Question

Professors Einstein, Newton, Bernoulli, Fermat, and Gauss teach the same math course at AB College. Students think that they are all good teachers. Suppose that you are deciding whether to take Professor Einstein’s or Professor Newton’s class next semester. You know that both Professor Einstein and Professor Newton use the same grading scale: A, 90 – 100%; B, 80 – 89%; C, 70 – 79%; D, 60 – 69%; and F, below 60 %. You also know that the distribution of scores on Professor Einstein’s final is approximately normal with mean 74 and standard deviation 7; the distribution of scores on Professor Newton’s final is also approximately normal with mean 78 and standard deviation 18.

a. Produce a sketch for both Professor Einstein’s and Professor Newton’s grade distributions to compare the distributions. To accomplish a comparison you should graph both in the same graph using identical scales.

b. Which professor (Einstein or Newton) gives the higher proportion of A’s (that is, has more students getting 90 or above)? Show the appropriate calculations to support your answer.

c. Which professor gives the higher proportion of F’s (below 60)? Show the appropriate calculations to support your answer.

d. Suppose Professor Bernoulli has a policy of giving A’s to the top 10% of the scores on his final, regardless of the actual scores. If the distribution of scores on his final turns out to be normal with mean 69 and standard deviation 9, how high does your score have to be to earn an A? Show the appropriate calculations to support your answer.

e. Professors Fermat and Gauss have final exam scores that are approximately normal with mean 75. The standard deviation of Fermat’s scores is 10 and that of Gauss’ is 5.

(1) With which professor is a score of 90 more impressive? Support your answer with appropriate calculations and with a sketch.

(2) With which professor is a score of 60 more discouraging? Support your answer with appropriate calculations and with a sketch.

Answer 1

a)

The normal distribution plot for Prof. Einstein and Prof. Newton is obtained in R.

R Code

R Output

The plot with the red line represents the scores by Prof Einstein and with blue dashed line represents the scores by Prof Newton.

From the plot, we can see that the scores given by Prof Newton are higher compared to Prof Einstein

b)

The proportion of grade A's (score > 90) is obtained by calculating the z score.

For Prof Einstein

From the z distribution table,

For Prof Newton

From the z distribution table,

The proportion for Prof Newton is higher (25.25%) compared to Prof Einstein (1.11%)

c)

The proportion of grade F's (score < 60) is obtained by calculating the z score.

For Prof Einstein

From the z distribution table,

For Prof Newton

From the z distribution table,

The proportion for Prof Newton is higher (215.87%) compared to Prof Einstein (15.87%)

d)

For Professor Bernoulli

Mean of scores = 69

The standard deviation of scores = 9

The z score for minimum score corresponds to grade A (top 10%) is obtained from the standard normal distribution table for probability = 0.90

From the standard normal distribution table,

The score should be 80.5340 to earn a grade A

e)

1)

Professors Fermat

mean = 75

standard deviation = 10

From the z distribution table,

Professors Gauss

mean = 75

standard deviation = 5

From the z distribution table,

Since the proportion of score = 90 is less for Professor Gauss compared to Professor Fermat, the score is more impressive for Professor Gauss (because fewer students getting that score)

2)

Professors Fermat

From the z distribution table,

Professors Gauss

From the z distribution table,

Since the proportion of score = 60 or less is less for Professor Gauss compared to Professor Fermat, the score is more discouraging for Professor Gauss (because fewer students getting that score)