Question

Question Help Assume that adults have IQ scores that are normally distributed with a mean of 101.1101.1 and a standard deviation 17.717.7. Find the first quartile Upper Q 1Q1​, which is the IQ score separating the bottom​ 25% from the top​ 75%

Answer 1

Q1 = 89.1525

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 = (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 = 101.1, \sigma = 17.7`

Find the first quartile Upper Q 1Q1​, which is the IQ score separating the bottom​ 25% from the top​ 75%

This is the value of X when Z has a pvalue of 0.25. So it is X when Z = -0.675.

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

`-0.675 = (X - 101.1)/(17.7)`

`X - 101.1 = -0.675*17.7`

`X = 89.1525`

So

Q1 = 89.1525