Question

In a normal distribution the theoretical mean is 30 and the theoretical standard deviation is 5. Find: a) the area below 24, b) the area between 24 and 36, c) the point that has 95% of the area below it.

Answer 1

a) P(X < 24) = 0.1151

b) P(24 < X < 36) = 0.7698

c) X value for P(x < X) = 0.95 is X = 38.225

Explanation:

a) P(X < 24)

Since μ = 30 and σ = 5 we have:

`\text {z value = } (24-30)/(5)} = -1.2`

P(X < 24) = P(Z < -1.2) = 0.1151
You can use a calculator or the standard normal tables to determine P(Z < -1.2)

b) P(24 < X < 36)

First find P(X < 36) using the technique detailed in part a)

For 36, the Z value is

`(36-30)/(5) = 1.2`

P(X < 36) = P(z < 1.2) = 0.8849

P(24 < X < 36) = P(X < 36) - P(X < 24) = 0.8849 - 0.1151 = 0.7698

c) X value below which 95% of the area is covered

Using a calculator (or tables) the z value corresponding to P(z < Z) = 0.95 is Z = 1.65(approx)

`\text{We have } (X - \mu)/(\sigma) = z\\\\(X - 30)/(5) = 1.645\\\\\text{Multiplying both sides by 5: }\\\\X - 30 = 1.645 * 5 = 8.225\\\\X = 30 + 8.225 = 38.225`

So X value is 38.225-