Question

You want to obtain a sample to estimate the proportion of a population that possess a particular genetic marker. Based on previous evidence, you believe approximately p*=59%
of the population have the genetic marker. You would like to be 98% confident that your estimate is within 5% of the true population proportion. How large of a sample size is required?

n =




Do not round mid-calculation. However, you may use a critical value accurate to three decimal places.

Answer 1

To determine the required sample size for estimating a population proportion \( p \) with a specified level of confidence and margin of error, you can use the formula for the sample size \( n \):

\[
n = \left( \frac{Z^2 \cdot p^* \cdot (1 - p^*)}{E^2} \right)
\]

Where:
- \( Z \) is the z-score corresponding to the desired confidence level.
- \( p^* \) is the estimated population proportion.
- \( E \) is the desired margin of error.

### Step 1: Find the Z-score for a 98% confidence level

For a 98% confidence level, the critical value \( Z \) can be found using a Z-table or calculator. The Z-score for 98% confidence (two-tailed) is approximately:

\[
Z \approx 2.326
\]

### Step 2: Substitute the known values into the formula

Given:
- \( p^* = 0.59 \)
- \( E = 0.05 \)

### Step 3: Calculate \( n \)

Substituting the values into the formula:

\[
n = \frac{(2.326)^2 \cdot 0.59 \cdot (1 - 0.59)}{(0.05)^2}
\]

Calculating each part:

1. Calculate \( Z^2 \):
\[
(2.326)^2 \approx 5.419
\]

2. Calculate \( p^*(1 - p^*) \):
\[
0.59 \cdot (1 - 0.59) = 0.59 \cdot 0.41 \approx 0.2419
\]

3. Calculate \( E^2 \):
\[
(0.05)^2 = 0.0025
\]

Putting these into the sample size formula:

\[
n = \frac{5.419 \cdot 0.2419}{0.0025}
\]

Now calculate the numerator:

\[
5.419 \cdot 0.2419 \approx 1.3106
\]

Now calculate \( n \):

\[
n = \frac{1.3106}{0.0025} \approx 524.24
\]

Since you cannot have a fraction of a sample, round up to the next whole number:

\[
n \approx 525
\]

### Final Answer:

\[
n = 525
\]

Thus, the required sample size is **525**.