Question

Question: Shopping on Black Friday. Refer to the International Journal of Retail and Distribution Management (Vol. 39, 2011) study of shopping on Black Friday (the day after Thanksgiving), Exercise 6.16 (p. 340). Recall that researchers conducted interviews with a sample of 38 women shopping on Black Friday to gauge their shopping habits. Two of the variables measured for each shopper were age (x) and number of years shopping on Black Friday (y). Data on these two variables for the 38 shoppers are listed in the accompanying table.

1. Fit the quadratic model, $\mathbf{\text{E(y)=}}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}\mathit{x}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}^{\mathbf{2}}$, to the data using statistical software. Give the prediction equation.
2. Conduct a test of the overall adequacy of the model. Use $\mathit{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{01}$.
3. Conduct a test to determine if the relationship between age (x) and number of years shopping on Black Friday (y) is best represented by a linear or quadratic function. Use $\mathit{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{01}$.

Short answer

Answer:

1. The prediction equation here becomes$\overbrace{\mathbf{y}}\mathbf{=}\mathbf{21}\mathbf{.}\mathbf{50519}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{877438}\mathit{x}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{027}{\mathit{x}}^{\mathbf{2}}$
2. At 99% confidence interval
3. At 99% level, ${\mathit{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$which means the model will be best represented by a linear function.

Step-by-step solution

Prediction equation

Age(y)	Years (x)	${\mathit{x}}^{\mathbf{2}}$
32	5	25
27	3	9
40	12	144
62	35	1225
47	20	400
53	30	900
24	8	64
27	2	4
47	24	576
40	25	625
45	11	121
22	11	121
25	5	25
60	35	1225
22	3	9
50	15	225
70	22	484
50	10	100
21	6	36
21	5	25
52	10	100
40	18	324
38	5	25
56	8	64
60	5	25
35	15	225
50	25	625
56	10	100
20	2	4
20	4	16
21	4	16
22	5	25
50	10	100
30	6	36
28	16	256
25	7	49
30	6	36
49	30	900

Using excel, the output generated is

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.668369
R Square	0.446718
Adjusted R Square	0.415102
Standard Error	11.1575
Observations	38
ANOVA
	df	SS	MS	F	Significance F
Regression	2	3517.937	1758.968	14.12942	3.17E-05
Residual	35	4357.142	124.4898
Total	37	7875.079
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	21.50519	5.010678	4.291872	0.000133	11.33297	31.67741	11.33297	31.67741
Years (x)	1.877438	0.775442	2.42112	0.020796	0.303207	3.451668	0.303207	3.451668
x2	-0.0257	0.021877	-1.17481	0.248002	-0.07011	0.018711	-0.07011	0.018711

The prediction equation here becomes ${\mathbf{\text{Therefore,β}}}_{\mathbf{1}}\mathbf{\text{≠}}{\mathit{\beta }}_{\mathbf{2}}\mathbf{\ne }\mathbf{0}$

Overall goodness of fit

${\mathit{H}}_{\mathbf{0}}\mathbf{}\mathbf{:}\mathbf{}{\mathit{\beta }}_{\mathbf{1}}\mathbf{=}{\mathit{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$

${\mathit{H}}_{\mathbf{a}}\mathbf{}\mathbf{}\mathbf{:}$At least one of the parameters ${\mathit{\beta }}_{\mathbf{1}}\mathbf{}or\mathbf{}{\mathit{\beta }}_{\mathbf{2}}$is non zero

Here, F test statistic localid="1649834135544" $\mathbf{=}\frac{\mathbf{SSE}}{\mathbf{n}\mathbf{-}\mathbf{k}\mathbf{+}\mathbf{1}}\mathbf{=}\mathbf{14}\mathbf{.}\mathbf{12942}$

${\mathit{H}}_{\mathbf{0}}$is rejected if p-value <$\mathit{\alpha }$. For $\mathit{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{01}$, since 0.0000317 < 0.01

Sufficient evidence to reject ${\mathit{H}}_{\mathbf{0}}$ at 99% confidence interval.

${\mathbf{\text{Therefore,β}}}_{\mathbf{1}}\mathbf{\text{≠}}{\mathit{\beta }}_{\mathbf{2}}\mathbf{\ne }\mathbf{0}$

Significance of β2

${\mathit{H}}_{\mathbf{0}}\mathbf{}\mathbf{:}{\mathit{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$, while,role="math" localid="1649834767365" ${\mathit{H}}_a\mathbf{}\mathbf{:}\mathbf{}{\mathit{\beta }}_{\mathbf{2}}\mathbf{\ne }\mathbf{0}$

The p-value of${\mathit{\beta }}_{\mathbf{3}}$is 0.248002 while$\mathit{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{01}$

${\mathit{H}}_{\mathbf{0}}$is rejected if P-value <$\mathit{\alpha }$. For ,$\mathit{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{01}$ since 0.248002 > 0.01

Not sufficient evidence to reject ${\mathit{H}}_{\mathbf{0}}$at 95% confidence interval.

Therefore, ${\mathit{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$which means the model will be best represented by a linear function.