College freshman typically spend a large amount of money purchasing pizza during the academic year. In order to estimate these expenditures, a sample of 36 college freshman, showed a sample mean of $850, with a sample standard deviation of $54. A 99% confidence interval estimate of the population mean is: 




Question 12Select one:

$823.72 to $876.28



$832.36 to $867.64


$826.82 to $873.18


$825.48 to $874.52

To calculate the 99% confidence interval for the population mean, we will use the following formula:

\[
\text{CI} = \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right)
\]

Where:
- $\bar{x}$ is the sample mean,
- $z$ is the critical value corresponding to the confidence level (99%),
- $s$ is the sample standard deviation, and
- $n$ is the sample size.

### Step 1: Identify the known values
- Sample mean ($\bar{x}$) = $850
- Sample standard deviation ($s$) = $54
- Sample size ($n$) = 36
- Confidence level = 99%

For a 99% confidence level, the critical value $z$ is approximately 2.576.

### Step 2: Calculate the margin of error
\[
\text{Margin of error} = z \times \left( \frac{s}{\sqrt{n}} \right)
\]

\[
\text{Margin of error} = 2.576 \times \left( \frac{54}{\sqrt{36}} \right)
\]

Let me calculate the result.The margin of error is approximately $23.18.

### Step 3: Calculate the confidence interval

Now, calculate the confidence interval by subtracting and adding the margin of error to the sample mean:

\[
\text{Lower limit} = 850 - 23.18 = 826.82
\]
\[
\text{Upper limit} = 850 + 23.18 = 873.18
\]

Thus, the 99% confidence interval estimate for the population mean is **$826.82 to $873.18**.

The correct answer is:
- **$826.82 to $873.18**

Would you like more details or further clarification?

### Related Questions:
1. How do you calculate the critical value for a given confidence level?
2. What is the impact of increasing the sample size on the confidence interval?
3. How would the confidence interval change with a different confidence level (e.g., 95%)?
4. What assumptions must be satisfied for using the z-distribution in confidence intervals?
5. How does the sample standard deviation influence the width of the confidence interval?

**Tip:** A higher confidence level increases the margin of error, leading to a wider confidence interval.

College freshman typically spend a large amount of money purchasing pizza during the academic year. In order to estimate these expenditures, a sample of 36 college freshman showed a sample mean of $850, with a sample standard deviation of $54. A 99% confidence interval estimate of the population mean is:

Select one:
$823.72 to $876.28
$832.36 to $867.64
$826.82 to $873.18
$825.48 to $874.52

Learn how to calculate a 99% confidence interval for the population mean based on a sample of college freshman expenditures on pizza. The sample mean is $850 with a standard deviation of $54. Detailed step-by-step explanation of the confidence interval calculation using the critical value for a 99% confidence level.

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