Question

on a standardized test, the mean is 37 and the standard deviation is 2.5. approximately what percent of the scores fall in the range 32 - 42?
95%
34%
99%
68%

Explanation:

Step1: Calculate the number of standard deviations from the mean

The formula for the number of standard deviations \(z=\frac{x - \mu}{\sigma}\), where \(\mu = 37\) (mean) and \(\sigma=2.5\) (standard deviation).
For \(x = 32\): \(z_1=\frac{32 - 37}{2.5}=\frac{- 5}{2.5}=-2\)
For \(x = 42\): \(z_2=\frac{42 - 37}{2.5}=\frac{5}{2.5}=2\)

Step2: Use the empirical rule

The empirical rule (68 - 95 - 99.7 rule) for a normal distribution states that approximately \(95\%\) of the data lies within \(z=-2\) and \(z = 2\) (i.e., within \(2\) standard deviations of the mean)

Answer:

\(95\%\)