For the standard normal random variable Z, compute the following probabilities. For the following problems, use four decimal places.

16.
P (0 less than or equal to Z less than or equal .9)

17.
P (-1.3 less than or equal to Z less than or equal to)

18.
P(Z > 2.5)

19.
P(Z greater than or equal to -.75)

20.
P(Z < .05)

21.
P (Z less than or equal to -.72)

Assuming that you mean the Z score (Z = (score-mean)/Standard deviation), find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the above Z scores.

To compute the probabilities for the standard normal random variable Z, we can use a standard normal distribution table or a calculator. Here are the step-by-step calculations for each problem:

16. P(0 ≤ Z ≤ 0.9)
First, find the area under the curve to the left of 0.9:
P(Z ≤ 0.9) = 0.8159
Then, find the area under the curve to the left of 0:
P(Z ≤ 0) = 0.5000
Finally, subtract the second probability from the first:
P(0 ≤ Z ≤ 0.9) = P(Z ≤ 0.9) - P(Z ≤ 0) = 0.8159 - 0.5000 = 0.3159

17. P(-1.3 ≤ Z ≤ ?.?)
To find this probability, we need the area under the curve to the left of -1.3 (P(Z ≤ -1.3)) and the area under the curve to the left of ?.? (P(Z ≤ ?.?)). However, the value for ?.? is missing in the question, so we cannot calculate the probability without that information.

18. P(Z > 2.5)
First, find the area under the curve to the left of 2.5:
P(Z ≤ 2.5) = 0.9938
Then, subtract this probability from 1 to find the area to the right of 2.5:
P(Z > 2.5) = 1 - P(Z ≤ 2.5) = 1 - 0.9938 = 0.0062

19. P(Z ≥ -0.75)
First, find the area under the curve to the left of -0.75:
P(Z ≤ -0.75) = 0.2266
Then, subtract this probability from 1 to find the area to the right of -0.75:
P(Z ≥ -0.75) = 1 - P(Z ≤ -0.75) = 1 - 0.2266 = 0.7734

20. P(Z < 0.05)
First, find the area under the curve to the left of 0.05:
P(Z ≤ 0.05) = 0.5199
This is the final probability as it represents the area to the left of 0.05.

21. P(Z ≤ -0.72)
This is similar to problem 19. Using the standard normal distribution table or calculator, find the area under the curve to the left of -0.72 (P(Z ≤ -0.72)), which is 0.2357.

To compute the probabilities for the standard normal random variable Z, we can use a standard normal distribution table or a calculator/statistical software. In this case, we will use a standard normal distribution table.

Here's how to compute each probability:

16. P (0 ≤ Z ≤ 0.9):
To find this probability, we need to look up the area under the standard normal curve between 0 and 0.9. Look up the value 0.9 in the standard normal distribution table, and you will find the corresponding area to be 0.8159. Therefore, the probability P (0 ≤ Z ≤ 0.9) is 0.8159.

17. P (-1.3 ≤ Z ≤ x):
To find this probability, we need to look up the area under the standard normal curve between -1.3 and x. As x is not specified in the question, we cannot determine the exact probability without knowing its value.

18. P (Z > 2.5):
To find this probability, we need to find the area under the standard normal curve to the right of 2.5. Since the standard normal distribution is symmetric around the mean of 0, we can find the area to the left of -2.5 and subtract it from 1. Look up the value -2.5 in the standard normal distribution table, and you will find the corresponding area to be 0.0062. Subtracting it from 1 gives us 1 - 0.0062 = 0.9938. Therefore, the probability P (Z > 2.5) is 0.9938.

19. P (Z ≥ -0.75):
To find this probability, we need to find the area under the standard normal curve to the right of -0.75. Using the standard normal distribution table, we find the corresponding area to be 0.7734. Therefore, the probability P (Z ≥ -0.75) is 0.7734.

20. P (Z < 0.05):
To find this probability, we need to find the area under the standard normal curve to the left of 0.05. Look up the value 0.05 in the standard normal distribution table, and you will find the corresponding area to be 0.5199. Therefore, the probability P (Z < 0.05) is 0.5199.

21. P (Z ≤ -0.72):
To find this probability, we need to find the area under the standard normal curve to the left of -0.72. Using the standard normal distribution table, we find the corresponding area to be 0.2357. Therefore, the probability P (Z ≤ -0.72) is 0.2357.

Remember to always round the probabilities to four decimal places, as specified in the question.