Question

Margin of error$=0.02$

Confidence level$=95\%$

Educated guess$=0.6$

(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 $2,305.$

(b) When compared to the sample size obtained in $11.32$, the sample size obtained in part an is less. because the educated guess is utilised in part, which produces a more accurate sample, whereas the constant $0.25$is used in , which provides a bigger sample.

Step-by-step solution

Part (a) Step 1: Given information

Margin of error$=0.02$

Confidence level$=95\%$

Educated guess$=0.6$

Part (a) Step 2: Explanation

From the given values

When the margin of error is $0.02$and the confidence level is $95\%$, calculate the sample size.

With a $95\%$confidence level, the required value of $z_{\frac{a}{2}}$from table areas under the standard normal curve is $1.96$.

The sample size is

$n=0.6(1-0.6){\left(\frac{z_{\frac{\alpha }{2}}}{E}\right)}^2$

$=0.6(1-0.6){\left(\frac{1.96}{0.02}\right)}^2$

$=0.24(9,604)$

$=2,305.96$

$\approx 2,305.$

Part (b) Step 1: Given information

Margin of error$=0.02$

Confidence level $=95\%$

Educated guess $=0.6$

Part (b) Step 2: Explanation

From the given data

When the margin of error is $0.02$and the confidence level is $95\%$, calculate the sample size.

With a $95\%$confidence level, the required value of $z_{\frac{a}{2}}$from table areas under the standard normal curve is $1.96$.

The sample size is

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

$=0.25{\left(\frac{1.96}{0.02}\right)}^2$

$=0.25(9,604)$

$=2,401.$

When compared to the sample size obtained in $11.32$, the sample size obtained in part an is less. because the educated guess is utilised in part, which produces a more accurate sample, whereas the constant $0.25$ is used in , which provides a bigger sample.