Question

1.) What is the probability that a randomly selected female student had a high school GPA lower than 3.75? Solve this problem using the Standard Normal Table (Z table). Show all work and provide the probability as a decimal rounded to four decimal places.

2.) ​​​​​​​If a female student had a high school GPA of 4.00, what percentile would this be for all female students? Solve this problem using the Standard Normal Table (Z table). Show all work and provide the probability as a decimal rounded to four decimal places.​​​​​​​

Student Group	Mean GPA (μ)	Standard deviation (σ) of GPA
All students	3.33	0.53
Female students	3.40	0.50
Male students	3.23	0.54

Answer 1

Let X denote the female student who had a high school

1) P(X<3.75) = P(3.75 -3.40/.5) { in this step we have changed the normal probability in standard normal probability to attain our result and the formula use is (x - mean(x))/ standard deviation(x) }

P(X<3.75) = standard normal probability (0.7)

taking the value from standard normal table , we get

P(X<3.75) = 0.75804

2) to compute the percentile we would have to compute the standard normal probability considering mean and standard deviation of all students as percentile would be computed when all students would be taken into consideration which goes as follows

Standard normal probabilty (4-3.33/0.53) = Standard normal probabilty(1.2641)

Standard normal probabilty (1.27) = 0.89796

Standard normal probabilty (1.26) = 0.89617

Standard normal probabilty (1.26415) = ? = y (assumed)

Now , by the method of interpolation we get

So femle student with 4 GPA , the female student will get a percentile of 89 among all students

Similarly , percentile of the female student considering all female students would be

Standard normal probabilty (4-3.4/0.5) = Standard normal probabilty(1.2)

Now taking the probabilities from standard normal table the female student would be 88.493 percentile considering all female students .