size needed to estimate the difference between two population proportions to within a margin of error \(E\) with a confidence level of \(1-\alpha\) can be found by using the following expression: $$ E=z_{\alpha / 2} \sqrt{\frac{p_{1} q_{1}}{n_{1}}+\frac{p_{2} q_{2}}{n_{2}}} $$ Replace \(n_{1}\) and \(n_{2}\) by \(n\) in the preceding formula (assuming that both samples have the same size) and replace each of \(p_{1}, q_{1}, p_{2},\) and \(q_{2}\) by 0.5 (because their values are not known). Solving for \(n\) results in this expression: $$n=\frac{z_{\alpha / 2}^{2}}{2 E^{2}}$$ Use this expression to find the size of each sample if you want to estimate the difference between the proportions of men and women who own smartphones. Assume that you want \(95 \%\) confidence that your error is no more than 0.03

Step by step solution

01

- Understand the Given Data

Identify the given information: confidence level (95%) and margin of error (0.03).

02

- Find the Z-value for the Confidence Level

Determine the z-value corresponding to a 95% confidence level, which is found in statistical tables or standard values. For a confidence level of 95%, the z-value is 1.96.

03

- Use the Provided Formula

Substitute the z-value and margin of error into the formula to solve for the sample size: Where: .

04

- Calculate Sample Size

Substitute the values into the formula n ≈ and solve for n.

05

- Simplify the Equation

Simplify the expression to find the value of n.

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Key Concepts

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

Confidence Level

In statistics, the confidence level represents the percentage of times that the confidence interval would contain the true population parameter if you repeated the study multiple times. For example, a 95% confidence level means that if you were to take 100 different samples and compute a confidence interval for each sample, we would expect that approximately 95 out of the 100 confidence intervals will contain the true population parameter.

When calculating the sample size, the confidence level determines the z-value, which is a critical component in the formula. Typically, common confidence levels include 90%, 95%, and 99%. The higher the confidence level, the larger the sample size required for a given margin of error, because we want to be more certain that our interval contains the true parameter.

Margin of Error

The margin of error (MOE) indicates the range within which we expect the true population parameter to lie. It gives you a sense of how much you can expect your sample estimate to deviate from the actual population parameter. A smaller margin of error requires a larger sample size to achieve, because you are demanding greater precision.

In the given exercise, the margin of error is specified as 0.03. This means we want our estimate of the difference in proportions to be within ±3% of the true difference. The formula for estimating sample size, as shown in the problem, includes the margin of error in the denominator, highlighting its inverse relationship to the required sample size.

Z-Value

The z-value, or z-score, corresponds to the number of standard deviations a data point is from the mean in a standard normal distribution. In the context of confidence intervals, the z-value defines the number of standard deviations we need to extend from the sample mean to find our interval estimate.

For a 95% confidence level, the z-value is 1.96. This is derived from standard normal distribution tables, indicating that 95% of the data lies within ±1.96 standard deviations of the mean. Thus, in the sample size formula, the z-value multiplies with the standard error to ensure that the confidence interval covers the desired proportion of the population.

Proportion Difference

Proportion difference measures the difference between two population proportions. In this context, it refers to the difference in proportions of men and women who own smartphones. When calculating sample size, we often start with an assumption of equal proportions, especially when no preliminary data is available. Hence, using 0.5 for both proportions makes the formula more conservative, ensuring that the sample size is adequate.

The formula assumes equal sample sizes for both groups being compared, simplifying calculations and application. This approach helps balance the margin of error evenly across both populations, providing a robust estimate of the proportion difference given the specified statistical confidence.