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An estimated 1.8 million students take on student loans to pay ever-rising tuition and room and board (The New York Times, April 17, 2009). It is also known that the average cumulative debt of recent college graduates is about $22,500. Let the cumulative debt among recent college graduates be normally distributed with a standard deviation of $7,000. Approximately how many recent college graduates have accumulated a student loan of more than $30,000

Sagot :

Answer:

Approximately 256,140 recent college graduates have accumulated a student loan of more than $30,000.

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.

Average cumulative debt of recent college graduates is about $22,500, standard deviation of $7,000.

This means that [tex]\mu = 22500, \sigma = 7000[/tex]

Proportion more than 30000.

1 subtracted by the pvalue of Z when X = 30000. So

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

[tex]Z = \frac{30000 - 22500}{7000}[/tex]

[tex]Z = 1.07[/tex]

[tex]Z = 1.07[/tex] has a pvalue of 0.8577

1 - 0.8577 = 0.1423

Approximately how many recent college graduates have accumulated a student loan of more than $30,000?

0.1423 out of 1.8 million.

0.1423*1.8 = 0.25614 million = 256,140

Approximately 256,140 recent college graduates have accumulated a student loan of more than $30,000.

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