Apex Solutions, a local moving and hauling firm, wants to determine their average daily expenditures (in dollars). A sample of the expenditures for 52 days revealed average expenditures of $136,000 and a sample standard deviation of $17,000. Provide a 90% confidence interval estimate for the average daily expenditures

To calculate a 90% confidence interval for the average daily expenditures of Apex Solutions, we use the following formula for the confidence interval of the mean when the population standard deviation is unknown:

\[
\text{Confidence Interval} = \bar{x} \pm t_{\alpha/2} \times \frac{s}{\sqrt{n}}
\]

Where:
- $\bar{x} = 136,000$ (the sample mean)
- $s = 17,000$ (the sample standard deviation)
- $n = 52$ (the sample size)
- $t_{\alpha/2}$ is the critical value from the t-distribution for a 90% confidence level and $n-1 = 51$ degrees of freedom.

### Step 1: Find the critical t-value for 90% confidence level.
For a 90% confidence level, $\alpha = 0.10$, and $\alpha/2 = 0.05$. Using a t-table or a calculator, the critical value $t_{0.05}$ for 51 degrees of freedom is approximately 1.676.

### Step 2: Calculate the margin of error.
The margin of error is given by:

\[
\text{Margin of Error} = t_{\alpha/2} \times \frac{s}{\sqrt{n}} = 1.676 \times \frac{17,000}{\sqrt{52}}
\]

Let's calculate that.The margin of error is approximately $3,951.13.

### Step 3: Calculate the confidence interval.

Now, we can compute the 90% confidence interval:

\[
\text{Confidence Interval} = \bar{x} \pm \text{Margin of Error} = 136,000 \pm 3,951.13
\]

So, the confidence interval is:

\[
(136,000 - 3,951.13, 136,000 + 3,951.13) = (132,048.87, 139,951.13)
\]

### Conclusion:
The 90% confidence interval for the average daily expenditures of Apex Solutions is approximately $ (132,049, 139,951) $.

Would you like further details or clarifications?

Here are 5 related questions:
1. What does the confidence interval tell us about the population mean?
2. How does increasing the confidence level affect the width of the confidence interval?
3. What assumptions must be met to use the t-distribution for this calculation?
4. How would the confidence interval change if the sample size were larger?
5. Can the population mean ever be outside the confidence interval?

**Tip**: A larger sample size generally leads to a narrower confidence interval, providing a more precise estimate.

Apex Solutions, a local moving and hauling firm, wants to determine their average daily expenditures (in dollars). A sample of the expenditures for 52 days revealed average expenditures of $136,000 and a sample standard deviation of $17,000. Provide a 90% confidence interval estimate for the average daily expenditures.

Learn how to calculate a 90% confidence interval for Apex Solutions' average daily expenditures. Using a sample mean of $136,000 and standard deviation of $17,000 from 52 days, this guide walks through the steps, including the t-distribution critical value and the margin of error calculation. Suitable for college and advanced high school students.

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