Question

A community college has a math placement exam that has a mean of 65 and a standard deviation of 8.2. If a student takes the exam and scores in the lower 2.5%, they are eligible for a free intensive course to help them succeed in their math class. Assume that the exam has a normal distribution. What is the score that cuts off the bottom 2.5%

Answer 1

The score that cuts off the bottom 2.5% is 48.93.

Explanation:

When the distribution is normal, we use the z-score formula.

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

`Z = (X - \mu)/(\sigma)`

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 question, we have that:

`\mu = 65, \sigma = 8.2`

What is the score that cuts off the bottom 2.5%

This is X when Z has a pvalue of 0.025, so X when Z = -1.96.

`Z = (X - \mu)/(\sigma)`

`-1.96 = (X - 65)/(8.2)`

`X - 65 = -1.96*8.2`

`X = 48.93`

The score that cuts off the bottom 2.5% is 48.93.