Question

A humanities professor assigns letter grades on a test according to the following scheme. A: Top 12% of scores B: Scores below the top 12% and above the bottom 64d% C: Scores below the top 366% and above the bottom 23#% D: Scores below the top 77w% and above the bottom 7%7% F: Bottom 7%7% of scores Scores on the test are normally distributed with a mean of 77.877.8 and a standard deviation of 8.58.5. Find the minimum score required for an A grade. Round your answer to the nearest whole number, if necessary.

Answer 1

Answer:

The minimum score required for an A grade is 88.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 77.8, \sigma = 8.5$

Find the minimum score required for an A grade.

Top 12%, which is at least the 100-12 = 88th percentile, which is the value of X when Z has a pvalue of 0.88. So it is X when Z = 1.175.

$Z = \frac{X - \mu}{\sigma}$

$1.175 = \frac{X - 77.8}{8.5}$

$X - 77.8 = 1.175*8.5$

$X = 87.8$

Rounding to the nearest whole number

The minimum score required for an A grade is 88.

Answer 2

Answer: the minimum score required for an A grade is 88.

Step-by-step explanation:

Since the scores on the test are normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = scores on the test.

µ = mean score

σ = standard deviation

From the information given,

µ = 77.8

σ = 8.5

The probability value for the top 12% of the scores would be (1 - 12/100) = (1 - 0.13) = 0.88

Looking at the normal distribution table, the z score corresponding to the probability value is 1.18

Therefore,

1.18 = (x - 77.8)/8.5

Cross multiplying by 8.5, it becomes

1.18 × 8.5 = x - 77.8

10.03 = x - 77.8

x = 10.03 + 77.8

x = 88 to the nearest whole number