To find the minimum sample size required, we need to use the formula for the sample size in a confidence interval for the mean when the population standard deviation is known. The formula is:

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

where:

• $n$ is the sample size,
• $Z_{\alpha/2}$ is the critical value from the standard normal distribution corresponding to the desired confidence level (99% in this case),
• $\sigma$ is the population standard deviation (0.43 ounces),
• $E$ is the margin of error (0.07 ounces).

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

For a 99% confidence level, the critical value $Z_{\alpha/2}$ corresponds to the value where 99% of the distribution is within this range. In terms of the standard normal distribution, we look for the critical value such that 0.5% is in each tail (since $1 - 0.99 = 0.01$, and half of that is 0.005). The critical value $Z_{\alpha/2}$ for 99% confidence is:

$Z_{\alpha/2} \approx 2.576$

Step 2: Plug the values into the formula

We have:

• $Z_{\alpha/2} = 2.576$,
• $\sigma = 0.43$,
• $E = 0.07$.

Now, substitute these into the sample size formula:

$n = \left( \frac{2.576 \cdot 0.43}{0.07} \right)^2$

Step 3: Simplify the expression

First, calculate the value inside the parentheses:

$\frac{2.576 \cdot 0.43}{0.07} = \frac{1.10768}{0.07} \approx 15.824$

Now, square this result:

$n = (15.824)^2 \approx 250.417$

Step 4: Round to the next whole number

Since the sample size must be a whole number and we need the minimum size, round 250.417 up to the next whole number:

$n = 251$

Thus, the minimum sample size needed is 251.

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Would you like further details or have any questions? Here are five related questions to deepen understanding:

1. How does the confidence level affect the sample size?
2. What is the effect of increasing the margin of error on the sample size?
3. What if the population standard deviation was unknown? How would the approach change?
4. Why do we square the result in the sample size formula?
5. How is the critical value $Z_{\alpha/2}$ determined for different confidence levels?

Tip: The margin of error decreases as the sample size increases, making your estimate of the population mean more precise.