Question

A set of final examination grades in an introductory statistics course was found to be normally distributed with a mean of 73 and a standard deviation of 8.

What is the probability of getting a grade of 79 or less?

What percentage of students scored between 61 and 81?

What is the probability of getting a grade no higher than 72 (less than 72) on this exam?

Answer 1

Solution :

Let X be a random variable which represents the grades of final examination in an introductory statistics course.

Given that, X ~ N(73, 82)

i.e. Mean (μ) = 73

Standard deviation (σ) = 8

a) We have to obtain P(X ≤ 79).

We know that if X ~ N(μ, σ​​​​​​2) then

Using "pnorm" function of R we get, P(Z ≤ 0.75) = 0.7734

Hence, the probability of getting a grade of 79 or less is 0.7734.

b) We have to obtain P(61 < X < 81).

We know that if X ~ N(μ, σ​​​​​​2) then

Using "pnorm" function of R we get,

P(Z < 1) = 0.8413 and P(Z < -1.5) = 0.0668

0.7745 = 77.45%

Hence, 77.45% of students scored between 61 and 81.

c) We have to obtain P(X < 72).

We know that if X ~ N(μ, σ​​​​​​2) then

Using "pnorm" function of R we get, P(Z < -0.125) = 0.4503

Hence, the probability of getting a grade no higher than 72 (less than 72) on this exam is 0.4503.

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