Question

Margin of error$=0.04$

Confidence level$=99\%$

Educated guess$=0.3$

(a) 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 $0.5$that of the educated guess.

(b). Compare your answer to the corresponding one and explain the reason for the difference, if any.

Short answer

(a) The required sample size is $1,037.$

(b) The sample size obtain in part a is smaller when compared to the sample size obtained in $11.36$because in part a the educated guess $0.25$is used which provides the more accurate sample, whereas in $11.36$the constant $0.25$is used which provides the larger sample.

Step-by-step solution

Part (a) Step 1: Given information

Margin of error$=0.04$

Confidence level$=99\%$

Educated guess $=0.5$

Part (a) Step 2: Explanation

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

With a $99\%$confidence level, the required value of $z_{\frac{a}{2}}$from table areas under the standard normal curve is $2.575$.

The sample size is

$n=0.25{\left(\frac{\frac{z_a}{2}}{E}\right)}^2$

$=0.25{\left(\frac{2.575}{0.04}\right)}^2$

$=0.25(4,144.1)$

$=1,036.025$

$\approx 1,037.$

Part (b) Step 1: Given information

Margin of error$=0.04$

Confidence level$=99\%$

Educated guess$=0.5$

Part (b) Step 2: Explanation

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

With a $99\%$confidence level, the required value of $z_{\frac{a}{2}}$from table areas under the standard normal curve is $2.575$.

The sample size is

$n=0.25{\left(\frac{z_{\underset{¯}{\alpha }}}{E}\right)}^2$

$=0.25{\left(\frac{2.575}{0.04}\right)}^2$

$=0.25(4,144.14)$

$=1,036.04$

$\approx 1,037.$

The sample size obtain in part a is smaller when compared to the sample size obtained in $11.36$. because in part a the educated guess $0.25$is used which provides the more accurate sample, whereas in $11.36$the constant$0.25$ is used which provides the larger sample.