Question

The mean student loan debt for college graduates in Illinois is $30000 with a standard deviation of $9000. Suppose a random sample of 100 college grads in Illinois is collected. What is the probability that the mean student loan debt for these people is between $31000 and $33000?

Answer 1

Answer:

the probability that the mean student loan debt for these people is between $31000 and $33000 is 0.1331

Step-by-step explanation:

Given that:

Mean = 30000

Standard deviation = 9000

sample size = 100

The probability that the mean student loan debt for these people is between $31000 and $33000 can be computed as:

$P(31000 < X < 33000) = P( X \leq 33000) - P (X \leq 31000)$

$P(31000 < X < 33000) = P( \dfrac{X - 30000}{\dfrac{\sigma}{\sqrt{n}}} \leq \dfrac{33000 - 30000}{\dfrac{9000}{\sqrt{100}}} )- P( \dfrac{X - 30000}{\dfrac{\sigma}{\sqrt{n}}} \leq \dfrac{31000 - 30000}{\dfrac{9000}{\sqrt{100}}} )$

$P(31000 < X < 33000) = P( Z \leq \dfrac{33000 - 30000}{\dfrac{9000}{\sqrt{100}}} )- P(Z \leq \dfrac{31000 - 30000}{\dfrac{9000}{\sqrt{100}}} )$

$P(31000 < X < 33000) = P( Z \leq \dfrac{3000}{\dfrac{9000}{10}}}) -P(Z \leq \dfrac{1000}{\dfrac{9000}{10}}})$

$P(31000 < X < 33000) = P( Z \leq 3.33)-P(Z \leq 1.11})$

From Z tables:

$P(31000 < X$

$P(31000 < X$

Therefore; the probability that the mean student loan debt for these people is between $31000 and $33000 is 0.1331