The management of a supermarket wants to find if there is a relationship between
ID: 3321146 • Letter: T
Question
The management of a supermarket wants to find if there is a relationship between the number of times a specific product is pro- moted on the intercom system in the store and the number of units of that product sold. To experiment, the management selected a product and promoted it on the intercom system for 7 days. The following table gives the number of times this product was promoted each day and the number of units sold.
Number of Promotions per Day. Number of Units Sold per Day (hundreds)
15 11
22 22
42 30
30 26
18 17
12 15
38 23
a. With the number of promotions as an independent variable and the number of units sold as a dependent variable, what do you expect the sign of B in the regression line y = A + Bx + will be?
b. Find the least squares regression line yˆ = a + bx. Is the sign of b the same as you hypothesized for B in part a?
c. Give a brief interpretation of the values of a and b calculated in part b.
d. Compute r and r2 and explain what they mean.
e. Predict the number of units of this product sold on a day with
35 promotions.
f. Compute the standard deviation of errors.
g. Construct a 98% confidence interval for B.
h. Testing at a 1% significance level, can you conclude that B is
positive?
Explanation / Answer
a.> cor(Promotions,Units.sold)
[1] 0.8861456
Since the correlation is positive, we expect the coefficient of Promotions (B) in the regression equation to be positive.
b.
> summary(lm(Units.sold~Promotions))
Call:
lm(formula = Units.sold ~ Promotions)
Residuals:
1 2 3 4 5 6 7
-4.38863 3.08419 1.00653 3.05312 0.09972 1.12301 -3.97794
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.8304 3.2380 2.418 0.06024 .
Promotions 0.5039 0.1178 4.276 0.00789 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 3.353 on 5 degrees of freedom
Multiple R-squared: 0.7853, Adjusted R-squared: 0.7423
F-statistic: 18.28 on 1 and 5 DF, p-value: 0.007894
Yes, the coefficient of Promotions (B) = 0.5039 which is positive, same as expected in a.
c. Here, a = 7.8304 indicates the number of units sold when there is no promotions at all, i.e zero number of promotions per day.
B = 0.5039 indicates the increase in the number of units sold when there is an increase of one promotion per day.
d. r = 0.886145 means that the two variables are highly linearly related.
R-squared: 0.7853 indicates that 78.53% of the total variation is explained by the linear regression model.
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