Our statistics students, as noted in Exercise 6.32, were asked to rate their admiration of Hillary Rodham Clinton
on a scale of 1 to 7. They also were asked to rate their admiration of Jennifer Lopez and Venus Williams
on a scale of 1 to 7. As noted earlier, the mean rating of Clinton was 4.06, with a standard deviation of 1.70.
The mean rating of Lopez was 3.72, with a standard deviation of 1.90. The mean rating of Williams was 4.58,with a standard deviation of 1.46. One of our students rated her admiration of Clinton and Williams at 5 and
her admiration of Lopez at 4.
a. What is her z score for her admiration rating of Clinton?
b. What is her z score for her admiration rating of Williams?
c. What is her z score for her admiration rating of Lopez?
d. Compared to the other statistics students in our sample, which celebrity does this student most admire?
(We can tell by her raw scores that she prefers Clinton and Williams to Lopez, but when we take into account the general perception of these celebrities, how does this student feel about them?)
To get the z-scores for the student's admiration ratings, we can use the formula:
z = (x - μ) / σ
where:
- z is the z-score
- x is the raw score (student's admiration rating)
- μ is the mean of the population (celebrity admiration ratings)
- σ is the standard deviation of the population
Given the mean and standard deviation for each celebrity's admiration ratings, we can calculate the z-scores for the student.
a. The z score for the student's admiration rating of Clinton:
x = 5
μ (mean rating of Clinton) = 4.06
σ (standard deviation of Clinton) = 1.70
z = (5 - 4.06) / 1.70 = 0.55294118
b. The z score for the student's admiration rating of Williams:
x = 5
μ (mean rating of Williams) = 4.58
σ (standard deviation of Williams) = 1.46
z = (5 - 4.58) / 1.46 = 0.28767123
c. The z score for the student's admiration rating of Lopez:
x = 4
μ (mean rating of Lopez) = 3.72
σ (standard deviation of Lopez) = 1.90
z = (4 - 3.72) / 1.90 = 0.14736842
d. To determine which celebrity the student most admires compared to the other students in the sample, we need to compare the z-scores. The higher the z-score, the more the student's admiration rating deviates from the average.
Based on the z-scores, this student's admiration rating for Clinton has a higher deviation from the mean compared to the admiration ratings for Williams and Lopez. Therefore, this student most admires Hillary Rodham Clinton.