Question

Question: Orange juice demand study. A chilled orange juice warehousing operation in New York City was experiencing too many out-of-stock situations with its 96-ounce containers. To better understand current and future demand for this product, the company examined the last 40 days of sales, which are shown in the table below. One of the company’s objectives is to model demand, y, as a function of sale day, x (where x = 1, 2, 3, c, 40).

1. Construct a scatterplot for these data.
2. Does it appear that a second-order model might better explain the variation in demand than a first-order model? Explain.
3. Fit a first-order model to these data.
4. Fit a second-order model to these data.
5. Compare the results in parts c and d and decide which model better explains variation in demand. Justify your choice.

Short answer

Answers:

1. Scatterplot
2. Looking at the data, a second-order model equation might be a better fit for the data as it can be seen that there is an upward curvature kind of relationship between x and y.
3. The first-order model equation can be written as$\hat{\mathbf{y}}\mathbf{=}\mathbf{2802}\mathbf{.}\mathbf{362}\mathbf{+}\mathbf{122}\mathbf{.}\mathbf{9507}\mathit{x}$
4. The second-order model equation for y on x is$\hat{\mathbf{y}}\mathbf{=}\mathbf{4944}\mathbf{.}\mathbf{221}\mathbf{-}\mathbf{189}\mathbf{.}\mathbf{029}\mathit{x}\mathbf{+}\mathbf{7}\mathbf{.}\mathbf{462924}{\mathit{x}}^{\mathbf{2}}$
5. The ${\mathit{R}}^{\mathbf{2}}$value for the first-order model equation is around 35% while for the second-order model equation the value of was around 50%. value gives an estimate about if the model is a better fit for the data. A higher value for a model denotes that the model is a better fit for the data. Since the value of is higher for the second-order equation, the second-order model equation is a better fit for the data.

Step-by-step solution

Scatterplot

Demand for containers, y	Sale day, x	$X^2$
4581	1	1
4239	2	4
2754	3	9
4501	4	16
4016	5	25
4680	6	36
4950	7	49
3303	8	64
2367	9	81
3055	10	100
4248	11	121
5067	12	144
5201	13	169
5133	14	196
4211	15	225
3195	16	256
5760	17	289
5661	18	324
6102	19	361
6099	20	400
5902	21	441
2295	22	484
2682	23	529
5787	24	576
3339	25	625
3798	26	676
2007	27	729
6282	28	784
3267	29	841
4779	30	900
9000	31	961
9531	32	1024
3915	33	1089
8964	34	1156
6984	35	1225
6660	36	1296
6921	37	1369
10005	38	1444
10153	39	1521
11520	40	1600

Model fit for the data

Looking at the data, a second-order model equation might be a better fit for the data as it can be seen that there is an upward curvature kind of relationship between x and y.

First-order model equation

Excel summary output

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.613045
R Square	0.375824
Adjusted R Square	0.359398
Standard Error	1876.566
Observations	40
ANOVA
	df	SS	MS	F	Significance F
Regression	1	80572885	80572885	22.88027	2.61E-05
Residual	38	1.34E+08	3521501
Total	39	2.14E+08
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	2802.362	604.7266	4.634096	4.13E-05	1578.156	4026.567	1578.156	4026.567
Sale day, x	122.9507	25.70397	4.783332	2.61E-05	70.91568	174.9856	70.91568	174.9856

The first-order model equation can be written as$\hat{\mathbf{y}}\mathbf{=}\mathbf{2802}\mathbf{.}\mathbf{362}\mathbf{+}\mathbf{122}\mathbf{.}\mathbf{9507}\mathit{x}$

Second-order model equation

Excel summary output

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.723292
R Square	0.523151
Adjusted R Square	0.497376
Standard Error	1662.232
Observations	40
ANOVA
	df	SS	MS	F	Significance F
Regression	2	1.12E+08	56079162	20.29636	1.12E-06
Residual	37	1.02E+08	2763016
Total	39	2.14E+08
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	4944.221	829.6005	5.959761	7.12E-07	3263.291	6625.151	3263.291	6625.151
Sale day, x	-183.029	93.31859	-1.96134	0.057398	-372.111	6.052169	-372.111	6.052169
	7.462924	2.207279	3.381051	0.001716	2.990552	11.9353	2.990552	11.9353

The second-order model equation for y on x is$\hat{\mathbf{y}}\mathbf{=}\mathbf{4944}\mathbf{.}\mathbf{221}\mathbf{-}\mathbf{189}\mathbf{.}\mathbf{029}\mathit{x}\mathbf{+}\mathbf{7}\mathbf{.}\mathbf{462924}{\mathit{x}}^{\mathbf{2}}$

Comparison between the models

The ${\mathit{R}}^{\mathbf{2}}$value for the first-order model equation is around 35% while for the second-order model equation the value of ${\mathit{R}}^{\mathbf{2}}$was around 50%. ${\mathit{R}}^{\mathbf{2}}$value gives an estimate about if the model is a better fit for the data. The higher value${\mathit{R}}^{\mathbf{2}}$ for a model denotes that the model is a better fit for the data. Since the value of is higher for the second-order equation, the second-order model equation is a better fit for the data.