Question

Let X be the height of a student to be randomly selected at the University. Assume (for the purpose of modeling) that X has the normal distribution with a mean of 68 inches and a standard deviation of 4.9 inches.

a.) What is the probability a randomly selected student is shorter than 65 inches?

b.) What is the probability a randomly selected student is taller than 73 inches?

c.) What is the probability a randomly selected student's height falls within one standard deviation of the mean?

Answer 1

Here Let X be the height of a student to be randomly selected at the University.

Here mean height = = 68 inches

standard deviation = = 4.9 inches

(a) Pr(x < 65 inches) = NORMDIST(x < 65 inches ; 68 inches ; 4.9 inches)

Z = (65 - 68)/4.9 = -0.6122

Now we look into Z table or use excel NORMSDIST function

Pr(x < 65 inches) = Pr(Z < -0.6122)

= 0.2702

(b) Pr(x > 73 inches) = 1 - Pr(x < 73 inches) = 1 - NORMDIST(x < 73 inches ; 68 inches ; 4.9 inches)

Z = (73 - 68)/4.9 = 1.0204

Now we look into Z table or use excel NORMSDIST function

Pr(x > 73 inches) = 1 - Pr(Z > 1.0204) = 1 - 0.8462 = 0.1538

(c) As here we have to find the probability a randomly selected student's height falls within one standard deviation of the mean

Pr( - < x < + ) = Pr(x < + ​​​​​) - Pr(x < - ​​​​​​​)

Z₂ = 1

Z₁ = -1

Pr( - < x < + ​​​​​​​) = Pr(Z < 1) - Pr(Z < -1)

looking into z table we find probability value for Z = 1 and Z = -1

Pr( - < x < + ​​​​​​​) = Pr(Z < 1) - Pr(Z < -1)

= 0.8413 - 0.1587 = 0.6827