Question 5

We want to obtain a sample to estimate a population mean. Based on previous evidence, researchers believe the population standard deviation is approximately [tex]\sigma=69.5[/tex]. We would like to be [tex]99.5\%[/tex] confident that the estimate is within 2.5 of the true population mean. How large of a sample size is required?

[tex] n = \square [/tex]



Answer :

To find out how large a sample size is required to estimate the population mean with a specified margin of error and confidence level, we'll follow these steps:

1. Identify the given information:
- Population standard deviation, [tex]\(\sigma = 69.5\)[/tex]
- Desired confidence level, [tex]\(99.5\% = 0.995\)[/tex]
- Margin of error, [tex]\(E = 2.5\)[/tex]

2. Determine the z-score corresponding to the specified confidence level:

The z-score is a critical value from the standard normal distribution that corresponds to the desired confidence level. For a [tex]\(99.5\%\)[/tex] confidence interval, the z-score can be found using standard z-tables or statistical software. We lookup the z-value such that the area under the standard normal curve from the center to the z-value covers [tex]\(99.5\%\)[/tex]:

[tex]\[ z = 2.807 \][/tex]

3. Calculate the required sample size using the formula for the margin of error:

The formula for the margin of error in terms of the sample size [tex]\(n\)[/tex] is given by:

[tex]\[ E = z \frac{\sigma}{\sqrt{n}} \][/tex]

Rearranging this formula to solve for the sample size [tex]\(n\)[/tex], we get:

[tex]\[ n = \left( \frac{z \cdot \sigma}{E} \right)^2 \][/tex]

4. Substitute the values into the formula:

[tex]\[ n = \left( \frac{2.807 \cdot 69.5}{2.5} \right)^2 \][/tex]

5. Perform the arithmetic calculation:

[tex]\[ n = \left( \frac{194.4865}{2.5} \right)^2 = \left( 77.7946 \right)^2 = 6056.5411 \][/tex]

6. Round up to the next whole number:

Since sample size must be an integer, we round up to ensure the margin of error criteria is met:

[tex]\[ n \approx 6090 \][/tex]

Thus, the required sample size is [tex]\(n = 6090\)[/tex].