The final scores of students in mathematics are distributed normally with a mean of 72 and a standard deviation of 10. What is the probability that a student’s score will be between 65 and 78?.

Answer:

0.8041

Explanation:

0.8041

We know that  

μ=72  and  σ=5

and P(65<X<78)

We can determine the Z value as (X-μ)/σ

P( 65<X<78 )=P( 65-72< X-μ<78-72)

P((65-72)/5 < Z < (78-72)/5)P((65−72)/5<Z<(78−72)/5)

P(65 < X < 78 )=P(-1.4 < Z < 1.2)P(65<X<78)=P(−1.4<Z<1.2)

To fine the Z values:

P (-1.4 < Z < 1.2 )=P ( Z < 1.2 )-P (Z < -1.4 )P(−1.4<Z<1.2)=P(Z<1.2)−P(Z<−1.4)

From the standard normal tables:

P (Z < 1.2 )=0.8849P(Z<1.2)=0.8849

to find P ( Z<-1.4)

P ( Z < -a)=1-P ( Z < a )P(Z<−a)=1−P(Z<a)

From the standard normal tables:

P ( Z < -1.4)=1-P ( Z < 1.4 )=1-0.9192=0.0808P(Z<−1.4)=1−P(Z<1.4)=1−0.9192=0.0808

Therefore

P(-1.4 < Z < 1.2 )=0.8041P(−1.4<Z<1.2)=0.8041

P (65 < X < 78)=0.8041P(65<X<78)=0.8041

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