Question

1. Suppose the random variable X follows a normal distribution with mean µ = 84 and standard deviation σ = 20. Calculate each of the following:

1. P(X > 100)
2. P(80 < X < 144)
3. P(124 < X < 160)
4. P(X < 50)
5. P(X > X*) = .0062. What is the value of X*?

Answer 1

Given,

$\mu$ = 84 , $\sigma$ = 20

We convert this to standard normal as

P(X < x) = P(Z < ( x - $\mu$ ) / $\sigma$ )

a)

P(X > 100) = P(Z > ( 100 - 84) / 20)

= P(Z > 0.8)

= 0.2119

b)

P( 80 < X < 144 ) = P(X < 144) - P(X < 80)

= P(Z < ( 144 - 84) / 20) - P(Z < ( 80 - 84) / 20)

= P(Z < 3) - P(Z < -0.2)

= 0.9987 - 0.4207

= 0.5780

c)

P( 124 < X < 160 ) = P(X < 160) - P(X < 124)

= P(Z < ( 160 - 84) / 20) - P(Z < ( 124 - 84) / 20)

= P(Z < 3.8 ) - P(Z < 2)

= 0.9999 - 0.9772

= 0.0227

d)

P(X < 50) = P(Z < (50 - 84) / 20)

= P(Z < -1.7)

= 0.0446

e)

P(X > x*) = 0.0062

So,

P(X < x*) = 0.9938

P(Z < (x* - $\mu$ ) / $\sigma$ ) = 0.9938

From Z table, z-score for the probability of 0.9938 is 2.50

So,

(x* - $\mu$ ) / $\sigma$ = 2.50

x* - 84 / 20 = 2.50

x* = 134