Question

Let X be normally distributed with mean μ = 4.3 and standard deviation σ = 2. [You may find it useful to reference the z table.]


a. Find P(X > 6.5). (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

b. Find P(5.5 ≤ X ≤ 7.5). (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

c. Find x such that P(X > x) = 0.0869. (Round "z" value and final answer to 3 decimal places.)

d. Find x such that P(x ≤ X ≤ 4.3) = 0.2088. (Negative value should be indicated by a minus sign. Round "z" value and final answer to 3 decimal places.)

Answer 1

Mean = 4.3

Standard deviation = 2

(a) P (X > 6.5) = 6.5 - 4.3/2 = 1.1

The value for this z score according to z score table is 0.8643

Required probability = 1 - 0.8643 = 0.1357​​​​​​

(b) P (5.5 < X < 7.5)

For x = 5.5, 5.5 - 4.3/2 = 0.6

For x = 7.5, 7.5 - 4.3/2 = 1.6

The value for this z scores 0.6 and 1.6 according to z score table is 0.7275 and 0.9452

Required probability = 0.9452 - 0.7275 = 0.2177​​​​​​

(c) P (X> x) = 0.0869

The z score for this value is -1.36

Required probability = 1 - 0.0869 = 0.913

(d) P (x < X < 4.3) = 0.2088

For x, x - 4.3/2 =?

For x = 4.3, 4.3 - 4.3/2 = 0

The value for this z score according to z score table is 0.5000

Now, 0.5000 - ? = 0.2088

0.5000 - 0.2088 = ?

? = 0.2912

If we look at the z score table then we can see that the value is for a z score of - 0.55

Z value = -0.55​​​​​​

And,

x - 4.3/2 = - 0.55

x - 4.3 = - 1.1

x = 4.3 - 1.1

x = 3.2