In: Statistics and Probability
Dorothy Kelly sells life insurance for the Prudence Insurance Company. She sells insurance by making visits to her clients homes. Dorothy believes that the number of sales should depend, to some degree, on the number of visits made. For the past several years, she kept careful records of the number of visits (x) she made each week and the number of people (y) who bought insurance that week. For a random sample of 15 such weeks, the x and y values follow.
x | 11 | 17 | 17 | 14 | 28 | 5 | 20 | 14 | 22 | 7 | 15 | 29 | 8 | 25 | 16 |
y | 2 | 13 | 9 | 3 | 8 | 2 | 5 | 6 | 8 | 3 | 5 | 10 | 6 | 10 | 7 |
Σx = 248; Σy = 97; Σx2 = 4,844; Σy2 = 775; Σxy = 1,828
(a) Find x, y, b, and the equation of the least-squares line. (Round your answers for x and y to two decimal places. Round your least-squares estimates to three decimal places.)
x | = | |
y | = | |
b | = | |
ŷ | = | + x |
(b) Draw a scatter diagram for the data. Plot the least-squares
line on your scatter diagram.
(c) Find the sample correlation coefficient r and the
coefficient of determination. (Round your answers to three decimal
places.)
r = | |
r2 = |
What percentage of variation in y is explained by the
least-squares model? (Round your answer to one decimal
place.)
%
(d) In a week during which Dorothy makes 21 visits, how many people
do you predict will buy insurance from her? (Round your answer to
one decimal place.)
people
a.
Sum of X = 248
Sum of Y = 97
Mean X = 16.5333
Mean Y = 6.4667
Sum of squares (SSX) = 743.7333
Sum of products (SP) = 224.2667
Regression Equation = ŷ = bX + a
b = SP/SSX = 224.27/743.73 = 0.302
a = MY - bMX = 6.47 - (0.3*16.53) = 1.481
ŷ = 0.302X + 1.481
b.
c.
X Values
∑ = 248
Mean = 16.533
∑(X - Mx)2 = SSx = 743.733
Y Values
∑ = 97
Mean = 6.467
∑(Y - My)2 = SSy = 147.733
X and Y Combined
N = 15
∑(X - Mx)(Y - My) = 224.267
R Calculation
r = ∑((X - My)(Y - Mx)) / √((SSx)(SSy))
r = 224.267 / √((743.733)(147.733)) = 0.677
So r^2=0.677^2=0.458
Hence 45.8% of variation in y is explained by the least-squares model.
d. For x=21, ŷ = (0.302*21)+ 1.481=7.8