To construct a confidence interval for the population mean [tex]\(\mu\)[/tex] of the attractiveness ratings using a 95% confidence level, we need to follow several steps:
### Step 1: Calculate the sample mean ([tex]\(\bar{x}\)[/tex])
The sample mean is the average of all the data points in the sample.
Given data: [tex]\(5, 9, 1, 10, 6, 5, 7, 8, 9, 10, 3, 8\)[/tex]
[tex]\[
\bar{x} = \frac{\sum x_i}{n} = \frac{5 + 9 + 1 + 10 + 6 + 5 + 7 + 8 + 9 + 10 + 3 + 8}{12} = \frac{81}{12} = 6.75
\][/tex]
### Step 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.
Using the detailed steps to compute the sample standard deviation:
The sample standard deviation [tex]\(s \approx 2.832\)[/tex]
### Step 3: Determine the sample size ([tex]\(n\)[/tex])
[tex]\[
n = 12
\][/tex]
### Step 4: Calculate the standard error of the mean ([tex]\(SE\)[/tex])
The standard error of the mean indicates how much the sample mean ([tex]\(\bar{x}\)[/tex]) is expected to vary from the population mean ([tex]\(\mu\)[/tex]).
[tex]\[
SE = \frac{s}{\sqrt{n}} = \frac{2.832}{\sqrt{12}} \approx 0.818
\][/tex]
### Step 5: Find the critical value (t-score) for a 95% confidence level
For a 95% confidence level and [tex]\(n - 1 = 12 - 1 = 11\)[/tex] degrees of freedom, the critical value (t-score) can be found from the t-distribution table or statistical software:
[tex]\[
t \approx 2.201
\][/tex]
### Step 6: Calculate the margin of error (ME)
The margin of error provides the range within which the population mean is expected to lie.
[tex]\[
ME = t \times SE = 2.201 \times 0.818 \approx 1.800
\][/tex]
### Step 7: Calculate the confidence interval
The confidence interval is given by the sample mean plus and minus the margin of error.
[tex]\[
\text{Lower bound} = \bar{x} - ME = 6.75 - 1.800 \approx 4.95
\][/tex]
[tex]\[
\text{Upper bound} = \bar{x} + ME = 6.75 + 1.800 \approx 8.55
\][/tex]
### Conclusion
The 95% confidence interval for the population mean [tex]\(\mu\)[/tex] of attractiveness ratings is:
[tex]\[
4.9 < \mu < 8.5
\][/tex]
This interval suggests that the mean attractiveness rating of the population of all adult females, as perceived by male dates in speed dating, is likely to be between 4.9 and 8.5 (rounding to one decimal place).
