To calculate the sample size needed, we can use the formula for the required sample size when estimating a population mean with a known standard deviation:

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

Where:

• $n$ = required sample size
• $Z_{\alpha/2}$ = critical value (z-score) corresponding to the desired confidence level
• $\sigma$ = population standard deviation
• $E$ = margin of error (the maximum allowable difference between the sample mean and the population mean)

Step 1: Determine the critical value $Z_{\alpha/2}$

For a 99% confidence level, the area in each tail of the standard normal distribution is $\frac{1 - 0.99}{2} = 0.005$. Using a z-table or calculator, the critical value for a 99% confidence interval is:

$Z_{\alpha/2} = 2.576$

Step 2: Set up the formula

Given:

• $\sigma = 77.2$
• $E = 4$
• $Z_{\alpha/2} = 2.576$

Substitute these values into the sample size formula:

$n = \left( \frac{2.576 \cdot 77.2}{4} \right)^2$

Step 3: Perform the calculations

First, calculate the expression inside the parentheses:

$\frac{2.576 \cdot 77.2}{4} = \frac{198.842}{4} = 49.7105$

Now square that result:

$n = (49.7105)^2 = 2471.12$

Step 4: Conclusion

The required sample size is:

$n = 2472 \quad \text{(rounded up to the nearest whole number)}$

Summary:

• Z-score: 2.576
• Margin of Error (MoE): 4
• Sample size required: 2472

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Would you like more details on any part of this calculation?

Here are 5 related questions to explore:

1. How does increasing the confidence level affect the required sample size?
2. How would the sample size change if the margin of error was reduced to 2?
3. What happens to the required sample size if the population standard deviation is unknown?
4. How does the population size influence the required sample size, if at all?
5. What if you wanted a 95% confidence level instead of 99%—how would that impact the result?

Tip: When planning a sample size, it's always a good idea to overestimate slightly to ensure precision, especially if the true population standard deviation is not well-known.