Answer :
The probabilities for the given scenarios in a standardized normal distribution are: a) Z < 1.57: approximately 0.9418, b) Z > 1.84: approximately 0.0336, c) 1.57 < Z < 1.84: approximately 0.0294, and d) Z < 1.57 or Z > 1.84: approximately 0.9754.
a. The probability that Z is less than 1.57 can be found by looking up the corresponding area under the standard normal distribution curve. Using a standard normal distribution table or a calculator, we find that the probability is approximately 0.9418.
b. The probability that Z is greater than 1.84 can also be obtained by looking up the area under the standard normal distribution curve. The probability is the complement of the area to the left of 1.84, which is 1 minus the probability that Z is less than or equal to 1.84. Using the table or calculator, we find this probability to be approximately 0.0336.
c. To find the probability that Z is between 1.57 and 1.84, we calculate the difference between the probabilities of Z being less than or equal to 1.84 and Z being less than or equal to 1.57. Using the standard normal distribution table or calculator, we find that the probability is approximately 0.0294.
d. To find the probability that Z is less than 1.57 or greater than 1.84, we can calculate the sum of the probabilities of Z being less than 1.57 and Z being greater than 1.84. Using the table or calculator, we find these individual probabilities to be approximately 0.9418 and 0.0336, respectively. Adding them together, we get a probability of approximately 0.9754.
In conclusion, the probabilities are as follows: a) 0.9418, b) 0.0336, c) 0.0294, and d) 0.9754. These probabilities represent the likelihood of observing Z values within the specified ranges in a standardized normal distribution.
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