Question

GPAs of freshman biology majors have approximately the normal distribution with mean 2.87 and

standard deviation .34. In what range do the middle 90% of all freshman biology majors' GPAs lie?

Answer 1

The middle 90% of all freshman biology majors' GPAs lie between 2.31 and 3.43.

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Mean 2.87 and standard deviation .34.

This means that
`\mu = 2.87, \sigma = 0.34`

Middle 90% of scores:

Between the 50 - (90/2) = 5th percentile and the 50 + (90/2) = 95th percentile.

5th percentile:

X when Z has a pvalue of 0.05. So X when Z = -1.645.

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

`-1.645 = (X - 2.87)/(0.34)`

`X - 2.87 = -1.645*0.34`

`X = 2.31`

95th percentile:

X when Z has a pvalue of 0.95. So X when Z = 1.645.

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

`1.645 = (X - 2.87)/(0.34)`

`X - 2.87 = 1.645*0.34`

`X = 3.43`

The middle 90% of all freshman biology majors' GPAs lie between 2.31 and 3.43.