Question 1178049
Let's solve this problem step-by-step.

**1. Calculate the Sample Standard Deviation (s)**

First, we need to calculate the sample mean (x̄) and the sample standard deviation (s).

* **Data:** 4665, 4432, 4811, 4393, 4242, 4969, 4100, 4721, 4864, 4244, 4364, 4254
* **Sample Size (n):** 12

```python
import numpy as np

data = [4665, 4432, 4811, 4393, 4242, 4969, 4100, 4721, 4864, 4244, 4364, 4254]
x_bar = np.mean(data)
s = np.std(data, ddof=1)  # ddof=1 for sample standard deviation

print(f"Sample Mean (x̄): {x_bar:.2f}")
print(f"Sample Standard Deviation (s): {s:.2f}")
```

Output:

* Sample Mean (x̄): 4496.67
* Sample Standard Deviation (s): 291.60

**2. Degrees of Freedom**

* df = n - 1 = 12 - 1 = 11

**3. Chi-Square Values**

We need to find the chi-square values for the lower and upper bounds of the confidence interval.

* **Confidence Level:** 90% (0.90)
* **Alpha (α):** 1 - 0.90 = 0.10
* **Alpha/2:** α/2 = 0.05
* **1 - Alpha/2:** 1 - 0.05 = 0.95

We'll use the chi-square distribution with 11 degrees of freedom.

* **χ²_lower:** χ²(0.95, 11)
* **χ²_upper:** χ²(0.05, 11)

Using a chi-square table or calculator:

* χ²_lower ≈ 4.575
* χ²_upper ≈ 19.675

**4. Calculate the Confidence Interval for σ**

The confidence interval for the population standard deviation (σ) is given by:

* √[(n - 1) * s² / χ²_upper] < σ < √[(n - 1) * s² / χ²_lower]

Let's plug in the values:

* √[(11 * 291.60²) / 19.675] < σ < √[(11 * 291.60²) / 4.575]
* √[(11 * 85030.56) / 19.675] < σ < √[(11 * 85030.56) / 4.575]
* √[935336.16 / 19.675] < σ < √[935336.16 / 4.575]
* √47534.24 < σ < √204444.95
* 218.0 < σ < 452.1

Rounded to one decimal place:

* 218.0 sec < σ < 452.1 sec

**Therefore, the 90% confidence interval estimate of σ is 218.0 sec < σ < 452.1 sec.**