4. Professor Kingsfield's class follows very strict grading guidelines. Assume that student averages in Kingsfield's class are normally distributed with a mean fo 62 points and a standard deviation of 12 points. Professor Kingsfield wants letter grades in his class to have a very specific distribution: 68 percent of students get Cs, 13.5 percent get Ds, 13.5 percent get Bs, 2.5 perent receive Fs, and the top 2.5 percent get As. What should be the numeric range for each letter grade

For grade A ,a student needs Z score of 2 or above, the required score is

X = 68 + 12Z =92, For grade B, a student needs Z score between 1 and 2 .

The required score is = 68 + (12 * 1) < X = 68 + ( 12 * 2) = 80 < X < 92, For grade C ,a student needs Z score between -1 and 1 . The  required score is = 68 + ( 12 * x - 1) < X = 68 + (12 * x 1) = 56 <X<80. For the  grade D, a student needs Z score between -2 and -1 .

The required score is = 68 + (12 * x -2) < X = 68 + (12 * x -1) = 44< X< 56

Step-by-step explanation:

From the given question,

let the rv X represent student's score.

Z = x - 68/12  ~ N (0,1 )

Now, we know for a standard normal distribution

Thus,

For grade A ,a student needs Z score of 2 or above, the required score is

X = 68 + 12Z =92

For grade B, a student needs Z score between 1 and 2 .

The required score is = 68 + (12 * 1) < X = 68 + ( 12 * 2) = 80 < X < 92

For grade C ,a student needs Z score between -1 and 1 .

The  required score is = 68 + ( 12 * x - 1) < X = 68 + (12 * x 1) = 56 <X<80

For the  grade D, a student needs Z score between -2 and -1 .

The required score is = 68 + (12 * x -2) < X = 68 + (12 * x -1) = 44< X< 56

For student  with score less 44 gets an F.

Note: find an attached image for the graphic sketch of the normal distribution.

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Step-by-step explanation:

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