A survey of 20 colleges found that seniors graduated with an average $28,950 in debt from student loans. The debt was normally distributed with a standard deviation of $3200. Find the probability that a senior graduated owing more than $15,000.

Answer 1

We can use the standard normal distribution to solve this problem by transforming the original data into z-scores:

z = (x - μ) / σ

where x is the value we are interested in (in this case, $15,000), μ is the mean debt ($28,950), and σ is the standard deviation ($3200).

z = (15000 - 28950) / 3200 = -4.078125

Note that the negative z-score indicates that $15,000 is more than 4 standard deviations below the mean.

Using a standard normal distribution table or calculator, we can find that the probability of a z-score less than -4.08 is very close to 0 (less than 0.0001). Therefore, the probability of a senior graduating with debt exceeding $15,000 is very close to 1 (or 100%). This suggests that the original data may have outliers or other unusual characteristics that make the distribution highly skewed or non-normal.