In: Statistics and Probability
Consider the following sample data for the relationship between advertising budget and sales for Product A:
Observation | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
Advertising ($) | 60,000 | 70,000 | 70,000 | 80,000 | 80,000 | 90,000 | 100,000 | 100,000 | 110,000 | 110,000 |
Sales ($) | 363,000 | 432,000 | 417,000 | 502,000 | 483,000 | 537,000 | 614,000 | 625,000 | 653,000 | 666,000 |
What is the slope of the "least-squares" best-fit regression line?
X | Y | XY | X² | Y² |
60000 | 363000 | 21780000000 | 3600000000 | 1.31769E+11 |
70000 | 432000 | 30240000000 | 4900000000 | 1.86624E+11 |
70000 | 417000 | 29190000000 | 4900000000 | 1.73889E+11 |
80000 | 502000 | 40160000000 | 6400000000 | 2.52004E+11 |
80000 | 483000 | 38640000000 | 6400000000 | 2.33289E+11 |
90000 | 537000 | 48330000000 | 8100000000 | 2.88369E+11 |
100000 | 614000 | 61400000000 | 10000000000 | 3.76996E+11 |
100000 | 625000 | 62500000000 | 10000000000 | 3.90625E+11 |
110000 | 653000 | 71830000000 | 12100000000 | 4.26409E+11 |
110000 | 666000 | 73260000000 | 12100000000 | 4.43556E+11 |
Ʃx = | Ʃy = | Ʃxy = | Ʃx² = | Ʃy² = |
870000 | 5292000 | 4.7733E+11 | 78500000000 | 2.90353E+12 |
Sample size, n = | 10 |
x̅ = Ʃx/n = | 87000 |
y̅ = Ʃy/n = | 529200 |
SSxx = Ʃx² - (Ʃx)²/n = | 2810000000 |
SSyy = Ʃy² - (Ʃy)²/n = | 1.03004E+11 |
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = | 16926000000 |
Slope, b = SSxy/SSxx = 6.0234875
y-intercept, a = y̅ -b* x̅ = 5156.58363
Regression equation :
ŷ = 5156.5836 + (6.0235) x