To construct a 95% confidence interval for a sampling distribution with a mean of 150, a standard deviation of 20, and a sample size of 16, follow these detailed steps:
1. Determine the z-score for the 95% confidence level:
- A 95% confidence level corresponds to a z-score that captures the central 95% of the distribution.
- The z-score for a 95% confidence interval is approximately 1.96.
2. Calculate the standard error of the mean (SEM):
- The standard error of the mean is calculated using the formula:
[tex]\[
\text{SEM} = \frac{\sigma}{\sqrt{n}}
\][/tex]
- Here, [tex]\(\sigma\)[/tex] (the population standard deviation) is 20, and [tex]\(n\)[/tex] (the sample size) is 16.
- Therefore:
[tex]\[
\text{SEM} = \frac{20}{\sqrt{16}} = \frac{20}{4} = 5
\][/tex]
3. Calculate the margin of error (MOE):
- The margin of error is obtained by multiplying the z-score by the standard error of the mean:
[tex]\[
\text{MOE} = z \cdot \text{SEM}
\][/tex]
- Given the z-score is approximately 1.96 and SEM is 5:
[tex]\[
\text{MOE} = 1.96 \cdot 5 \approx 9.80
\][/tex]
4. Calculate the lower and upper bounds of the confidence interval:
- The lower bound is calculated as:
[tex]\[
\text{Lower bound} = \text{Mean} - \text{MOE}
\][/tex]
[tex]\[
\text{Lower bound} = 150 - 9.80 \approx 140.20
\][/tex]
- The upper bound is calculated as:
[tex]\[
\text{Upper bound} = \text{Mean} + \text{MOE}
\][/tex]
[tex]\[
\text{Upper bound} = 150 + 9.80 \approx 159.80
\][/tex]
5. State the final confidence interval:
- The 95% confidence interval for the sampling distribution is approximately:
[tex]\[
(140.20, 159.80)
\][/tex]
So, the 95% confidence interval for a sampling distribution with a mean of 150, a standard deviation of 20, and a sample size of 16 is [tex]\((140.20, 159.80)\)[/tex].
