Question

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.

Answer 1

Answer:

a. As college debt increases current investment decreases.

b. Y= 68778.2406 - 1.9112X

Every time the college debt increases one dollar, the estimated mean of the current investments decreases 1.9112 dollars.

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

d. Y= $59222.2406

e. R²= 0.9818

Step-by-step explanation:

Hello!

You have the information on a random sample of 20 individuals who graduated from college five years ago. The variables of interest are:

Y: Current investment of an individual that graduated from college 5 years ago.

X: Total debt of an individual when he graduated from college 5 years ago.

a)

To see the relationship between the information about the debt and the investment is it best to make a scatterplot with the sample information.

As you can see in the scatterplot (attachment) there is a negative relationship between the current investment and the debt after college, this means that the greater the debt these individuals had, the less they are currently investing.

The statement that best describes it is: As college debt increases current investment decreases.

b)

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

To develope the regression equation you have to estimate alpha and beta:

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

Means:

Y[bar]= ∑Y/n= 884971/20= 44248.55

X[bar]= ∑X/n= 256594/20= 12829.70

The estimated regression equation is:

Y= 68778.2406 - 1.9112X

Every time the college debt increases one dollar, the estimated mean of the current investments decreases 1.9112 dollars.

c)

The hypotheses to test if there is a linear regression between the two variables are two tailed:

H₀: β = 0

H₁: β ≠ 0

α: 0.01

To make this test you can use either a Student t or the Snedecor's F (ANOVA)

Using t=  b - β  =  -1.91 - 0  = -31.83

                 Sb         0.06

The critical region and the p-value for this test are two tailed.

The p-value is: 0.0001

The p-value is less than the level of signification, the decision is to reject the null hypothesis.

Using the

$F= \frac{MSTr}{MSEr}= \frac{4472537017.96}{4400485.72} =1016.37$

The rejection region using the ANOVA is one-tailed to the right, and so is the p-value.

The p-value is: 0.0001

Using this approach, the decision is also to reject the null hypothesis.

The conclusion is that at a 1% significance level, there is a linear regression between the current investment and the college debt.

The correct statement is:

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

d)

To predict what value will take Y to a given value of X you have to replace it in the estimated regression equation.

Y/X=$5000

Y= 68778.2406 - 1.9112*5000

Y= $59222.2406

The current investment of an individual that had a $5000 college debt is $59222.2406.

e)

To estimate the proportion of variation of the dependent variable that is explained/ given by the independent variable you have to calculate the coefficient of determination R².

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

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

R²= 0.9818

This means that 98.18% of the variability of the current investments are explained by the college debt at graduation under the estimated regression model: Y= 68778.2406 - 1.9112X

I hope it helps!