Answer :
Final answer:
To find the probability that Z is less than 1.57 or greater than 1.84 in a standardized normal distribution, use a Z-table to find P(Z < 1.57) and 1 - P(Z < 1.84), then sum these probabilities.
Explanation:
The question inquires about the probability of a random variable Z in a standardized normal distribution being either less than 1.57 or greater than 1.84. To find this probability, we can use a Z-table to determine the area under the curve for each condition and then add these probabilities together.
Firstly, the probability that Z is less than 1.57 is found by looking up the value of Z = 1.57 in the Z-table which gives us an area, or probability, to the left of Z = 1.57. Let's denote this value P(Z < 1.57).
For finding the probability that Z is greater than 1.84, we look up Z = 1.84 in the Z-table to get the area to the left of Z = 1.84. Since the table gives the area to the left, and we want the area to the right (greater than 1.84), we subtract this value from 1 to get P(Z > 1.84).
The final step is to add these probabilities: The probability that Z is less than 1.57 or greater than 1.84 equals P(Z < 1.57) + P(Z > 1.84). Remember that in a normal distribution, the total area under the curve is 1, representing a 100% probability.
