The following data relate the sales figures of the bar in Mark Kaltenbach’s smal

Question

The following data relate the sales figures of the bar in Mark Kaltenbach’s small bed-and-breakfast inn in Portand, to the number of guests registered that week, perform a linear regression that relates bar sales to guests (not to time) (Show your calculation):

A)What is the slope for regression equation?

B)What is the intercept for regression equation?

C)If the forecast is for 13 guests next week, what are the sales expected to be? (

D)What is the standard error of the estimate? How to interpret it?

E)What is the correlation coefficient for this regression? How to interpret it?

F)What is the coefficient of determination for this regression? How to interpret it?

Week Guests Sale 1 5 60 2 10 120 3 15 220 4 20 300

Explanation / Answer

Equation: S = a + b G + e

Where, a and b are the regression coefficients, and e is the residual of which expected or average value is zero.

You need to find sample standard deviation.

Slope is given by

b = Cov(S, G) / Var(G) = Corr(S, G)*STDEV(S) / STDEV(G) = 16.40

Intercept is given by

Substitute the average values in equation

E(S) = a + b*E(G)

a = E(S) - b*E(G) because E(e) = 0

a = -30

The equation is S = -30 + 16.40 G

When G = 13, S = -30 + 16.40*13 = 183.20

Degree of freedom: df = n - 2 = difference between number of observation and parameters needed to estimate

df = 4 - 2 = 2

Standard Error of Estimate: SEE = (280/2) = 140 = 11.83

As standard error of estimate is also equal to the standard deviation of the regression residuals, Smaller SEE results in to more accurate prediction.

Correlation coefficient = CORREL = 0.995861643

It is almost equal to 1, indication that relation highly positively correlated.

Coefficient of determination:

R-square = Explained variation / Total variation = 1 - (Unexplained variation)/Total Variation

R-square = 1 - 280/33900 = 0.991740413

It indicates how much variation in dependent variable (S) is explained by independent variable (G). Here, 99.17% are explained.

Week G S Predicted S (S - Predicted S)^2 (S - Mean S)^2 1 5 60 52 64 13225 2 10 120 134 196 3025 3 15 220 216 16 2025 4 20 300 298 4 15625 SUM = 50 700 700 280 33900 Mean = 12.5 175 STDEV = 6.454972 106.3015 CORREL = 0.995862 b = 16.40 a = -30.00

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