Question 1182729: In a study of speed dating, male subjects were asked to rate the attractiveness of their female dates, and a sample of the results is listed below (1 = not attractive; 10 = extremely attractive). Construct a confidence interval using a 95% confidence level. What do the results tell about the mean attractiveness ratings of the population of all adult females?
6, 7, 1, 10, 4, 6, 7, 8, 7, 9, 5, 9
What does the confidence interval tell about the population of all adult females? Select the correct choice below and, if necessary, fill in the blanks to complete your choice.
A. We are 95% confident that the interval from __ to __ actually contains the true mean attractiveness rating of all adult females. (Round to one decimal place as needed.)
B. We are confident that 95% of all adult females have attractiveness ratings between __ and __. (Round to one decimal place as needed.)
C. The results tell nothing about the population of all adult females, because participants in speed dating are not a representative sample of the population of all adult females.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to construct the confidence interval and interpret the results:
1. **Calculate the sample mean (x̄):**
(6 + 7 + 1 + 10 + 4 + 6 + 7 + 8 + 7 + 9 + 5 + 9) / 12 = 72 / 12 = 6
2. **Calculate the sample standard deviation (s):**
* Find the squared difference of each value from the mean, sum them, divide by (n-1), and then take the square root. Using a calculator or software is recommended.
* s ≈ 2.50
3. **Find the critical t-value:**
* Degrees of freedom (df) = n - 1 = 12 - 1 = 11
* For a 95% confidence level and 11 degrees of freedom, the critical t-value (two-tailed) is approximately 2.201 (you'll need a t-table or calculator).
4. **Calculate the margin of error (E):**
E = t * (s / √n) = 2.201 * (2.50 / √12) ≈ 1.59
5. **Construct the confidence interval:**
Lower bound = x̄ - E = 6 - 1.59 ≈ 4.4
Upper bound = x̄ + E = 6 + 1.59 ≈ 7.6
6. **Interpretation:**
The correct choice is **A. We are 95% confident that the interval from 4.4 to 7.6 actually contains the true mean attractiveness rating of all adult females.**
**Important Note:** The confidence interval *only* tells us about the population from which the *sample* was drawn. As the problem states, the sample consists of ratings from *male* subjects in a *speed dating* context. This is *not* a representative sample of all adult females. Therefore, we *cannot* generalize these results to the population of all adult females. The confidence interval is valid, but its applicability is limited. Option C is also correct. The results tell nothing about the population of all adult females, because participants in speed dating are not a representative sample of the population of all adult females.
|
|
|
