Question

Suppose the grade on a Math test is normally distributed with mean 78 and standard deviation 10. (a) Compute the z-scores (5 points) (a-1) If Bob got 70 on the test, what is his z-score? (a-2) If Jane got 90 on the test, what is her z-score? (b) Compute the actual grades (5 points) (b-1) Suppose David achieved a grade 1.8 standard deviation above the mean (? = 1.8), what was his actual grade? (b-2) Suppose Lily achieved a grade 0.5 standard deviation below the mean (? = −0.5), what was her actual grade? (c) Rob achieved a grade that exceeded 95% of all grades. Find Rob’s actual grade. (6 points) (d) Suppose 32% of students did better than Mei. Find Mei’s actual grade. (6 points)

Answer 1

Solution:

Given in the question
Mean = 78
Standard deviation = 10

SOlution(a)
If Bob Got 70 on the test, what is his Z score
Z-score = (70-78)/10 = -0.8
If jan got 90 on the test what is her Z-Score
Z-Score = (90-78)/12 = 1.2

Solution(b-1)
Given that Z =1.8 than Xbar can be calculated as
1.8 = (Xbar -78)/10
18 = Xbar-78
Xbar = 96

Solution(b-2)
Given Z = -0.5
-0.5 = (Xbar-78)/10
Xbar = 78-5 =73

Solution(c)
Given p-value = 0.95
Z-score from Z table is 1.645
1.645 = (Xbar-78)/10
Xbar = 78+16.45 = 94.45

Solution(d)
Here P-value = 0.68 so Z -score = 0.47
0.47 = (Xbar-78)/10
4.7 = Xbar-78
Xbar = 82.7