Question

9.A standard IQ test produces normally distributed results with a mean of 100 and a standard deviation of 15. A class of high school science students is grouped homogeneously by excluding students with IQ scores in either the top 15% or the bottom 15%. Find the lowest and highest possible IQ scores of students remaining in the class.(7)

Answer 1

The lowest possible IQ scores of students remaining in the class is 84.46.

The highest possible IQ scores of students remaining in the class is 115.54.

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 of 100 and a standard deviation of 15.

This means that
`\mu = 100, \sigma = 15`

Find the lowest and highest possible IQ scores of students remaining in the class.

Lowest:

The 15th percentile, which is X when Z has a pvalue of 0.15, so X when Z = -1.036.

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

`-1.036 = (X - 100)/(15)`

`X - 100 = -1.036*15`

`X = 84.46`

Highest:

The 100 - 15 = 85th percentile, which is X when Z has a pvalue of 0.85, so X when Z = 1.036.

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

`1.036 = (X - 100)/(15)`

`X - 100 = 1.036*15`

`X = 115.54`

The lowest possible IQ scores of students remaining in the class is 84.46.

The highest possible IQ scores of students remaining in the class is 115.54.