Answer to Question #320028 in Statistics and Probability for Mirylle Anne

Question

The heights of 1000 students are normally distributed with a mean of 174.5 centimeters and a standard deviation is 6.9 centimeters. Assuming that the heights are recorded to the nearest half-centimeter, how many of these students would you expect to have heights.

a) less than 160.0 centimeters?

b) between 171.5 and 182.0 centimeters inclusive?

c) equal to 175.0 centimeters?

d) greater than or equal to 188.0 centimeters?

Answer 1

$a:\\P\left( X<159.75 \right) =P\left( \frac{X-174.5}{6.9}<\frac{159.75-174.5}{6.9} \right) =\\=P\left( Z<-2.1377 \right) =\varPhi \left( -2.1377 \right) =0.0163\Rightarrow 16 students\\b:\\P\left( 171.25<X<182.25 \right) =P\left( \frac{171.25-174.5}{6.9}<\frac{X-174.5}{6.9}<\frac{182.25-174.5}{6.9} \right) =\\=P\left( -0.4710<Z<1.1232 \right) =\varPhi \left( 1.1232 \right) -\varPhi \left( -0.4710 \right) =0.8693-0.3188=0.5505\Rightarrow \\\Rightarrow 551 students\\c:\\P\left( 174.75<X<175.25 \right) =P\left( \frac{174.75-174.5}{6.9}<\frac{X-174.5}{6.9}<\frac{175.25-174.5}{6.9} \right) =\\=P\left( 0.0362<Z<0.1087 \right) =\varPhi \left( 0.1087 \right) -\varPhi \left( 0.0362 \right) =0.5433-0.5144=0.0289\Rightarrow \\\Rightarrow 29 students\\d:\\P\left( X>187.75 \right) =P\left( \frac{X-174.5}{6.9}>\frac{187.75-174.5}{6.9} \right) =\\=P\left( Z>1.9203 \right) =\varPhi \left( -1.9203 \right) =0.0274\Rightarrow 27 students$