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:
[tex]P(31000 < X < 33000) = P( X \leq 33000) - P (X \leq 31000)[/tex]
[tex]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}}} )[/tex]
[tex]P(31000 < X < 33000) = P( Z \leq \dfrac{33000 - 30000}{\dfrac{9000}{\sqrt{100}}} )- P(Z \leq \dfrac{31000 - 30000}{\dfrac{9000}{\sqrt{100}}} )[/tex]
[tex]P(31000 < X < 33000) = P( Z \leq \dfrac{3000}{\dfrac{9000}{10}}}) -P(Z \leq \dfrac{1000}{\dfrac{9000}{10}}})[/tex]
[tex]P(31000 < X < 33000) = P( Z \leq 3.33)-P(Z \leq 1.11})[/tex]
From Z tables:
[tex]P(31000 < X <33000) = 0.9996 -0.8665[/tex]
[tex]P(31000 < X <33000) = 0.1331[/tex]
Therefore; the probability that the mean student loan debt for these people is between $31000 and $33000 is 0.1331
