A philosophy 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 58% C: Scores below the top 42% and above the bottom 20% D: Scores below the top 80% and above the bottom 9% F: Bottom 9% of scores Scores on the test are normally distributed with a mean of 67.3 and a standard deviation of 7.3. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessary.

Answer:

The numerical limits for a D grade are 58(lower limit) and 61(upper limit).

Step-by-step explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

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 p-value, we get the probability that the value of the measure is greater than X.

Scores on the test are normally distributed with a mean of 67.3 and a standard deviation of 7.3.

This means that [tex]\mu = 67.3, \sigma = 7.3[/tex]

D: Scores below the top 80% and above the bottom 9%

This means that lower bound is the 9th percentile and the upper bound is the 100 - 80 = 20th percentile.

9th percentile:

This is X when Z has a pvalue of 0.09. So X when Z = -1.34.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-1.34 = \frac{X - 67.3}{7.3}[/tex]

[tex]X - 67.3 = -1.34*7.3[/tex]

[tex]X = 57.52[/tex]

Rounding to the nearest whole number, 58.

20th percentile:

This is X when Z has a pvalue of 0.2. So X when Z = -0.84

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-0.84 = \frac{X - 67.3}{7.3}[/tex]

[tex]X - 67.3 = -0.84*7.3[/tex]

[tex]X = 61.17[/tex]

Rounding to the nearest whole number, 61.

The numerical limits for a D grade are 58(lower limit) and 61(upper limit).

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