To construct a confidence interval for the population mean [tex]\(\mu\)[/tex] using a [tex]\(95\%\)[/tex] confidence level, let's follow these steps with the given data set: [tex]\(6, 8, 3, 10, 7, 6, 8, 9, 9, 8, 4, 9\)[/tex].
1. Calculate the sample mean [tex]\(\bar{x}\)[/tex]:
The sample mean [tex]\(\bar{x}\)[/tex] is the average of all the values in the sample.
[tex]\[
\bar{x} \approx 7.25
\][/tex]
2. Calculate the sample standard deviation [tex]\(s\)[/tex]:
The sample standard deviation measures the dispersion of the sample data points around the sample mean.
[tex]\[
s \approx 2.137
\][/tex]
3. Determine the sample size [tex]\(n\)[/tex]:
The number of values in the sample.
[tex]\[
n = 12
\][/tex]
4. Calculate the standard error of the mean (SE):
The standard error of the mean is given by:
[tex]\[
\text{SE} = \frac{s}{\sqrt{n}} \approx \frac{2.137}{\sqrt{12}} \approx 0.617
\][/tex]
5. Find the critical value for the [tex]\(95\%\)[/tex] confidence level:
Since the sample size [tex]\(n\)[/tex] is small (less than 30), we use the [tex]\(t\)[/tex]-distribution. The critical value [tex]\(t_{\alpha/2}\)[/tex] for a [tex]\(95\%\)[/tex] confidence level and [tex]\(df = n - 1 = 11\)[/tex] is:
[tex]\[
t_{\alpha/2} \approx 2.201
\][/tex]
6. Calculate the margin of error (ME):
The margin of error is given by:
[tex]\[
\text{ME} = t_{\alpha/2} \times \text{SE} \approx 2.201 \times 0.617 \approx 1.358
\][/tex]
7. Determine the confidence interval:
The confidence interval is given by:
[tex]\[
\bar{x} - \text{ME} < \mu < \bar{x} + \text{ME}
\][/tex]
Substituting the values calculated:
[tex]\[
7.25 - 1.358 < \mu < 7.25 + 1.358
\][/tex]
[tex]\[
5.9 < \mu < 8.6
\][/tex]
Thus, the confidence interval for the population mean [tex]\(\mu\)[/tex] is:
[tex]\[
5.9 < \mu < 8.6
\][/tex]
### Interpretation:
The confidence interval [tex]\(5.9 < \mu < 8.6\)[/tex] means that we are [tex]\(95\%\)[/tex] confident that the true mean attractiveness rating of all female dates, as rated by male subjects in this speed dating study, lies between [tex]\(5.9\)[/tex] and [tex]\(8.6\)[/tex] on the attractiveness scale.
