To construct a [tex]\(90\%\)[/tex] confidence interval for a sampling distribution with the given sample means, we follow these steps:
1. List the Sample Means:
The sample means given are:
[tex]\[
\{4, 4, 4, 4.2, 4.2, 4.3, 4.3, 4.3, 4.4, 4.4, 4.4, 4.4, 4.5, 4.5, 4.6, 4.7, 4.7, 4.7, 4.8, 4.8, 4.8, 4.9, 4.9, 4.9, 4.9\}
\][/tex]
2. Calculate the Sample Mean ([tex]\(\bar{x}\)[/tex]):
The sample mean is calculated as:
[tex]\[
\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}
\][/tex]
Given:
[tex]\[
\bar{x} = 4.504
\][/tex]
3. Calculate the Sample Standard Deviation (s):
Using the formula for sample standard deviation:
[tex]\[
s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}
\][/tex]
Given:
[tex]\[
s = 0.2979
\][/tex]
4. Determine the Sample Size (n):
The number of samples given is:
[tex]\[
n = 25
\][/tex]
5. Determine the Degree of Freedom (df):
Degrees of freedom for the t-distribution is:
[tex]\[
df = n - 1 = 25 - 1 = 24
\][/tex]
6. Find the t-Critical Value (t_*):
For a [tex]\(90\%\)[/tex] confidence level and [tex]\(24\)[/tex] degrees of freedom, the critical t-value is:
[tex]\[
t_* = 1.7109
\][/tex]
7. Calculate the Standard Error of the Mean (SE):
The standard error of the mean is given by:
[tex]\[
SE = \frac{s}{\sqrt{n}}
\][/tex]
Given:
[tex]\[
SE = 0.0596
\][/tex]
8. Determine the Margin of Error (ME):
The margin of error is calculated as:
[tex]\[
ME = t_* \times SE
\][/tex]
Given:
[tex]\[
ME = 0.1019
\][/tex]
9. Construct the Confidence Interval:
The [tex]\(90\%\)[/tex] confidence interval for the mean is:
[tex]\[
\left( \bar{x} - ME, \bar{x} + ME \right)
\][/tex]
Substituting the values:
[tex]\[
\left( 4.504 - 0.1019, 4.504 + 0.1019 \right)
\][/tex]
Thus:
[tex]\[
\left( 4.4021, 4.6059 \right)
\][/tex]
Therefore, the [tex]\(90\%\)[/tex] confidence interval for the given sampling distribution is [tex]\(\left( 4.4021, 4.6059 \right)\)[/tex].
