Answer to Question #311011 in Statistics and Probability for Supremo

Question

A region-wide aptitude test in Mathematics was conducted to 500 pupils. The mean of the test is 85 and the standard deviation is 15. The scores also approximate a normal distribution. Estimate the range of the scores that will include the middle 75%of the distribution.

Answer 1

$P(x_1<X<x_2)=0.75.$

$P(\frac{x_1-\mu}{\sigma}<Z<\frac{x_2-\mu}{\sigma})=0.75.$

$P(Z<\frac{x_2-\mu}{\sigma})=\frac{0.75}{2}+0.5=0.875.$

$\frac{x_2-\mu}{\sigma}=1.15.$

$\frac{x_2-85}{15}=1.15.$

$x_2=1.15*15+85\approx102.$

$P(Z<\frac{x_1-\mu}{\sigma})=0.5-\frac{0.75}{2}=0.125.$

$\frac{x_1-85}{15}=-1.15.$

$x_1=-1.15*15+85\approx 68.$

The range of the scores that will include the middle 75%of the distribution: (68, 102).