Question

The distribution of course grades in a very large class is (approximately) Normal with mean 57 and standard deviation 12 . The minimum possible grade is 0 and the maximum possible grade is 120 . A grade of 90 or more is required to get "A+". The course instructor is considering the possibility of a "bell curve" that will increase the mean grade to 77 and decrease the standard deviation to 8 . What is the proportion of students that will get "A+" if the bell curve is applied? Z table: P(Z≤Z) (You do not have to interpolate.)

Answer 1

The proportion of students that will get an A+ if the bell curve is applied is approximately 0.28%.

Step-by-step explanation:

To determine the proportion of students who will get an "A+" grade after applying the bell curve, we need to find the Z-score corresponding to a grade of 90 under the original distribution and then find the corresponding percentile in the new distribution.

For the original distribution with mean

�

1

=

57

μ

1

​

=57 and standard deviation

�

1

=

12

σ

1

​

=12, we calculate the Z-score for a grade of 90 using the formula:

�

=

�

−

�

1

�

1

Z=

σ

1

​

X−μ

1

​

​

where

�

X is the grade. Substituting in the values:

�

=

90

−

57

12

=

2.75

Z=

12

90−57

​

=2.75

Now, we use the Z-table to find the percentile corresponding to a Z-score of 2.75. The Z-table provides the probability that a standard normal random variable is less than or equal to a given Z-score.

Looking up the Z-score of 2.75 in the Z-table, we find the corresponding probability, let's call it

�

1

P

1

​

.

Now, for the new distribution with mean

�

2

=

77

μ

2

​

=77 and standard deviation

�

2

=

8

σ

2

​

=8, we find the Z-score for a grade of 90 using the same formula:

�

=

�

−

�

2

�

2

Z=

σ

2

​

X−μ

2

​

​

Substituting in the values:

�

=

90

−

77

8

=

1.625

Z=

8

90−77

​

=1.625

Now, using the Z-table, we find the corresponding probability, let's call it

�

2

P

2

​

.

The proportion of students who will get "A+" after applying the bell curve is the difference between

�

1

P

1

​

and

�

2

P

2

​

:

Proportion of A+

=

�

1

−

�

2

Proportion of A+=P

1

​

−P

2

This proportion represents the percentage of students who will get an "A+" grade under the new distribution. Note that we do not need to interpolate since the Z-table provides probabilities for specific Z-scores.