Answer to Question #324524 in Statistics and Probability for Oswald

Question

1. The heights of a group of boys are normally distributed with a mean of 54 inches and standard deviation of 2.5 inches.

What percentage of the population would have heights between 53 inches and 56 inches?

If a boy is chosen at random from this population, what is the probability that he is taller than 52 inches?

If all possible samples of size 25 are drawn from this population, what percentage of them would have means between 53 inches and 55 inches?

Answer 1

$a)\;P(53<x< 56)=P(\frac{53-54}{2.5}<z<\frac{56-54}{2.5})\\ =P(-0.4<z<0.8)\\ =P(0<z<0.4)+P(0<z<0.8)\\ =0.1554+0.2881=0.4435\\ the\;percentage\;is\;44.35\% \\ b)\;P(x> 52)=P(z> \frac{52-54}{2.5})\\ =P(z> -0.8)=0.5+P(0<z<0.8)\\ =0.5+0.2881=0.7881\\ c)\;P(53<\bar x< 55)=P(\frac{53-54}{\frac{2.5}{\sqrt{25}}}<z<\frac{55-54}{\frac{2.5}{\sqrt{25}}})\\ =P(-2<z<2)\\ =2 \times P(0<z<2) =2(0.4772)=0.9544\\ the\;percentage\;is\;95.44\% \\$