The statistics below summarize 806 Stat 100 students\' survey responses to the 3

Question

The statistics below summarize 806 Stat 100 students' survey responses to the 3 questions asked on Survey 2x (from Fall 2010):
1. How many drinks per week do you typically consume?
2. How many people have you slept with in your life? and
3. How many classes have you skipped so far this semester?

Here's the summary statistics:

Here's the correlation matrix describing r between the 3 variables:

a. Why is the diagonal of the correlation matrix equal to 1?
Because the average of the residuals=0
Because R=1
Because the correlation of X and Y is the same as the correlation of Y and X.
Because the correlation of a variable with itself =1.
All of the above.
None of the above.

b. Why are the 3 correlations in the upper right the same as the 3 correlations in the lower left?
Because the average of the residuals=0
Because R=1
Because the correlation of X and Y is the same as the correlation of Y and X.
Because the correlation of a variable with itself =1.
All of the above.
None of the above.

c. What is the slope for predicting skipped classes from # of sex partners? (Round to 2 decimal places.) (Hint: Find the correct r and SDy and SDx from tables above.)

d. The multiple regression equation predicting skipped classes from both # of drinks per week and # sexual partners is:
Skipped Classes = 0.1044 (Drinks) + 0.1317 (# Sex Partners) + 3.084

The slope is only 0.1317 which is less than the slope for # of Sex Partners you got in part (c) above. Why?
(Assume all slopes are positive.)
The slopes in the multiple regression equation are always less than the slopes in the simple regression equation when all the variables are negatively correlated.
The slopes in the multiple regression equation are always less than the slopes in the simple regression equation when all the variables are positively correlated.
The slopes in the multiple regression equation are always less than the slopes in the simple regression equation when all the predictor variables are uncorrelated.
The slopes in the multiple regression equation are always less in the simple regression equations.

e. Derive the slope for Drinks in the simple regression from the slope for Drinks in the multiple regression. In other words show how you get from:
# Skipped Classes= 0.1044 (Drinks) + 0.1317 (# Sex Partners) + Intercept # Skipped Classes=slope × (Drinks)+ Intercept

Do it in 2 steps:

i) First find the slope for predicting sex partners from drinks: Sex partners =  (Drinks) + Intercept.

ii) Then plug that slope in for Sex Partners in the multiple regression equation to find the slope for predicting skipped classes from drinks:

# Skipped Classes =  (Drinks) + Intercept

Part II

The multiple regression equation predicting skipped classes from both # of drinks per week and # sexual partners is:
Predicted # of Skipped Classes SC^SC^ = 0.1044 (Drinks/week) + 0.1317 (# Sex Partners) + 3.084

f. The above equation describes the best fitting ______ through all the points so as to minimize the squared errors in the _______.

Select one for the first blank:
line
plane
ellipsoid
cube

Select one for the second blank:
Number of Skipped Classes
Number of Sex Partners
Number of Drinks

g. What does the Drinks slope 0.1044 mean in the multiple regression equation above?
It means that if you compare 2 students who differ by 1 drink per week, your best prediction for how they'd differ on their skipped classes is 0.1044.
It means that if you compare 2 students who are the same on all relevant characteristics (sex partners, recreational drug use, party hours, fraternity/sorority membership, etc.) who differ by 1 drink per week then, your best prediction for how they'd differ on their number of skipped classes is 0.1444.
It means that if you compare 2 students who differ by 0.1044 on their drinks per week, your best prediction for how they'd differ on their skipped classes is 1.
It means that if you compare 2 students with the same number of sex partners who differ by 1 drink per week, your best prediction for how they differ on their number of skipped classes is 0.1044.

h. The multiple regression equation for all students with Sex Partners = 0 simplifies to

# of Skipped Classes SC^SC^ =  +  × drinks/week

i. The multiple regression equation for all students with Sex Partners = 10 simplifies to (round to 2 decimal places)

SC^SC^ =  +  × drinks/week

j. Suppose the slopes in the 2 simple regression equations: Drinks/week predicting Skipped Classes, and Sex Partners predicting Skipped Classes are the same as they are in the multiple regression equation you can conclude that
All 3 variables are positively correlated
All 3 variables are uncorrelated
Drinks/week and Sex Partners are uncorrelated
None of the above

Average Median SD Min Max n drinks_per_wk 10.57 7 11.47 0 50 806 sex_partners_num 3.107 1 5.843 0 50 806 skipped_class_num 4.597 4 4.152 0 15 806

Explanation / Answer

a. Why is the diagonal of the correlation matrix equal to 1?

Because the correlation of a variable with itself =1.

b. Why are the 3 correlations in the upper right the same as the 3 correlations in the lower left?

Because the correlation of X and Y is the same as the correlation of Y and X.

c. What is the slope for predicting skipped classes from # of sex partners? (Round to 2 decimal places.) (Hint: Find the correct r and SDy and SDx from tables above.)

0.2668*(4.152 /5.843)=0.19

d. The multiple regression equation predicting skipped classes from both # of drinks per week and # sexual partners is:
Skipped Classes = 0.1044 (Drinks) + 0.1317 (# Sex Partners) + 3.084

The slopes in the multiple regression equation are always less than the slopes in the simple regression equation when all the predictor variables are uncorrelated.

e. Derive the slope for Drinks in the simple regression from the slope for Drinks in the multiple regression.

i) First find the slope for predicting sex partners from drinks: Sex partners =  (Drinks) + Intercept.

slope = 0.2822*(5.843/11.47) =  0.1438

ii) Then plug that slope in for Sex Partners in the multiple regression equation to find the slope for predicting skipped classes from drinks:

# Skipped Classes =  0.1044 (Drinks) + 0.1317 *( 0.1438Drinks) + Intercept=0.1233Drinks+Intercept

f. The above equation describes the best fitting ______ through all the points so as to minimize the squared errors in the _______.

Select one for the first blank: plane

Select one for the second blank:

Number of Skipped Classes

g. What does the Drinks slope 0.1044 mean in the multiple regression equation above?

It means that if you compare 2 students who are the same on all relevant characteristics (sex partners, recreational drug use, party hours, fraternity/sorority membership, etc.) who differ by 1 drink per week then, your best prediction for how they'd differ on their number of skipped classes is 0.1444.

Skipped Classes SC^SC^ = 3.084 +  0.1044× drinks/week

. The multiple regression equation for all students with Sex Partners = 10 simplifies to (round to 2 decimal places)

SC = 4.40 + 0.10 × drinks/week

j. Suppose the slopes in the 2 simple regression equations: Drinks/week predicting Skipped Classes, and Sex Partners predicting Skipped Classes are the same as they are in the multiple regression equation you can conclude that

Drinks/week and Sex Partners are uncorrelated

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