To construct a 93% confidence interval for a sampling distribution with a mean of 150, a standard deviation of 20, and a sample size of 16, follow these steps:
1. Determine the Confidence Level and Corresponding Z-value:
- For a 93% confidence level, we need to find the z-value that corresponds to the middle 93% of the standard normal distribution.
- The confidence level implies that 7% of the distribution lies in the tails, split evenly, so each tail has 3.5% (or 0.035) in it.
- The Z critical value (z_critical) for 93% confidence level is approximately 1.8119.
2. Calculate the Standard Error:
- The standard error (SE) is calculated using the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
where [tex]\(\sigma\)[/tex] is the population standard deviation and [tex]\(n\)[/tex] is the sample size.
- Given [tex]\(\sigma = 20\)[/tex] and [tex]\(n = 16\)[/tex]:
[tex]\[
SE = \frac{20}{\sqrt{16}} = \frac{20}{4} = 5
\][/tex]
3. Calculate the Margin of Error:
- The margin of error (ME) is calculated using the formula:
[tex]\[
ME = z_{\text{critical}} \times SE
\][/tex]
- Using the z_critical value of 1.8119 and the SE of 5:
[tex]\[
ME = 1.8119 \times 5 \approx 9.0596
\][/tex]
4. Construct the Confidence Interval:
- The confidence interval (CI) is calculated using the formula:
[tex]\[
\text{CI} = \left( \bar{x} - ME, \bar{x} + ME \right)
\][/tex]
where [tex]\(\bar{x}\)[/tex] is the sample mean.
- Given [tex]\(\bar{x} = 150\)[/tex]:
[tex]\[
\text{Lower Bound} = 150 - 9.0596 \approx 140.9404
\][/tex]
[tex]\[
\text{Upper Bound} = 150 + 9.0596 \approx 159.0596
\][/tex]
Therefore, the 93% confidence interval for the sampling distribution is approximately [tex]\((140.9404, 159.0596)\)[/tex]. This means we are 93% confident that the true population mean lies within this interval.
