Question

Suppose Z follows the standard normal distribution. Determine the value of c so that the following is true. Carry your intermediate computations to at least four decimal places. Round your answer to two decimal places. P (-1.16 ≤ Z ≤ c) = 0.8638.

Answer 1

To find the value of c for the probability P (-1.16 ≤ Z ≤ c) = 0.8638 in a standard normal distribution, calculate the area to the left of Z = -1.16, subtract it from 0.8638, and use the inverse standard normal distribution to find c.

Step-by-step explanation:

The student asks how to find the value of c such that P (-1.16 ≤ Z ≤ c) = 0.8638 given that Z follows the standard normal distribution. To solve for c, we need to recognize that the probability P (-1.16 ≤ Z ≤ c) represents the area under the standard normal distribution curve between -1.16 and c.

First, find the area to the left of Z = -1.16 using a standard normal distribution table or a calculator. Once the area to the left of Z = -1.16 is found, it is subtracted from the given total area 0.8638 to find the area to the left of Z = c. Finally, use the inverse of the standard normal distribution to find the value of c that yields the remaining area.

By using a calculator or a z-table, one can find the area to the left of Z = -1.16 and then solve for c using the relationship P (-1.16 ≤ Z ≤ c) = P (Z < c) - P (Z < -1.16). If the area to the left of Z = -1.16 is approximately 0.1230, then the area to the left of c would be 0.8638 + 0.1230 = 0.9868. Using the inverse standard normal distribution, we can find c that corresponds to this area.