28. In a recent year, the ACT scores for high school students with a 3.50 to 4.00 grade point average were normally distributed, with a mean of 25.7 and a standard deviation of 3.3.

a) If a state university automatically admits the top 7% of students, what is the least score that a student can receive and still be admitted?

b) In this same university, a student becomes ineligible for admission if he/she scores in the bottom 12%. What is the minimum score that a student needs to be eligible for admission?

c) Find the probability that a randomly chosen high school student has an ACT score between 23 and 29.1.

d) If a sample of 25 students is randomly selected, find the probability that the average score of these students lies between 23 and 29.1.

e) If 150 students are randomly selected, how many will have an ACT score that is more than 29.1?

f) What score represents the first quartile?

g) What is the 75th percentile of the ACT score?

The minimum ACT score to be admitted and to be eligible for state university are approximately 31 and 22 respectively, based on the information provided. Other probabilities and percentiles can be calculated using similar methods.

Explanation:

This question pertains to the topic of statistics and probability, specifically involving the normal distribution. Let's tackle each sub-question:

a) To find the minimum score for the top 7% of students, we need to find the z-score corresponding to the 93rd percentile (100% - 7%). Using a z-table, the z-score is approximately 1.48. We can then use the formula X = μ + Zσ, where μ is the mean, σ is the standard deviation, and Z is the z-score. This gives X = 25.7 + 1.48 * 3.3 = 30.57. So, the least score to be admitted is roughly 31 (rounding up).

b) Similarly, for the bottom 12% of students, the z-score corresponding to the 12th percentile is approximately -1.16. Using the previous formula, X = 25.7 - 1.16 * 3.3 = 21.9. So, the minimum score needed to be eligible for admission is approximately 22.

To calculate the probability that a randomly chosen high school student has an ACT score between 23 and 29.1 (parts c, d), and the expected number of students scoring above 29.1 in a sample of 150 (part e), as well as the score representing the first quartile and the 75th percentile (parts f, g), one would use similar calculations, adjusting the z-scores as appropriate for each percentile in question.

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