Let Y = ex where X is normally distributed with μ = 2.0 and σ = 0.9. Compute the following values. [You may find it useful to reference the z table.]

Question

a. Compute P(Y ≤ 8.7). (Round your intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)



b. Compute P(8.7 < Y < 9.6). (Round your intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)



c. Compute the 60th percentile of Y. (Round your intermediate calculations to at least 4 decimal places and final answer to the nearest whole number.)

Answer 1

To solve these problems, we need to standardize the values and use the standard normal distribution table.

a. To compute P(Y ≤ 8.7), we need to find the z-value corresponding to x = 8.7 first. We can standardize x using the formula:
z = (x - μ) / σ = (8.7 - 2.0) / 0.9 = 7.77

Using the z-table, the probability corresponding to a z-value of 7.77 is very close to 1.0. Therefore, P(Y ≤ 8.7) is approximately 1.0000.

b. To compute P(8.7 < Y < 9.6), we need to find the z-values corresponding to x = 8.7 and x = 9.6.

For x = 8.7:
z1 = (8.7 - 2.0) / 0.9 = 7.77

For x = 9.6:
z2 = (9.6 - 2.0) / 0.9 = 9.33

Using the z-table, the probability corresponding to z1 = 7.77 is very close to 1.0 and the probability corresponding to z2 = 9.33 is very close to 1.0. Therefore, P(8.7 < Y < 9.6) is approximately 1.0000.

c. To compute the 60th percentile of Y, we need to find the corresponding z-value. The 60th percentile means there is a 60% probability that the value of Y is below this percentile.

Using the z-table, we need to find the z-value that corresponds to a cumulative probability of 0.60. The closest value to 0.60 in the table is 0.5793, which corresponds to a z-value of 0.25.

To find the x-value, we can use the formula for standardizing a variable:
z = (x - μ) / σ

Rearranging the formula to solve for x:
x = z * σ + μ
x = 0.25 * 0.9 + 2.0
x ≈ 2.225

Therefore, the 60th percentile of Y is approximately 2.225.