Question

The following data were collected during a study of consumer buying patterns: V Observation Observation y 74 83 86 82 99 95 84 16 23 45 15 18 13 23 20 31 79 74 67 89 10 11 12 13 31 52 50 28 92 87 b. Obtain a linear regression line for the data. (Round your intermediate calculations and final answers to 3 decimal places.) 0.517 x 69.407 y= + c. What percentage of the variation is explained by the regression line? (Do not round intermediate calculations. Round your answer to the nearest whole percent. Omit the "%" sign in your response.) 59 % of the variation in the dependent variable is explained by the Approximately independent variable d. Use the equation determined in part b to predict the expected value of y for x 48. (Round your intermediate calculations and final answers to 3 decimal places.) 94.223 1234t 567

Answer 1

The regression equation to forecast the sales is of the form y = a+bx, where the value of a and b is calculate as per below formulas

ΣΣ-ΣΣεν] Σ ) (Στ) α n(Σrε)- (Σ a

ΠΣα-ΣΗΣΥ n(Σx2)-(Σ

where y is the dependent variable, x is the independent variable and n is the number of observations

Observation x^2 X y xy 16 74 256 1184 529 2 23 83 1909 3 45 86 2025 3870 961 4 31 82 2542 2704 5 52 99 5148 2500 50 95 4750 2352 7 28 84 784 8 15 79 225 1185 18 74 324 1332 871 10 13 67 169 2047 11 23 89 529 12 20 92 400 1840 2697 13 31 87 961 365 1091 12367 31727

The above image shows the values required for calculation of a and b. We can see that

$\sum x = 365, \sum y=1091, \sum x^{2} = 12367, \sum xy = 31727$

Using these values in the above equations, we get

$a = \frac{1091*12367-365*31727}{13*12367-365^{2}} = \frac{1912042}{27546} = 69.413$

$b = \frac{13*31727-365*1091}{13*12367-365^{2}} = \frac{14236}{27546} = 0.517$

Hence the regression equation is y= 69.413+0.517x

b) The formula for R² or percentage of variation explained by regression line is given by below equation

$R^2 = 1-\frac{SS_{RES}}{SS_{TOT}}$

where SS_RES is the sum of squares of residuals and given using below formula

$SS_{RES} = \sum (y_i-f_i)^2$

where y_i is the given value of y for the i^th observation and f_i is the predicted value of i^th observation as per the regression equation above

$SS_{TOT} = \sum (y_i-\bar y)^2$

where $\bar y$ is the mean of given observations for y

The below table tabulates the required values

(yi-fi) 2 (yi-mean y) (yi-mean y) 2 Observation f yi-fi y 77.685 1 74 -3.685 13.5792 -9.9231 98.4675 81.304 2 83 1.696 2.8764 -0.9231 0.8521 44.5957 86 92.678 -6.678 2.0769 4.3136 82 11.8336 3.6982 85.44 -3.44 -1.9231 96.297 227.3136 99 2.703 7.3062 15.0769 0.0692 11.0769 6 95 95.263 -0.263 122.6982 0.0769 7 84 83.889 0.111 0.0123 0.0059 77.168 3.3562 8 79 1.832 -4.9231 24.2367 -4.719 22.2690 74 78.719 -9.9231 98.4675 10 76.134 -16.9231 67 -9.134 83.4300 286.3905 5.0769 25.7751 11 89 81.304 7.696 59.2284 12 92 79.753 12.247 149.9890 8.0769 65.2367 13 87 85.44 1.56 2.4336 3.0769 9.4675 400.9788 966.9231

From above table we have SS_RES = 400.9788 and SS_TOT = 966.9231.

Thus R² = 1-(400.9788/966.9231) = 1-0.4147 = 0.5853 $\approx$ 59%

Hence approximately 59% of the variation in the dependent variable is explained by the independent variable.

d) Expected value of y for x=48 is as below

y=69.413+0.517*48 = 69.413+24.816 = 94.229

​​​​​​​Hence expected value of y is 94.229