omework: 5.3 - 5.5 Question 1, 5.3.1-T HW Score: $0 \%$ Points: 0 Use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area 0.1962. $z=\square($ Type an integer or decimal rounded to two decimal places as needed.)
Solution
Step 1 ：The question is asking for the z-score that corresponds to the cumulative area 0.1962 in a standard normal distribution. The z-score is a measure of how many standard deviations an element is from the mean. In a standard normal distribution, the mean is 0 and the standard deviation is 1. The cumulative area mentioned in the question is the area under the curve of the distribution to the left of the z-score.
Step 2 ：To find the z-score, we can use the Percent Point Function (PPF), which is the inverse of the Cumulative Distribution Function (CDF). The CDF gives the cumulative area to the left of a z-score, and the PPF gives the z-score for a given cumulative area.
Step 3 ：We need to input the cumulative area into the PPF function. The cumulative area is 0.1962.
Step 4 ：After calculating, we find that the z-score corresponding to the cumulative area 0.1962 is approximately -0.8552730564153933.
Step 5 ：This means that the value is approximately 0.855 standard deviations below the mean in a standard normal distribution.
Step 6 ：Final Answer: The z-score that corresponds to the cumulative area 0.1962 is approximately \(\boxed{-0.855}\).
