Answer to Question #336128 in Statistics and Probability for jo

Question

If Scores are normally distributed with a mean of 30 and a standard deviation of 5. What percent score is.

a.) Equal or greater than 30?

b.) Equal or greater than 37?

c.) From 28 to 34?

Answer 1

We have a normal distribution, $\mu=30, \sigma=5.$

Let's convert it to the standard normal distribution,

$z=\cfrac{x-\mu}{\sigma}.$

$\text{a) }z_1=\cfrac{30-30}{5}=0; \\P(X\ge30)=P(Z\ge0)=\\ =1-P(Z<0)=\\ =1-0.5=0.5=\\ =50\%\text{ (from z-table)}.$

$\text{b) }z_2=\cfrac{37-30}{5}=1.40; \\P(X\ge37)=P(Z\ge1.40)=\\ =1-P(Z<1.40)=\\ =1-0.9192=0.0808=\\ =8.08\%\text{ (from z-table)}.$

$\text{c) }z_3=\cfrac{28-30}{5}=-0.40;\\ z_4=\cfrac{34-30}{5}=0.80;\\ P(28<X<34)=\\ =P(-0.40<Z<0.80)=\\ =P(Z<0.80)-P(Z<-0.40)=\\ =0.7881-0.3446=0.4435=\\ =44.35\%\text{ (from z-table)}.$