Answer :
To construct a confidence interval for the population standard deviation [tex]\(\sigma\)[/tex] based on the sample data, we need a few key statistics and steps. Here, I'll break down the entire process for you.
### Step 1: Understanding the Data
Given the data set:
[tex]\[ 4546, 4797, 4126, 4589, 4336, 4092, 4665, 4777, 3848, 4060, 4572, 4277 \][/tex]
The sample size ([tex]\(n\)[/tex]) is 12.
### Step 2: Calculating the Sample Variance
The sample variance is used as an estimator for the variance of the population. The formula for sample variance [tex]\(s^2\)[/tex] is:
[tex]\[ s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \][/tex]
where [tex]\( \bar{x} \)[/tex] is the sample mean.
Here, the sample variance is given as:
[tex]\[ s^2 = 96813.72 \][/tex]
### Step 3: Degrees of Freedom
The degrees of freedom ([tex]\( \text{df} \)[/tex]) for calculating the variance and the subsequent chi-square distribution is:
[tex]\[ \text{df} = n - 1 = 12 - 1 = 11 \][/tex]
### Step 4: Confidence Level
We are constructing a [tex]\(95\%\)[/tex] confidence interval. The confidence level [tex]\( (1 - \alpha) \)[/tex] is [tex]\(0.95\)[/tex], hence:
[tex]\[ \alpha = 1 - 0.95 = 0.05 \][/tex]
### Step 5: Chi-Square Critical Values
We use the chi-square distribution to get the critical values for our confidence interval. They are:
- Lower critical value ([tex]\( \chi^2_{\frac{\alpha}{2}, \text{df}} \)[/tex]): [tex]\(3.816\)[/tex]
- Upper critical value ([tex]\( \chi^2_{1-\frac{\alpha}{2}, \text{df}} \)[/tex]): [tex]\(21.920\)[/tex]
### Step 6: Calculating the Confidence Interval for Variance
The formula for the confidence interval for the variance [tex]\(\sigma^2\)[/tex] is:
[tex]\[ \left( \frac{(n-1)s^2}{\chi^2_{1-\frac{\alpha}{2}, \text{df}}}, \frac{(n-1)s^2}{\chi^2_{\frac{\alpha}{2}, \text{df}}} \right) \][/tex]
Plugging in:
[tex]\[ \left( \frac{(11) \cdot 96813.72}{21.920}, \frac{(11) \cdot 96813.72}{3.816} \right) \][/tex]
[tex]\[ = (48583.42, 279093.60) \][/tex]
### Step 7: Calculating the Confidence Interval for Standard Deviation
To find the confidence interval for the standard deviation [tex]\(\sigma\)[/tex], we take the square root of the interval bounds for the variance:
[tex]\[ \left( \sqrt{48583.42}, \sqrt{279093.60} \right) \][/tex]
[tex]\[ \approx (220.4 \text{ seconds}, 528.3 \text{ seconds}) \][/tex]
### Step 8: Final Confidence Interval
Therefore, the [tex]\(95\% \)[/tex] confidence interval for the population standard deviation [tex]\(\sigma\)[/tex] based on the sample data is:
[tex]\[ 220.4 \text{ seconds} < \sigma < 528.3 \text{ seconds} \][/tex]
This is the bounded interval that estimates the standard deviation of the duration of times of gameplay featuring substance use in this study.
### Step 1: Understanding the Data
Given the data set:
[tex]\[ 4546, 4797, 4126, 4589, 4336, 4092, 4665, 4777, 3848, 4060, 4572, 4277 \][/tex]
The sample size ([tex]\(n\)[/tex]) is 12.
### Step 2: Calculating the Sample Variance
The sample variance is used as an estimator for the variance of the population. The formula for sample variance [tex]\(s^2\)[/tex] is:
[tex]\[ s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \][/tex]
where [tex]\( \bar{x} \)[/tex] is the sample mean.
Here, the sample variance is given as:
[tex]\[ s^2 = 96813.72 \][/tex]
### Step 3: Degrees of Freedom
The degrees of freedom ([tex]\( \text{df} \)[/tex]) for calculating the variance and the subsequent chi-square distribution is:
[tex]\[ \text{df} = n - 1 = 12 - 1 = 11 \][/tex]
### Step 4: Confidence Level
We are constructing a [tex]\(95\%\)[/tex] confidence interval. The confidence level [tex]\( (1 - \alpha) \)[/tex] is [tex]\(0.95\)[/tex], hence:
[tex]\[ \alpha = 1 - 0.95 = 0.05 \][/tex]
### Step 5: Chi-Square Critical Values
We use the chi-square distribution to get the critical values for our confidence interval. They are:
- Lower critical value ([tex]\( \chi^2_{\frac{\alpha}{2}, \text{df}} \)[/tex]): [tex]\(3.816\)[/tex]
- Upper critical value ([tex]\( \chi^2_{1-\frac{\alpha}{2}, \text{df}} \)[/tex]): [tex]\(21.920\)[/tex]
### Step 6: Calculating the Confidence Interval for Variance
The formula for the confidence interval for the variance [tex]\(\sigma^2\)[/tex] is:
[tex]\[ \left( \frac{(n-1)s^2}{\chi^2_{1-\frac{\alpha}{2}, \text{df}}}, \frac{(n-1)s^2}{\chi^2_{\frac{\alpha}{2}, \text{df}}} \right) \][/tex]
Plugging in:
[tex]\[ \left( \frac{(11) \cdot 96813.72}{21.920}, \frac{(11) \cdot 96813.72}{3.816} \right) \][/tex]
[tex]\[ = (48583.42, 279093.60) \][/tex]
### Step 7: Calculating the Confidence Interval for Standard Deviation
To find the confidence interval for the standard deviation [tex]\(\sigma\)[/tex], we take the square root of the interval bounds for the variance:
[tex]\[ \left( \sqrt{48583.42}, \sqrt{279093.60} \right) \][/tex]
[tex]\[ \approx (220.4 \text{ seconds}, 528.3 \text{ seconds}) \][/tex]
### Step 8: Final Confidence Interval
Therefore, the [tex]\(95\% \)[/tex] confidence interval for the population standard deviation [tex]\(\sigma\)[/tex] based on the sample data is:
[tex]\[ 220.4 \text{ seconds} < \sigma < 528.3 \text{ seconds} \][/tex]
This is the bounded interval that estimates the standard deviation of the duration of times of gameplay featuring substance use in this study.
