Question

In a survey, the planning value for the population proportion is p* = 0.35. How large a sample should be taken to provide a 95% confidence interval with a margin of error of 0.05? Round your answer to next whole number.

Answer 1

Concepts and reason

Population proportion:

If the proportion is computed for population it is termed as population proportion and is denoted by p.

Sample proportion:

If the proportion is computed for sample it is termed as sample proportion and is denoted by $\hat p$ .

Sample size:

In statistics, the sample size is defined as the number of subjects included in a sample or it is a group of observations which is coming from the population and is considered a representative of the true population.

Fundamentals

The formula of sample size is,

$n = {\left( {\frac{{{Z_{\frac{\alpha }{2}}}}}{E}} \right)^2} \times p \times \left( {1 - p} \right)$

Here, p is the population proportion and E is the margin of error.

The ${Z_{\frac{\alpha }{2}}}$ value is obtained by using standard normal table as shown below:

Consider,

$\begin{array}{c}\\{\rm{For}}\left( {1 - \alpha } \right) = 0.95\\\\\alpha = 0.05\\\\\frac{\alpha }{2} = 0.025\\\end{array}$

From standard normal table, the required ${Z_{0.025}}$ value for 95% confidence level is 1.96

From the given information, the confidence level is 0.95.

The margin of error (E) is 0.05 and the population proportion (p) is 0.35.

The sample size for the 95% confidence interval is obtained below:

$\begin{array}{c}\\n = {\left( {\frac{{{Z_{\frac{\alpha }{2}}}}}{E}} \right)^2} \times p \times \left( {1 - p} \right)\\\\ = {\left( {\frac{{{Z_{\frac{{0.05}}{2}}}}}{{0.05}}} \right)^2} \times 0.35 \times \left( {1 - 0.35} \right)\\\\ = {\left( {\frac{{1.96}}{{0.05}}} \right)^2} \times 0.35 \times 0.65\\\\ = 1536.64 \times 0.35 \times 0.65\\\end{array}$

$\begin{array}{c}\\ = 1536.64 \times 0.2275\\\\ = 349.5856\\\\ \approx 350\\\end{array}$

Ans:

The sample taken to provide a 95% confidence interval with a margin of error of .05 is 350.

Answer 2