Question

The data show the systolic and diastolic blood pressure readings for 12 women.

 

      X=Systolic      123      116      82        126      116      138      109      119      155      106      178      84

      Y=Diastolic    66      71      62        84      70      91      67      66      80       70       89        72

 

      a….What is the Y-INTERCEPT of the Least-Squares regression equation?

      b…..Predict the diastolic reading for a woman whose systolic reading is 165.

      c…..What percent of the variation in diastolic pressure is due to factors other than one’s systolic pressure?

            A study from the U.S. Naval Academy in Annapolis, Maryland found that the one best single predictor for first-semester Grade Point Average was a student’s SAT-Math score.

            The correlation between these two variables was r = +0.64.

            The mean first semester GPA was 3.00 with a Standard Deviation of 0.48, and the mean SAT-Math score

for these freshmen was 670 with Standard Deviation of 80 points.

d……..Use the information above to predict a student’s first-term GPA if his SAT-Math score was 610.

Answer 1

1) Based on the given data,

Running a simple linear regression, the fitted regression equation can be expressed as:

where estimated intercept coefficient and estimated slope coefficient

Substituting the values,

= 0.263688

  

= 42.0937

Hence, fitted the regression line can be expressed as:

a. Hence, the y intercept of the least square regression line (when x = 0)

= 42.0937 + 0.263688 (0)

= 42.0937

b. For a woman whose systolic reading is 165, (when x = 165)

= 85.602

The predicted diastolic reading for a woman with systolic reading 165 is 85.602.

c.  To find the percent of the variation in diastolic pressure is due to factors other than one’s systolic pressure, we may use the measure 'Coefficient of determination' (r², ranging from 0 to 1) which is nothing but the square of the correlation between the two.

The correlation coefficient can be computed as:

  

= 0.742

r² = (0.742)² = 0.55

Here, the percent of the variation (explained variation) in diastolic pressure that is due to one’s systolic pressure is 55%.Now, the percent of unexplained variation i.e. variation due to factor other than this would simply be the complementary value

= 1 - 0.55

= 0.45

Hence, the percent of the variation in diastolic pressure due to factors other than one’s systolic pressure is 45%

Given:

Let  First-term GPA = Y (Response variable) SAT-Math score = X (Predictor)

To find the predicted Y value, we need to obtain the fitted regression line equation of the form:

where, Estimated slope can be computed using the formula:

= 0.00384

and the estimated intercept can be computed using the formula:

= 0.4272

Thus,

First-term GPA = 0.4272 + 0.00384 (SAT-Math score)

d. When SAT-Math score = 610

Predicted  First-term GPA = 0.4272 + 0.00384 (610)

= 2.7696

Hence, a student's first-term GPA when his SAT-Math score is 610 is 2.7696