Math Snap

STEP 1

What is this asking?
We're looking at a bunch of graduates and want to figure out, on average, what proportion of them have student loans, and how much that proportion varies from sample to sample.
Watch out!
Don't mix up the proportion of all students with loans and the proportion of a sample with loans.
Also, remember the difference between standard deviation and variance.

STEP 2

1. Find the mean proportion.
2. Calculate the standard deviation.

STEP 3

The problem tells us that 65% of college students graduate with student loan debt.
This is our population proportion, which we write as $p = 0.65$.
When we take a random sample of graduates, we expect the sample proportion ($\hat{p}$) to be the same as the population proportion, on average.

STEP 4

So, the mean of the sample proportion ($\mu_{\hat{p}}$) is simply equal to the population proportion:
$$\mu_{\hat{p}} = p = \mathbf{0.65}$$ This means that if we took tons of samples of graduates, the average of the proportions with loans in each sample would be around 0.65.

STEP 5

The standard deviation of the sample proportion ($\sigma_{\hat{p}}$) tells us how spread out the sample proportions are likely to be.
The formula for this is:
$$\sigma_{\hat{p}} = \sqrt{\frac{p \cdot (1-p)}{n}}$$ where $p$ is the population proportion, and $n$ is the sample size.

STEP 6

We know that $p = 0.65$ and our sample size is $n = 90$.
Let's plug these values into the formula:
$$\sigma_{\hat{p}} = \sqrt{\frac{0.65 \cdot (1-0.65)}{90}}$$

STEP 7

First, let's calculate $1 - 0.65 = 0.35$.
This represents the proportion of students without loans.
Then, we multiply this by $0.65$ to get $0.65 \cdot 0.35 = 0.2275$.

STEP 8

Next, we divide this by our sample size of 90:
$$\frac{0.2275}{90} \approx 0.00252778$$

STEP 9

Finally, we take the square root of this result:
$$\sqrt{0.00252778} \approx \mathbf{0.050277}$$ So, the standard deviation of the sample proportion is approximately 0.0503 when rounded to four decimal places.

The mean of the sample proportion ($\mu_{\hat{p}}$) is 0.65.
The standard deviation of the sample proportion ($\sigma_{\hat{p}}$) is approximately 0.0503.