Let's go through the solution step by step:
1. Sample Statistic and Point Estimate:
- The sample statistic for the average waist size is given by the mean. In this case, the mean is [tex]\( 41.39 \)[/tex].
- Therefore, the point estimate for the average waist size is also [tex]\( 41.39 \)[/tex].
2. Confidence Level as a Decimal:
- The confidence level is given as 95.0%, which needs to be converted to a decimal. To do this, divide the percentage by 100:
- [tex]\( \frac{95.0}{100} = 0.95 \)[/tex]
- The level of confidence as a decimal is [tex]\( 0.95 \)[/tex].
3. Margin of Error:
- The margin of error can be calculated using the formula: [tex]\( \text{Margin of Error} = \text{Standard Error} \times Z \)[/tex]
- For a 95% confidence level, the [tex]\( Z \)[/tex]-score is typically 1.96 (assuming a normal distribution).
- Given the standard error is [tex]\( 1.033 \)[/tex] and using the [tex]\( Z \)[/tex]-score of 1.96:
- [tex]\( \text{Margin of Error} = 1.033 \times 1.96 \approx 2.02468 \)[/tex].
4. Confidence Interval:
- A confidence interval is given by the formula: [tex]\( \text{Confidence Interval} = (\bar{x} - \text{Margin of Error}, \bar{x} + \text{Margin of Error}) \)[/tex]
- Using the mean [tex]\( 41.39 \)[/tex] and the margin of error [tex]\( 2.02468 \)[/tex]:
- Lower bound: [tex]\( 41.39 - 2.02468 \approx 39.36532 \)[/tex]
- Upper bound: [tex]\( 41.39 + 2.02468 \approx 43.41468 \)[/tex]
- Therefore, the confidence interval is [tex]\( (39.36532, 43.41468) \)[/tex].
Thus, the detailed solution is as follows:
- Sample Statistic: [tex]\( 41.39 \)[/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]\( 2.02468 \)[/tex]
- Confidence Interval: [tex]\( (39.36532, 43.41468) \)[/tex]
