Answer to Question #206550 in Statistics and Probability for dyo

Question

2000 Grade 11 students of MORMS took a Statistics test. The scores were distributed normally with a mean of 70 and Standard deviation of 5. Label the mean and three standard deviation from the mean.

1. What percentage of scores are between 65 and 75?

2. What percentage of scores are between 60 and 70?

3. What percentage of scores less than a score of 60?

4. What percentage of scores greater than a score of 80?

5. What percentage of score are between 70 and 75?

Answer 1

1.

$z_1=\frac{x_1-\mu}{\sigma/\sqrt{n}}=\frac{65-70}{5/\sqrt{2000}}=-0.02$

$z_2=\frac{x_2-\mu}{\sigma/\sqrt{n}}=\frac{75-70}{5/\sqrt{2000}}=0.02$

$P(65<x<75)=P(z_2<0.02)-P(z_1<-0.02)=0.50798-0.49202=0.01596$

2.

$z_1=\frac{x_1-\mu}{\sigma/\sqrt{n}}=\frac{60-70}{5/\sqrt{2000}}=-0.04$

$z_2=\frac{x_2-\mu}{\sigma/\sqrt{n}}=\frac{70-70}{5/\sqrt{2000}}=0$

$P(60<x<70)=P(z_2<0)-P(z_1<-0.04)=0.5-0.48405=0.01595$

3.

$z=\frac{x-\mu}{\sigma/\sqrt{n}}=\frac{60-70}{5/\sqrt{2000}}=-0.04$

$P(x<60)=P(z<-0.04)=0.48405$

4.

$z=\frac{x-\mu}{\sigma/\sqrt{n}}=\frac{80-70}{5/\sqrt{2000}}=0.04$

$P(x>80)=1-P(z<0.04)=1-0.51595=0.48405$

5.

$z_1=\frac{x_1-\mu}{\sigma/\sqrt{n}}=\frac{70-70}{5/\sqrt{2000}}=0$

$z_2=\frac{x_2-\mu}{\sigma/\sqrt{n}}=\frac{75-70}{5/\sqrt{2000}}=0.02$

$P(70<x<75)=P(z_2<0.02)-P(z<0)=0.50798-0.5=0.00798$