A person scores 81 on a test of verbal ability and 6.4 on a test of quantitative ability. For the verbal ability test, the mean for people in general is 50 and the standard deviation is 20. For the quantitative ability, the mean for people in general is 0 and the standard deviation is 5. Which is this person’s stronger ability: verbal or quantitative? Explain your answer to a person who has never had a course in statistics.
Z score is the raw score in terms of standard deviations from the mean. The Z score cuts off the same proportion for any normal distribution.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability for each Z score.
To determine which ability is stronger for the person - verbal or quantitative - we can compare their scores relative to the mean and standard deviation for each test.
Let's start with the verbal ability test. We have the person's score of 81, the mean of 50, and the standard deviation of 20.
To put the person's score in context, we need to calculate their z-score. The z-score tells us how many standard deviations the person's score is away from the mean. The formula to calculate the z-score is:
z = (x - mean) / standard deviation
Where:
- z = z-score
- x = person's score
- mean = mean score for people in general
- standard deviation = standard deviation of scores for people in general
For the verbal ability test:
z = (81 - 50) / 20
z = 31 / 20
z = 1.55
Now let's move to the quantitative ability test. We have the person's score of 6.4, the mean of 0, and the standard deviation of 5.
For the quantitative ability test:
z = (6.4 - 0) / 5
z = 6.4 / 5
z = 1.28
Comparing the z-scores for both tests, we can determine which ability is stronger.
Since the z-score for verbal ability is higher (1.55) compared to the z-score for quantitative ability (1.28), it means that the person's performance in verbal ability is stronger relative to the performance of others in general.
Therefore, the person's stronger ability is verbal, as their score in verbal ability is further above the mean compared to their score in quantitative ability.
In summary, by calculating the z-scores for the person's scores and comparing them to the mean and standard deviation of each test, we can determine that their stronger ability is verbal.