Question

Find a value of the standard normal random variable z, call it z₀, such that the following probabilities are satisfied.

A) P (z ≤ z₀) = 0.0129
B) P(−z₀ ≤ z ≤ z₀) = 0.09
C) P(−z₀ ≤ z ≤ z₀) = 0.95
D) P(−z₀ ≤ z ≤ z₀) = 0.8182
E) P(−z₀ ≤ z ≤ 0) = 0.3125
F) P(−3 < z < z₀) = 0.9877
G) P(z > z₀) = 0.5
H) P(z ≤ z₀) = 0.0097

Answer 1

To find the value of the standard normal random variable z, we can use a calculator, a computer, or a probability table. We can find the values of z that satisfy each of the given probabilities by using a z-table or calculator.

Step-by-step explanation:

In order to find the value of the standard normal random variable z, we can use a calculator, a computer, or a probability table. Let's go through each of the probabilities one by one:

1. A) P (z ≤ z₀) = 0.0129: By using a z-table or calculator, we can find that z₀ is approximately -2.32.
2. B) P(−z₀ ≤ z ≤ z₀) = 0.09: This means that the area between -z₀ and z₀ under the standard normal curve is 0.09. By using a z-table or calculator, we can find that z₀ is approximately 1.6449.
3. C) P(−z₀ ≤ z ≤ z₀) = 0.95: This means that the area between -z₀ and z₀ under the standard normal curve is 0.95. By using a z-table or calculator, we can find that z₀ is approximately 1.9599.
4. D) P(−z₀ ≤ z ≤ z₀) = 0.8182: This means that the area between -z₀ and z₀ under the standard normal curve is 0.8182. By using a z-table or calculator, we can find that z₀ is approximately 1.7258.
5. E) P(−z₀ ≤ z ≤ 0) = 0.3125: This means that the area between -z₀ and 0 under the standard normal curve is 0.3125. By using a z-table or calculator, we can find that z₀ is approximately 0.4603.
6. F) P(−3 < z < z₀) = 0.9877: This means that the area between -3 and z₀ under the standard normal curve is 0.9877. By using a z-table or calculator, we can find that z₀ is approximately 2.7478.
7. G) P(z > z₀) = 0.5: This means that the area to the right of z₀ under the standard normal curve is 0.5. By using a z-table or calculator, we can find that z₀ is approximately 0.
8. H) P(z ≤ z₀) = 0.0097: This means that the area to the left of z₀ under the standard normal curve is 0.0097. By using a z-table or calculator, we can find that z₀ is approximately -2.3816.