Question

Scores (x) on a college entrance examination are normally distributed with µ=560 and σ=110. (i) What proportion of scores are less than 585?

(ii) University U will consider for admission only those applicants whose scores exceed the 83rd percentile of the distribution. Find the minimum score an applicant must achieve in order to receive consideration for admission.

Answer 1

The proportion of scores less than 585 is approximately 59.08%. The minimum score an applicant must achieve for admission consideration is 666.03.

Step-by-step explanation:

(i) To find the proportion of scores that are less than 585, we need to convert the score to a z-score using the formula: z = (x - µ) / σ. Plugging in the values, we get z = (585 - 560) / 110 = 0.2273. Using a z-table or a calculator, we can find that the proportion of scores less than 0.2273 is approximately 0.5908, or 59.08%.

(ii) To find the minimum score an applicant must achieve in order to receive consideration for admission at University U, we need to find the score that corresponds to the 83rd percentile of the distribution. Using a z-table or a calculator, we can find that the z-score associated with the 83rd percentile is approximately 0.9673. Plugging this z-score into the z-score formula and solving for x, we get x = (0.9673 * 110) + 560 = 666.03.

Therefore, an applicant must score at least 666.03 to receive consideration for admission.