Question

A study wants to look at the correlation between sugar consumption and the development of cavities. The table below shows the average daily intake of sugar (g) and the total number of cavities per patient over the one-year study period.

Daily Sugar Intake / Number of Cavities

(X)                                      (Y)         

30                                         2

40                                         3

150                                      3

90                                        0

75                                       1

25                                        1

110                                     4

4. What is the sample correlation coefficient given Σ(??−?̅)27?=1=12821.4, Σ(??−?̅)27?=1=12, and Σ(?−?̅)(?−?̅)=130?

a. 0.33 b. 0.70 c. 0.87 d. -0.45

5. What type of correlation does this represent?

a. Strong positive b. Strong inverse c. Weak positive d. Weak inverse

The investigator wants to construct a regression equation based on his current sample to be able to predict the number of cavities that a patient develops based only on their sugar intake given the standard deviation for the daily sugar intake is 43.25 and the standard deviation for the number of cavities is 1.41.

6. What is the slope of the line (i.e. what is b1)? a. 0.87 b. 0.01 c. 1.41   d. 0.50

7. What is the y-intercept (i.e. what is b0)? a. 1.26 b. 0.50 c. 0.01 d. 1.15

8. What is the predicted number of cavities for someone who consumes on average 45 grams of sugar a day?

Answer 1

4. X Values
∑ = 520
Mean = 74.286
∑(X - M_x)² = SS_x = 12821.429

Y Values
∑ = 14
Mean = 2
∑(Y - M_y)² = SS_y = 12

X and Y Combined
N = 7
∑(X - M_x)(Y - M_y) = 130

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 130 / √((12821.429)(12)) = 0.33

Hence answer is a. 0.33

5. As it is near to zero and positive hence answer is c. Weak positive

6.

So Sum of X = 520
Sum of Y = 14
Mean X = 74.2857
Mean Y = 2
Sum of squares (SS_X) = 12821.4286
Sum of products (SP) = 130

Regression Equation = ŷ = bX + a

b = SP/SS_X = 130/12821.43 = 0.01

Hence answer is b. 0.01

7. Y intercept is a = M_Y - bM_X = 2 - (0.01*74.29) = 1.26

Hence answer is a. 1.26

8. Now regression line is y=1.26+0.01x, for x=45, y=1.26+0.01*45=1.71=2