Answer to Question #327304 in Statistics and Probability for Tarek

Question

The weekly income of managers are normally distributed with a mean of $10000.

And a standard deviation of 100S.

1: What is the z value for an income x of 1100 and for 900?

2. What is the probability that the weekly income is less than 1100?

3. What is the probability that the weekly income is more than 1100?

4. What is the probability that the weekly income between 800S and 1100$?

Answer 1

It seems to be a typo here, the mean is $1000; otherwise all the answers will be equal to 0 or 1.

We have a normal distribution, $\mu=10000, \sigma=100.$

Let's convert it to the standard normal distribution,

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

$1. \ \ z_1=\cfrac{1100-1000}{100}=1;\\ z_2=\cfrac{900-1000}{100}=-1.$

$2.\ \ P(X<1100)=P(Z<1)=0.8413\\ \text{ (from z-table).}$

$3.\ \ P(X>1100)=1- P(X<1100)=\\ =1-P(Z<1)=1-0.8413=0.1587.\\$

$4.\ \ z_3=\cfrac{800-1000}{100}=-2;\\ P(800<X<1100)=P(-2<Z<1)=\\ =P(Z<1)-P(Z<-2)=\\ =0.8413-0.0228=0.8185\\ \text{ (from z-table).}$