A college requires applicants to have an ACT score in the top 12% of all test scores. The ACT scores are normally distributed, with a mean of 21.5 and a standard deviation of 4.7.
i) type of distribution: _sampling distributions of mean
ii) find the mean: __21.5 points and standard deviation: 4.7 points
iii) find the following:

a. Find the lowest test score that a student could get and still meet the colleges requirement.
b. If 1300 students are randomly selected, how many would be expected to have a test score that would meet the colleges requirement?
c. How does the answer to part (a) change if the college decided to accept the top 15% of all test scores?

a) The lowest test score that a student could get and still meet the colleges requirement is 27.0225.

b) 156 would be expected to have a test score that would meet the colleges requirement

c) The lowest score that would meet the colleges requirement would be decreased to 26.388.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

a. Find the lowest test score that a student could get and still meet the colleges requirement.

This is the value of X when Z has a pvalue of 1 - 0.12 = 0.88. So it is X when Z = 1.175.

The lowest test score that a student could get and still meet the colleges requirement is 27.0225.

b. If 1300 students are randomly selected, how many would be expected to have a test score that would meet the colleges requirement?

Top 12%, so 12% of them.

0.12*1300 = 156

156 would be expected to have a test score that would meet the colleges requirement

c. How does the answer to part (a) change if the college decided to accept the top 15% of all test scores?

It would decrease to the value of X when Z has a pvalue of 1-0.15 = 0.85. So X when Z = 1.04.

The lowest score that would meet the colleges requirement would be decreased to 26.388.

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step-by-step explanation: