Question

A psychology professor assigns letter grades on a test according to the following scheme.

A: Top 10%
of scores

B: Scores below the top 10%
and above the bottom 56%
C: Scores below the top 44%
and above the bottom 23%
D: Scores below the top 77%
and above the bottom 8%
F: Bottom 8%
of scores

Scores on the test are normally distributed with a mean of 81.6
and a standard deviation of 7.2
. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessary.

Answer 1

Explanation:

we need to use the standard normal distribution table for mean = 0, standard deviation = 1.

to use it for our normal distribution with mean = 81.6 and standard deviation = 7.2 we need to find the transformation from the standard normal distribution to our normal distribution.

this is done by the z-scores of the interval limits (the z-score tells us how many standard deviations the desired value is away from the mean value).

and to correlate the given %-values to the distribution interval limits, we need to look up the corresponding p-value (probability value) in the table.

and then we need to calculate backwards to get our specific interval limits.

for the D-grade we are looking for the upper limit to "below the top 77%", which is "below the bottom 23%".

we have to assume this means "below or equal to the bottom 23%", just in case.

the lower level will be "above the bottom 8%".

so, the p-value for 23% = 0.2300.

that gives us the z-score ≈ -0.74.

z-score = (interval-limit - mean-value)/standard-deviation

-0.74 = (interval-limit - 81.6)/7.2

-0.74×7.2 = interval-limit - 81.6

-5.328 = interval-limit - 81.6

interval-limit = 76.272 ≈ 76

the p-value for 8% = 0.0800.

that gives us the z-score ≈ -1.41

-1.41 = (interval-limit - 81.6)/7.2

-1.41×7.2 = interval-limit - 81.6

-10.152 = interval-limit - 81.6

interval-limit = 71.448 ≈ 71

the students with a score s

71 < s <= 76

get a D.

(greater than 71 but lower or equal to 76).

Answer 2

Explanation:

To find the numerical limits for a D grade, we need to determine the cutoff points for the top 77% and bottom 8% of scores.

Using a standard normal distribution table or a calculator, we can find that the z-score cutoff points for the top 77% and bottom 8% are approximately 0.61 and -1.41, respectively.

To find the corresponding raw scores, we use the formula:

raw score = (z-score x standard deviation) + mean

For the top 77%, we have:

raw score = (0.61 x 7.2) + 81.6 = 86.392, which we can round up to 87

For the bottom 8%, we have:

raw score = (-1.41 x 7.2) + 81.6 = 70.848, which we can round down to 70

Therefore, the numerical limits for a D grade are scores below 87 and above 70.