Question

A

Assume a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded region of the graph. Assume the variable x is normally distributed Standardized Test Composite Scores 26 < x < 32 u = 20.8 o= 5.2 6 26 32 The probability that the member selected at random is from the shaded area of the graph is (Round to four decimal places as needed.)

Answer 1

Solution :

Given that, X ~ N(20.8, 5.2²)

μ = 20.8 and  σ = 5.2

The probability that the member selected at random is from the shaded area of the graph would be given by, P(26 < X < 32).

P(26 < X < 32) = P(X < 32) - P(X ≤ 26)

We know that if X ~ N(μ, σ²) then,

X-M Z=1 ~ N(0,1) o

X-H 32 - MI X- 26 - :: P(26 < X < 32) = P -P < o 0 0

$\large \therefore P(26 < X < 32) = P\left (Z < \frac{32-20.8}{5.2} \right ) - P\left (Z \leq \frac{26-20.8}{5.2} \right )$

:: P(26 < X < 32) = P(Z < 2.1538) - P(Z < 1)

Using "pnorm" function of R we get,

P(Z < 2.1538) = 0.9843 and P(Z ≤ 1) = 0.8413

$\large \therefore P(26 < X < 32) = 0.9843 - 0.8413$

$\large \therefore P(26 < X < 32) = 0.1430$

Hence, the probability that the member selected at random is from the shaded area of the graph is 0.1430.

Please rate the answer. Thank you.