Question 1178049: Twelve different video games showing substance use were observed and the duration of times of gameplay (in seconds) are listed below. The design of the study justifies the assumption that the sample can be treated as a simple random sample. Use the sample data to construct a 90%
confidence interval estimate of σ, the standard deviation of the duration times of gameplay. Assume that this sample was obtained from a population with a normal distribution.
4,665 4,432 4,811 4,393 4,242 4,969
4,100 4,721 4,864 4,244 4,364 4,254
The confidence interval estimate is
____ sec<σ<__sec.
(Round to one decimal place as needed.)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.**
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