Answer:
a = 200 + 0.253 * 50 = 212.65
So the value of height that separates the bottom 60% of data from the top 40% is 212.65. And rounded would be 212.
Step-by-Step Explanation:
The normal distribution is a "probability distribution that is symmetric around the mean, demonstrating that data near the mean occur more frequently than data distant from the mean."
The Z-score is defined as "a numerical measurement used in statistics of a value's relationship to the mean (average) of a set of values, expressed in standard deviations from the mean."
Let X be the random variable that represents how a loan officer scores credit applications from a population, and we know that the distribution for X is provided by:
X ~ N (200,50)
Where u = 200 and o = 50
And the best way to solve this problem is using the normal standard distribution and the z score given by:
Z = x - u/o
For this part we want to find a value a, such that we satisfy this condition:
P (X > a) = 0.40 (a)
P (X<a) = 0.60 (b)
In this example, both conditions are equal.
The z value that meets the criterion with 0.60 of the area on the left and 0.40 of the area on the right is z=0.253, as shown in the attached figure. P(Z0.253)=0.60 and P(z>0.253)=0.4 in this scenario.
Using the prior condition (b), we get:
P (X < a) = P (X-u/a < a - u/o) = 0.6
P (Z < a-u/o) = 0.6
But we know which value of z satisfy the previous equation so then we can do this:
z = 0.253 < a -200/50
And if we solve for a we got
a = 200 + 0.253 * 50 = 212.65
So the value of height that separates the bottom 60% of data from the top 40% is 212.65. And rounded would be 212.7.
