You are trying to estimate the average amount a family spends on food during a year. In the past, the standard deviation of the amount a family has spent on food during a year has been σ = $1200. If you want to be 99% sure that you have estimated average family food expenditures within $60, how many families do you need to survey? Place your answer, a whole number, do not use any decimals, in the blank___.

For example, 1234 would be a legitimate entry.

To solve this problem, we use the formula for determining the required sample size $ n $ for estimating a population mean:

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

Where:
- $ Z_{\alpha/2} $ is the Z-value corresponding to the desired confidence level (99% confidence in this case),
- $ \sigma $ is the population standard deviation,
- $ E $ is the margin of error.

### Step 1: Identify the parameters
- $ Z_{\alpha/2} $ for a 99% confidence level: From Z-tables, $ Z_{\alpha/2} = 2.576 $,
- $ \sigma = 1200 $ (the standard deviation),
- $ E = 60 $ (the margin of error).

### Step 2: Substitute the values into the formula
\[
n = \left( \frac{2.576 \cdot 1200}{60} \right)^2
\]

Now let's compute this.The required sample size is 2,654 families. 

Do you want further details or have any other questions? Here are some related questions to consider:

1. How would the sample size change with a lower confidence level, like 95%?
2. What happens to the required sample size if the margin of error is increased to $100?
3. How does increasing the standard deviation impact the sample size?
4. Why is it important to have a lower margin of error in this context?
5. Can the sample size be reduced by increasing the number of surveys?

**Tip:** A smaller margin of error or higher confidence level requires a larger sample size to maintain precision.

You are trying to estimate the average amount a family spends on food during a year. In the past, the standard deviation of the amount a family has spent on food during a year has been σ = $1200. If you want to be 99% sure that you have estimated average family food expenditures within $60, how many families do you need to survey? Place your answer, a whole number, do not use any decimals, in the blank___.

Learn how to calculate the required sample size to estimate the average family food expenditures with 99% confidence. This problem involves key statistical concepts like sample size determination, confidence intervals, and the application of the formula n = (Zα/2 * σ / E)^2. Suitable for college-level statistics students.

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