Question

IQ scores are known to be normally distributed. The mean IQ score is 100 and the standard deviation is 15. What percent of the population has an IQ between 85 and 105?

Answer 1

Answer:

47.06% of the population has an IQ between 85 and 105.

Step-by-step explanation:

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

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

$Z = \frac{X - \mu}{\sigma}$

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 problem, we have that:

$\mu = 100, \sigma = 15$

What percent of the population has an IQ between 85 and 105?

This is the pvalue of Z when X = 105 subtracted by the pvalue of Z when X = 85. So

X = 105

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{105 - 100}{15}$

$Z = 0.33$

$Z = 0.33$ has a pvalue of 0.6293.

X = 85

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{85 - 100}{15}$

$Z = -1$

$Z = -1$ has a pvalue of 0.1587

So 0.6293 - 0.1587 = 0.4706 = 47.06% of the population has an IQ between 85 and 105.