Answer to Question #318957 in Statistics and Probability for lyn

Question

The average time it takes college freshmen to take the standardized comprehension test is 24.6 minutes. The standard deviation is 5.8 minutes. Assume the variable is normally distributed. Find the following probabilities:

a. It will take between 15 and 30 minutes to complete the test.

b. It will take the student less than 18 minutes to complete the test.

c. It will take the student more than 28 minutes to complete the test.

Answer 1

$a:\\P\left( 15\leqslant X\leqslant 30 \right) =P\left( \frac{15-24.6}{5.8}\leqslant \frac{X-24.6}{5.8}\leqslant \frac{30-24.6}{5.8} \right) =\\=P\left( -1.65517\leqslant X\leqslant 0.931034 \right) =\varPhi \left( 0.931 \right) -\varPhi \left( -1.655 \right) =\\=0.8241-0.0490=0.7751\\b:\\P\left( X<18 \right) =P\left( \frac{X-24.6}{5.8}<\frac{18-24.6}{5.8} \right) =P\left( Z<-1.1379 \right) =\\=\varPhi \left( -1.1379 \right) =0.1276\\c:\\P\left( X>28 \right) =P\left( \frac{X-24.6}{5.8}>\frac{28-24.6}{5.8} \right) =\\=P\left( Z>0.5862 \right) =1-\varPhi \left( 0.5862 \right) =1-0.7211=0.2789$