Question

1. Suppose a professor gives an exam to a class of 40 students and the scores are as follows. (Type the data set in StatCrunch.)

35	44	46	47	47	48	49	51	53	54
55	55	57	57	57	58	59	59	59	59
60	60	60	60	60	62	62	62	64	68
69	70	72	73	73	75	75	77	82	88

Top 10% of scores receive an A
Bottom 10% of scores receive an F
Scores between the 70th and 90th percentile receive a B
Scores between the 30th and 70th percentile receive a C
Scores between the 10th and 30th percentile receive a D

Find the range of scores that would qualify for each grade under this plan.
(Assume all scores are normally distributed with a mean of 60.53 and standard deviation of 10.96.)

Answer 1

We have been given different percentile which are like proportions or probabilties. Using normal percentage tables we can get the values of ranges.

Since the percentage values are given for greater than values and are for below 50%, we need to convert everything, > x and p < 50%.

Top 10%

P(X >x) = 10%

P(Z > ) = 10%

= 1.2816

x = 74.58 75

Therefore students scoring above 75 get A

P(X <x) = 10% ...score F

P(X > -x) = 10%

P(Z > -) = 10%

= -1.2816 .....................due to difference of '<' there is a difference of '-' in the z-score

x = 46.48 46

Students scoring below 46 get F

70th percentile

P(X < x) = 70%

P(X > x) = 30%

P( X > ) = 30%

= 0.5244

x = 66.28 66

30th percentile

PX < x) = 30%

P(X > -x) = 30%

= -0.5244

x = 54.78 55

90th percentile

P(X <x) = 90%

P(X > x ) = 10%

x = 75 ....from above

Therefore students scoring between 70 and 90th percentile have the range 66 - 75 that get B

Therefore students scoring between 30 and 70th percentile have the range 55- 66 that get C

Therefore students scoring between 10 and 30th percentile have the range 46- 55 that get D

Grade	Marks
A	< 46
B	46- 55
C	55 - 66
D	66 - 75
F	> 75