Math Snap

STEP 1

What is this asking?
We're looking at the probability that less than 52% of a group of 90 recent graduates have student loan debt, knowing that 65% of all graduates typically do.
Watch out!
Don't forget to convert percentages to decimals before calculating, and make sure to use the sample size correctly in your calculations!

STEP 2

1. Calculate the z-score.
2. Find the probability associated with the z-score.

STEP 3

We're interested in the probability that less than 52% of the sample has debt.
So our target proportion is $0.52$.

STEP 4

From the problem, we know the mean proportion of graduates with debt is $\mu_{\hat{p}} = 0.65$, and the standard deviation is $\sigma_{\hat{p}} = 0.0503$.

STEP 5

The z-score tells us how many standard deviations our target proportion is away from the mean.
The formula is:
$$z = \frac{\text{target proportion} - \text{mean}}{\text{standard deviation}}$$ Plugging in our values:
$$z = \frac{0.52 - 0.65}{0.0503}$$ $$z = \frac{-0.13}{0.0503}$$$$z \approx -2.58$$So our z-score is approximately $-2.58$.
This means our target proportion of $0.52$ is $2.58$ standard deviations below the mean of $0.65$.

STEP 6

A z-score of $-2.58$ corresponds to a probability.
We can look this up in a z-table (negative z-scores are usually in a separate table) or use a calculator with a built-in z-score function.

STEP 7

Looking up $-2.58$ in the z-table, we find a probability of approximately $0.0049$.
This is the probability of getting a sample with a proportion of graduates in debt less than $0.52$.

The probability that less than 52% of the people in the sample were in debt is approximately $0.0049$.