Question

We have specified a margin of error, a confidence level, and a likely range for the observed value of the sample proportion. For each exercise, obtain a sample size that will ensure a margin of error of at most the one specified (provided of course that the observed value of the sample proportion is further from $0.5$than the educated guess).Obtain a sample size that will ensure a margin of error of at most the one specified.

$$\begin{gathered} marginoferror=0.04; \\ confidencelevel=99\%; \\ likelyrange=0.7orless \end{gathered}$$

Short answer

A sample size that will ensure a margin of error of at most the one specified.is approximately$1,037$

Step-by-step solution

step1: Given information

When the margin of error is 0.04 and the confidence level is 99% , calculate the sample size.

calculation

When the margin of error is $0.04$and the confidence level is $99\%$, we can calculate the sample size.

With a $99\%$confidence level, the required value of localid="1651496589910" $z_{\alpha /2}$from table areas under the standard normal curve is 2.575

Use localid="1651496594165" $p{\text{'}}_g$$=.05$Because the value in the range is close to $0.5.$

The sample size for this study is,

localid="1651496599039" $$\begin{gathered} n=0.25{\left(\frac{z_{\underset{\_}{a}}^2}{E}\right)}^2 \\ =0.25{\left(\frac{2.575}{0.04}\right)}^2 \\ =0.25(4,144.14) \\ =1,036.04 \\ \approx 1,037 \end{gathered}$$