Question

The score obtained on an emotional intelligence test is normally distributed, with an average score (μ) of 115 and a standard deviation (σ) of 9.5. What is the probability that a randomly selected person will: Q.1.3.1 Obtain a score between 110 and 118? Interpret your answer. (5) Q.1.3.2 Obtain a score between 109 and 113? Interpret your answer. (5)

Answer 1

$\mu=115 \\ \sigma=9.5$

1. P(110<X<118) = P(X<118) -P(X<110)

$=P(Z< \frac{118-115}{9.5}) -P(Z< \frac{110-115}{9.5}) \\ = P(Z<0.315) -P(Z< -0.526) \\ = 0.6236 -0.2994 \\ = 0.3242$

The probability that a randomly selected person will obtain a score between 110 and 118 is 32.42%.

2. P(109<X<113) = P(X<113) -P(X<109)

$=P(Z< \frac{113-115}{9.5}) -P(Z< \frac{109-115}{9.5}) \\ = P(Z< -0.210) -P(Z< -0.631) \\ = 0.4168 -0.2640 \\ = 0.1528$

The probability that a randomly selected person will obtain a score between 109 and 113 is 15.28%.