Question

The total scores on the Medical College Admission Test (MCAT) follow a Normal distribution with mean 25.3 and standard deviation 6.7.

(a) What are the median and the first and third quartiles of the MCAT scores?

Answer 1

The median of the MCAT scores is 25.3, the first quartile is approximately 20.788, and the third quartile is approximately 29.812.

Step-by-step explanation:

In order to calculate the median and the first and third quartiles of the MCAT scores, we need to use the properties of the standard normal distribution and convert the scores to z-scores.

To find the median, we need to find the z-score that corresponds to the cumulative probability of 0.5. Using a standard normal distribution table, we can find that the z-score for a cumulative probability of 0.5 is 0.

To find the first quartile, we need to find the z-score that corresponds to the cumulative probability of 0.25. Using a standard normal distribution table, we can find that the z-score for a cumulative probability of 0.25 is -0.675.

To find the third quartile, we need to find the z-score that corresponds to the cumulative probability of 0.75. Using a standard normal distribution table, we can find that the z-score for a cumulative probability of 0.75 is 0.675.

Now, we can convert the z-scores back to the original MCAT scores using the formula: x = mean + (z * standard deviation).

For the median:

x = 25.3 + (0 * 6.7) = 25.3

For the first quartile:

x = 25.3 + (-0.675 * 6.7) = 20.788

For the third quartile:

x = 25.3 + (0.675 * 6.7) = 29.812