For the standard normal random variable Z, compute the following probabilities. For the following problems, use four decimal places.
16.
P (0 ¡Ü Z ¡Ü .9)
17.
P (-1.3 ¡Ü Z ¡Ü 0)
18.
P(Z > 2.5)
19.
P(Z ¡Ý -.75)
20.
P(Z < .05)
21.
P (Z ¡Ü -.72)
To compute probabilities for the standard normal random variable Z, you need to use a Z-table or a statistical calculator like Excel or R. I will explain how to find these probabilities using a Z-table.
A Z-table gives you the area under the standard normal curve to the left of a given Z-value. To find the area between two Z-values, you subtract the area to the left of the lower Z-value from the area to the left of the higher Z-value.
Let's compute the probabilities for the given problems:
16. P(0 ≤ Z ≤ 0.9)
First, find the area to the left of Z = 0 and Z = 0.9 from the Z-table. Subtract the former from the latter to get the desired probability.
Using the Z-table, the area to the left of Z = 0 is 0.5000, and the area to the left of Z = 0.9 is 0.8159.
So, the probability P(0 ≤ Z ≤ 0.9) is 0.8159 - 0.5000 = 0.3159.
17. P(-1.3 ≤ Z ≤ 0)
First, find the area to the left of Z = -1.3 and Z = 0 from the Z-table. Subtract the former from the latter to get the desired probability.
Using the Z-table, the area to the left of Z = -1.3 is 0.0968, and the area to the left of Z = 0 is 0.5000.
So, the probability P(-1.3 ≤ Z ≤ 0) is 0.5000 - 0.0968 = 0.4032.
18. P(Z > 2.5)
To find the probability P(Z > 2.5), subtract the area to the left of Z = 2.5 from 1 (the total area under the curve).
Using the Z-table, the area to the left of Z = 2.5 is 0.9938.
So, the probability P(Z > 2.5) is 1 - 0.9938 = 0.0062.
19. P(Z ≥ -0.75)
To find the probability P(Z ≥ -0.75), subtract the area to the left of Z = -0.75 from 1.
Using the Z-table, the area to the left of Z = -0.75 is 0.2266.
So, the probability P(Z ≥ -0.75) is 1 - 0.2266 = 0.7734.
20. P(Z < 0.05)
Using the Z-table, find the area to the left of Z = 0.05.
The area to the left of Z = 0.05 is 0.5199.
So, the probability P(Z < 0.05) is 0.5199.
21. P(Z ≤ -0.72)
Using the Z-table, find the area to the left of Z = -0.72.
The area to the left of Z = -0.72 is 0.2357.
So, the probability P(Z ≤ -0.72) is 0.2357.
Please note that these values are rounded to four decimal places based on the information provided.