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A psychology professor assigns letter grades on a test according to the following scheme. A: Top 9% of scores B: Scores below the top 9% and above the bottom 56% C: Scores below the top 44% and above the bottom 20% D: Scores below the top 80% and above the bottom 6% F: Bottom 6% of scores Scores on the test are normally distributed with a mean of 79.3 and a standard deviation of 8.4. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessary.

Sagot :

Answer:

You get a D if you have a grade between 66 and 72.

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 zscore 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 pvalue, 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 79.3 and a standard deviation of 8.4.

This means that [tex]\mu = 79.3, \sigma = 8.4[/tex]

D: Scores below the top 80% and above the bottom 6%

So between the 6th and the 20th percentile.

6th percentile:

X when Z has a pvalue of 0.06. So X when Z = -1.555.

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

[tex]-1.555 = \frac{X - 79.3}{8.4}[/tex]

[tex]X - 79.3 = -1.555*8.4[/tex]

[tex]X = 66.2[/tex]

Rounds to 66

20th percentile:

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 - 79.3}{8.4}[/tex]

[tex]X - 79.3 = -0.84*8.4[/tex]

[tex]X = 72.2[/tex]

Rounds to 72

You get a D if you have a grade between 66 and 72.

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