Math / Data & Statistics

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Student loans: The Institute for College Access and Success reported that $65 \%$ of college students in a recent year graduated with student loan debt. A random sample of 90 graduates is drawn. Round your answers to at least four decimal places if necessary.
Part 1 of 6 (a) Find the mean $\mu_{\hat{p}}$ .
The mean $\mu_{\hat{p}}$ is 0.65 .
Part 2 of 6 (b) Find the standard deviation $\sigma \hat{p}$ .
The standard deviation $\sigma_{\hat{p}}$ is 0.0503 . $p$
Part: $2 / 6$
Part 3 of 6 (c) Find the probability that less than $52 \%$ of the people in the sample were in debt.
The probability that less than $52 \%$ of the people in the sample were in debt is $\square$ . Skip Part Check Save For Later Submit A Q 2024 McGraw Hill LLC All Rights Reserved. Terms of Uso - Fivacy Center

Studdy Solution

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 $0.52$ .

STEP 8

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

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Studdy Solution

STEP 1

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

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 $0.52$ .

STEP 8

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

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Studdy Solution

STEP 1

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 0.520.520.52.

STEP 4

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

STEP 5

STEP 6

A z-score of −2.58-2.58−2.58 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 −2.58-2.58−2.58 in the z-table, we find a probability of approximately 0.00490.00490.0049.This is the probability of getting a sample with a proportion of graduates in debt less than 0.520.520.52.

STEP 8

The probability that less than 52% of the people in the sample were in debt is approximately 0.00490.00490.0049.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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

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

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

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.

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 $0.0049$ .