You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately
σ

To determine the required sample size for estimating a population mean, we use the formula for the margin of error:

$E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

Where:

• $E$ is the margin of error (2.5 in this case),
• $z_{\alpha/2}$ is the critical z-value corresponding to the desired confidence level (94%),
• $\sigma$ is the population standard deviation (37.4),
• $n$ is the sample size, which we need to find.

Step 1: Find the critical z-value for a 94% confidence level.

For a 94% confidence level, the remaining area in the tails is 6% (since 100% - 94% = 6%), and half of that is in each tail (3% in each tail). So, the z-value corresponding to 97% cumulative probability (since 94% + 3% = 97%) is required.

Using a standard z-table or calculator, the z-value for 97% cumulative probability is approximately:

$z_{\alpha/2} = 1.88$

Step 2: Rearranging the formula to solve for $n$:

$n = \left(\frac{z_{\alpha/2} \cdot \sigma}{E}\right)^2$

Substitute the values into the formula:

$n = \left(\frac{1.88 \cdot 37.4}{2.5}\right)^2$

Step 3: Calculate the sample size.

First, compute the numerator:

$1.88 \cdot 37.4 = 70.232$

Then divide by the margin of error:

$\frac{70.232}{2.5} = 28.093$

Finally, square the result:

$n = (28.093)^2 = 788.23$

Since the sample size must be an integer, we round up (since you can't sample a fraction of an individual), so the required sample size is:

$n = 789$

Final Answer:

The required sample size is 789.

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Would you like a more detailed explanation or have additional questions?

Here are five follow-up questions you might find interesting:

1. How do you calculate the sample size when the population standard deviation is unknown?
2. What happens to the required sample size if the confidence level increases to 99%?
3. How does the margin of error affect the sample size requirement?
4. How would you interpret the z-value if the confidence level were 90% instead of 94%?
5. What are the assumptions made in this sample size calculation?

Tip: If you ever encounter a problem with unknown standard deviation, consider using the sample standard deviation in place, but use a t-distribution instead of the z-distribution.