Question

Assume X is normally distributed with a mean of 10 and a standard deviation of 2. Determine the value for x

that solves each of the following:

(a) ( X> x) = 0.5

(b) ( X> x) = 0.95

(c) ( x< X < 10) = 0.2

(d) (− x< X− 10 < x) = 0.95

(e) (−x < X − 10

Answer 1

To solve each of the above probability statements involving the standard normal distribution (X ~ N(0, 1)), we need to perform inverse standard normal calculations. First, we'll convert the given values to standard normal (Z) values, and then we'll find the corresponding Z values using the inverse standard normal (Z-inverse) table or a calculator. Finally, we'll convert the Z values back to the original X values.

(a) P(X > x) = 0.5

We want to find the Z value for which the area to the right is 0.5 (since P(X > x) = 1 - P(X ≤ x) = 0.5).

Using the standard normal table or calculator, the Z value is approximately 0. This means:

x = mean + (Z * standard deviation) = 10 + (0 * 2) = 10.

(b) P(X > x) = 0.95

We want to find the Z value for which the area to the right is 0.95.

Using the standard normal table or calculator, the Z value is approximately 1.645. This means:

x = mean + (Z * standard deviation) = 10 + (1.645 * 2) ≈ 13.29.

(c) P(x < X < 10) = 0.2

We want to find the Z values for which the area between them is 0.2.

Using the standard normal table or calculator, the Z values are approximately -0.842 and 0.842.

Now we can find the corresponding X values:

x₁ = mean + (Z₁ * standard deviation) = 10 + (-0.842 * 2) ≈ 8.316.

x₂ = mean + (Z₂ * standard deviation) = 10 + (0.842 * 2) ≈ 11.684.

(d) P(−x < X−10 < x) = 0.95

We want to find the Z values for which the area between them is 0.95.

Using the standard normal table or calculator, the Z values are approximately -1.96 and 1.96.

Now we can find the corresponding X values:

x₁ = mean + (Z₁ * standard deviation) = 10 + (-1.96 * 2) ≈ 5.08.

x₂ = mean + (Z₂ * standard deviation) = 10 + (1.96 * 2) ≈ 14.92.

(e) P(−x < X − 10 < x) = 0.9

We want to find the Z values for which the area between them is 0.9.

Using the standard normal table or calculator, the Z values are approximately -1.645 and 1.645.

Now we can find the corresponding X values:

x₁ = mean + (Z₁ * standard deviation) = 10 + (-1.645 * 2) ≈ 6.71.

x₂ = mean + (Z₂ * standard deviation) = 10 + (1.645 * 2) ≈ 13.29.

In summary:

(a) x = 10.

(b) x ≈ 13.29.

(c) x₁ ≈ 8.316 and x₂ ≈ 11.684.

(d) x₁ ≈ 5.08 and x₂ ≈ 14.92.

(e) x₁ ≈ 6.71 and x₂ ≈ 13.29.

Explanation: