To construct a 90% confidence interval for a sampling distribution with a given mean, standard deviation, and sample size, we need to follow several steps. Here they are, broken down in detail:
1. Identify the parameters given:
- Mean ([tex]\(\mu\)[/tex]) = 150
- Standard deviation ([tex]\(\sigma\)[/tex]) = 20
- Sample size ([tex]\(n\)[/tex]) = 16
- Confidence level = 90%
2. Determine the Z-score for the 90% confidence level:
The Z-score corresponds to the point on a standard normal distribution where the area under the curve to the left of the Z-score is equal to the confidence level plus half of the remaining area outside the confidence level. For a 90% confidence level, the Z-score is approximately [tex]\(1.645\)[/tex].
3. Calculate the standard error of the mean (SEM):
The standard error measures the dispersion of the sample mean around the population mean, calculated as follows:
[tex]\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Plugging in the given values:
[tex]\[
\text{Standard Error} = \frac{20}{\sqrt{16}} = \frac{20}{4} = 5
\][/tex]
4. Calculate the margin of error (MOE):
The margin of error combines the Z-score and the standard error to provide a range for the confidence interval:
[tex]\[
\text{Margin of Error} = Z \times \text{Standard Error}
\][/tex]
Plugging in the Z-score and standard error:
[tex]\[
\text{Margin of Error} = 1.645 \times 5 = 8.225
\][/tex]
5. Determine the confidence interval:
Finally, we calculate the lower and upper bounds of the confidence interval by subtracting and adding the margin of error from/to the mean:
[tex]\[
\text{Lower Bound} = \mu - \text{Margin of Error} = 150 - 8.225 = 141.775
\][/tex]
[tex]\[
\text{Upper Bound} = \mu + \text{Margin of Error} = 150 + 8.225 = 158.225
\][/tex]
Thus, the 90% confidence interval for the sampling distribution is approximately [tex]\( (141.776, 158.224) \)[/tex].
These steps provide a clear method of constructing a confidence interval using the given data.
