Let's go through the problem step-by-step to find the required values.
1. Sample Statistic ([tex]\(\bar{x}\)[/tex])
This is the mean of the sample data. Based on the given information:
[tex]\(\bar{x} = 41.39\)[/tex]
2. Point Estimate for the Average Waist Size
The point estimate for the population parameter (the average waist size) is also typically the sample mean:
[tex]\(\text{Point Estimate} = 41.39\)[/tex]
3. Level of Confidence as a Decimal
The given confidence level is 95.0%. Expressed as a decimal, this is:
[tex]\(\text{Level of Confidence} = 0.95\)[/tex]
4. Margin of Error
The margin of error is calculated using the formula:
[tex]\[
\text{Margin of Error} = \text{Confidence Level} \times \text{Standard Error}
\][/tex]
Using the data provided:
[tex]\[
\text{Confidence Level} = 2.119
\][/tex]
[tex]\[
\text{Standard Error} = 1.033
\][/tex]
[tex]\[
\text{Margin of Error} = 2.119 \times 1.033 = 2.188927
\][/tex]
5. Confidence Interval
The confidence interval is calculated as:
[tex]\[
\text{Confidence Interval} = (\text{Mean} - \text{Margin of Error}, \text{Mean} + \text{Margin of Error})
\][/tex]
Using the given mean and the calculated margin of error:
[tex]\[
\text{Lower Bound} = 41.39 - 2.188927 = 39.201073
\][/tex]
[tex]\[
\text{Upper Bound} = 41.39 + 2.188927 = 43.578927
\][/tex]
So, the confidence interval is:
[tex]\[
(39.201073, 43.578927)
\][/tex]
Putting it all together, we have:
- Sample Statistic [tex]\(\bar{x} = 41.39\)[/tex]
- Point Estimate for the Average Waist Size [tex]\(= 41.39\)[/tex]
- Level of Confidence as a Decimal [tex]\(= 0.95\)[/tex]
- Margin of Error [tex]\(= 2.188927\)[/tex]
- Confidence Interval [tex]\(= (39.201073, 43.578927)\)[/tex]
Hence, the detailed solution for each part is:
- Sample Statistic: [tex]\(\bar{x} = 41.39\)[/tex]
- Point Estimate for the Average Waist Size: [tex]\(41.39\)[/tex]
- Level of Confidence as a decimal: [tex]\(0.95\)[/tex]
- Margin of Error: [tex]\(2.188927\)[/tex]
- Confidence Interval: [tex]\((39.201073, 43.578927)\)[/tex]
