Answer :
Final answer:
The probabilities for the given Z-scores in the standardized normal distribution are approximately 94.18% for Z less than 1.57, 3.29% for Z greater than 1.84, 2.53% for Z between 1.57 and 1.84, and 97.47% for Z less than 1.57 or greater than 1.84.
Explanation:
The question is asking for the probability of Z values from a standard normal distribution, calculated manually. The Z score is a measure of how many standard deviations an element is from the mean. Remember, for a standardized normal distribution, the mean is 0, and the standard deviation is 1.
- For a Z value less than 1.57, we would look at a standard normal distribution table which would give us a value of approximately 0.9418. This means there is a 0.9418 or 94.18% probability that Z is less than 1.57.
- To find the probability of a Z value greater than 1.84, we first find the cumulative probability for 1.84, which is 0.9671 from the table. However, since we need the probability that Z is greater than 1.84, we subtract this value from 1. So, 1 - 0.9671 = 0.0329 or 3.29%.
- For finding the probability that Z is between 1.57 and 1.84, subtract the cumulative probability for 1.57 from one for 1.84. i.e., 0.9671 - 0.9418 = 0.0253 or 2.53%.
- Finally, the probability that Z is less than 1.57 OR greater than 1.84, you add the probabilities calculated for each of these scenarios from steps 1&2. So, 0.9418 + 0.0329 = 0.9747 or 97.47%.
Learn more about Standardized Normal Distribution here:
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