To construct a [tex]\(95\%\)[/tex] confidence interval for the sampling distribution given the sample means and the known population standard deviation, follow these steps:
1. Calculate the Sample Mean ([tex]\(\overline{X}\)[/tex]):
The sample mean is the average of the sample means provided.
[tex]\[
\overline{X} = \frac{\sum X_i}{n}
\][/tex]
Given the sample means: [tex]\(\{4, 4, 4, 4.2, 4.2, 4.3, 4.3, 4.3, 4.4, 4.4, 4.4, 4.4, 4.5, 4.5, 4.6, 4.7, 4.7, 4.7, 4.8, 4.8, 4.8, 4.9, 4.9, 4.9, 4.9, 5, 5, 5, 5, 5, 5, 5.1, 5.1, 5.1, 5.2, 5.2\}\)[/tex]:
[tex]\[
\overline{X} = 4.675
\][/tex]
2. Determine the Sample Size ([tex]\(n\)[/tex]):
The sample size is the number of sample observations.
[tex]\[
n = 36
\][/tex]
3. Calculate the Standard Error of the Mean (SE):
The standard error of the mean is given by:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
Where [tex]\(\sigma = 0.357\)[/tex] and [tex]\(n = 36\)[/tex]:
[tex]\[
SE = \frac{0.357}{\sqrt{36}} = \frac{0.357}{6} = 0.0595
\][/tex]
4. Determine the Z-Score for a [tex]\(95\%\)[/tex] Confidence Level:
For a [tex]\(95\%\)[/tex] confidence level, the Z-score (z*) corresponds to the value dividing the central [tex]\(95\%\)[/tex] of the standard normal distribution.
[tex]\[
z^* \approx 1.96
\][/tex]
5. Calculate the Margin of Error (ME):
The margin of error is calculated by:
[tex]\[
ME = z^* \times SE
\][/tex]
Given [tex]\(z^* \approx 1.96\)[/tex] and [tex]\(SE = 0.0595\)[/tex]:
[tex]\[
ME = 1.96 \times 0.0595 \approx 0.1166
\][/tex]
6. Construct the Confidence Interval:
The confidence interval is given by:
[tex]\[
(\overline{X} - ME, \overline{X} + ME)
\][/tex]
Using the previously calculated values:
[tex]\[
\left(4.675 - 0.1166, 4.675 + 0.1166\right)
\][/tex]
[tex]\[
\left(4.5584, 4.7916\right)
\][/tex]
Therefore, the [tex]\(95\%\)[/tex] confidence interval for the sampling distribution is [tex]\((4.5584, 4.7916)\)[/tex].
In summary:
- Sample Mean ([tex]\(\overline{X}\)[/tex]): [tex]\(4.675\)[/tex]
- Standard Error (SE): [tex]\(0.0595\)[/tex]
- Margin of Error (ME): [tex]\(0.1166\)[/tex]
- Confidence Interval: [tex]\((4.5584, 4.7916)\)[/tex]
