On a test whose distribution is approximately normal with a mean of 50 and a standard deviation of 10, the results for three students were reported as follows:
Student Opie has a T-score of 60.
Student Paul has a z-score of -1.00.
Student Quincy has a z-score of +2.00.
Obtain the z-score and T-score for EACH student. Show your calculations. Who did better on the test? How many standard deviation units is each score from the mean? Compare the results of the three students.
Since the Z and T score are standard for any normal distribution, you don't even need to know the mean or standard deviation of that distribution to answer your questions.
Z score = standard deviations away from the mean.
T score = 50 + 10Z
To obtain the z-score for each student, we will use the formula:
z = (x - μ) / σ
Where:
- x is the student's score
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
To obtain the T-score, we need to standardize the z-score using the formula:
T = 10 * z + 50
Now let's calculate the z-score and T-score for each student:
1. Student Opie:
z = (60 - 50) / 10 = 1.00
T = 10 * 1.00 + 50 = 60
2. Student Paul:
z = (-1.00) = -1.00
T = 10 * (-1.00) + 50 = 40
3. Student Quincy:
z = (2.00) = 2.00
T = 10 * 2.00 + 50 = 70
To determine who did better on the test, we compare the T-scores. In this case, Student Quincy (T-score of 70) performed the best, followed by Student Opie (T-score of 60), and finally by Student Paul (T-score of 40).
To determine how many standard deviation units each score is from the mean, we look at the z-scores. Each z-score represents the number of standard deviations a score is above or below the mean.
1. Student Opie: The z-score of 1.00 indicates that Opie's score is 1 standard deviation above the mean.
2. Student Paul: The z-score of -1.00 indicates that Paul's score is 1 standard deviation below the mean.
3. Student Quincy: The z-score of 2.00 indicates that Quincy's score is 2 standard deviations above the mean.
Comparing the results of the three students, Quincy performed the best with the highest T-score and the farthest distance from the mean. Opie performed better than Paul, as Opie's T-score is higher and closer to the mean than Paul's score.