A random sample of 20 individuals who graduated from college five years ago were asked to report the total amount of debt (in $)

Question

12 days ago

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A random sample of 20 individuals who graduated from college five years ago were asked to report the total amount of debt (in $)

they had when they graduated from college and the total value of their current investments (in $) resulting in the data set below.
Debt,Invested
16472,37226
19048,33930
4033,66292
22575,24887
12020,44976
4731,59924
4571,59901
18224,33154
1995,66794
21884,26700
18741,34926
14425,42960
22088,25054
24123,22511
2999,60091
11798,47106
8507,49350
2627,59963
21951,26687
3782,62539
a) Which statement best describes the relationship between these two variables?

As college debt decreases current investment decreases.

College debt is not associated with current investment.

As college debt increases current investment decreases.

As college debt increases current investment increases.

b) Develop a regression equation for predicting current investment based on college debt. What is the expected change in current investment for each additional dollar of college debt? Give your answer to four decimal places.

c) When testing for a significant linear relationship in your regression analysis, what is the proper conclusion at the 0.1 level of significance?

We fail to reject the claim of no linear relationship between college debt and current investment because the P-value is less than 0.1.

There is a significant linear relationship between college debt and current investment because the P-value is greater than 0.1.

We fail to reject the claim of no linear relationship between college debt and current investment because the P-value is greater than 0.1.

There is a significant linear relationship between college debt and current investment because the P-value is less than 0.1.

d) What is the predicted current investment for an individual who had a college debt of $5000? Give your answer to two decimal places.

e) What proportion of the variation in current investment is explained by college debt? Give your answer to four decimal places.

Mathematics

1 answer:

AnnZ [12.3K] · Answer 1 · 2026-04-07T16:58:54+03:00

Response:

a. As student debt rises, current investment diminishes.

b. Y= 68778.2406 - 1.9112X

For each dollar increase in college debt, the average current investments decrease by 1.9112 dollars.

c. A substantial linear correlation exists between college debt and current investment as the P-value falls below 0.1.

d. Y= $59222.2406

e. R²= 0.9818

Step-by-step breakdown:

Hello!

Data has been gathered on a random sample of 20 individuals who completed their college education five years ago. The variables under consideration are:

Y: Current investment by an individual who graduated from college five years prior.

X: Total debt of an individual upon graduating five years ago.

a)

To explore the relationship between debt and investment, creating a scatterplot with the sample data is ideal.

The scatterplot demonstrates a negative correlation, indicating that as these individuals' debt increases, their current investments decrease.

Therefore, the statement that accurately describes this is: As college debt rises, current investment decreases.

b)

The population regression equation is Y= α + βX +Ei

To develop this equation, estimates for alpha and beta are required:

a= Y[bar] -bX[bar]

a= 44248.55 - (-1.91)*12829.70

a= 68778.2406

b= $\frac{sumXY-\frac{(sumX)(sumY)}{n} }{sumX^2-\frac{(sumX)^2}{n} }$

b= $\frac{9014653088-\frac{(256594)(884971)}{20} }{4515520748-\frac{(256594)^2}{20} }$

b= -1.9112

∑X= 256594

∑X²= 4515520748

∑Y= 884971

∑Y²= 43710429303

∑XY= 9014653088

n= 20

Averages:

Y[bar]= ∑Y/n= 884971/20= 44248.55

X[bar]= ∑X/n= 256594/20= 12829.70

The estimated regression equation becomes:

Y= 68778.2406 - 1.9112X

For every dollar increase in college debt, the average current investments drop by 1.9112 dollars.

c)

To evaluate if there's a linear regression between these variables, the following null hypotheses are formulated:

H₀: β = 0

H₁: β ≠ 0

α: 0.01

Testing can be performed utilizing either a Student t-test or Snedecor's F (ANOVA)

Using t= b - β = -1.91 - 0 = -31.83

Sb 0.06

The critical area and P-value for this test is two-tailed. The P-value equals: 0.0001

Since this P-value is underneath the significance level, we reject the null hypothesis.

In the case of ANOVA, the rejection area is also one-tailed to the right, corresponding to the P-value.

The P-value remains: 0.0001

Using this method, we similarly reject the null hypothesis. $F= \frac{MSTr}{MSEr}= \frac{4472537017.96}{4400485.72} =1016.37$

In conclusion, at a significance level of 1%, there exists a linear relationship linking current investment to college debt.

The accurate statement is:

There exists a significant linear association between college debt and current investment since the P-value is less than 0.1.

d)

To forecast the value of Y when X is set, it is essential to substitute X in the estimated regression equation.

Y/$5000

Y= 68778.2406 - 1.9112*5000

Y= $59222.2406

The anticipated investment for someone with a college debt of $5000 is $59222.2406.

e)

To determine the proportion of variation in the dependent variable that the independent variable accounts for, the coefficient of determination R² must be calculated.

R²= 0.9818

$R^2= \frac{b^2[sumX^2-\frac{(sumX)^2}{n} ]}{sumY^2-\frac{(sumY)^2}{n} }$

$R^2= \frac{-1.9112^2[4515520748-\frac{(256594)^2}{20} ]}{43710429303-\frac{(884971)^2}{20} }$

This indicates that 98.18% of the variability in current investments relates to college graduation debt within the projected regression model: Y= 68778.2406 - 1.9112X

I trust this is beneficial!