find the z- score location of a vertical line that spearates a normal distribution as described in each of the following.
a. 20% in the tail on the left
b 40% in the tail on the right
c 75% in the body on the left
d. 99% in the body on the right
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find Z scores that correspond to these proportions.
To find the z-score location of a vertical line that separates a normal distribution according to the provided descriptions, we need to perform a process called z-score calculation.
The z-score is a measure of how many standard deviations an individual data point is away from the mean. We can use the cumulative distribution function (CDF) of the standard normal distribution (mean = 0, standard deviation = 1) to calculate the z-score.
a. 20% in the tail on the left:
To find the z-score that corresponds to 20% in the tail on the left, we need to find the value that has an area of 0.20 to the left of it in the standard normal distribution. Using a standard normal distribution table or statistical software, we can find that the z-score corresponding to this area is approximately -0.842.
b. 40% in the tail on the right:
To find the z-score that corresponds to 40% in the tail on the right, we need to find the value that has an area of 0.40 to the right of it in the standard normal distribution. However, since z-scores are typically used to find values to the left of them, we can convert this problem to finding the z-score with an area of 0.60 to the left (1 - 0.40 = 0.60). Using the standard normal distribution table or statistical software, we can find that the z-score corresponding to this area is approximately 0.253.
c. 75% in the body on the left:
To find the z-score that corresponds to 75% in the body on the left, we need to find the value that has an area of 0.75 to the left of it in the standard normal distribution. Using the standard normal distribution table or statistical software, we can find that the z-score corresponding to this area is approximately -0.674.
d. 99% in the body on the right:
To find the z-score that corresponds to 99% in the body on the right, we need to find the value that has an area of 0.99 to the right of it in the standard normal distribution. However, as mentioned earlier, z-scores are typically used to find values to the left of them, so we can convert this problem to finding the z-score with an area of 0.01 to the left (1 - 0.99 = 0.01). Using the standard normal distribution table or statistical software, we can find that the z-score corresponding to this area is approximately 2.326.
By using these z-score values, you can locate the vertical lines that separate the normal distribution according to the given descriptions.