Question 1179445
**Step 1: Find the critical value**

Since the population standard deviation is unknown, we use the t-distribution.  

* Degrees of freedom (df) = n - 1 = 20 - 1 = 19
* Confidence level = 90%, so alpha (α) = 10% = 0.10
* We want the t-value that leaves 5% (α/2) in each tail.

Using a t-table or calculator with df = 19 and α/2 = 0.05, the critical value is approximately **1.729**.


**Step 2: Construct the confidence interval**

The formula for a confidence interval is:

```
Confidence Interval = Sample Mean ± (Critical Value) * (Standard Error)
```

Where the Standard Error (SE) = Sample Standard Deviation / √(Sample Size)

* Sample Mean = $57.22
* Critical Value = 1.729
* Sample Standard Deviation = $25.76
* Sample Size = 20

Let's calculate:

1. **Standard Error:** SE = $25.76 / √20 ≈ $5.77

2. **Margin of Error:** Margin of Error = (Critical Value) * (Standard Error) = 1.729 * $5.77 ≈ $9.97

3. **Confidence Interval:**
   $57.22 ± $9.97

   Lower Bound: $57.22 - $9.97 = $47.25
   Upper Bound: $57.22 + $9.97 = $67.19

**Therefore, the 90% confidence interval for the mean repair cost is approximately ($47.25, $67.19).**