Question

1.

perform Levene’s test for equal variance. Note, this is a one‐way ANOVA testing for the equality of 16 variances (each combination of promotion/discount). 0.1 signigicance

2. perform 2-way anova with replication

To answer these questions, an experiment was designed using laundry detergent pods. For ten weeks, 160 subjects received information about the products. The factors under consideration were the number of promotions (1, 3, 5, or 7) that were described during this ten‐ week period and the percent that the product was discounted (10%, 20%, 30%, or 40%) off the average non‐promotional price. Ten individuals were randomly assigned to each of the sixteen combinations. The data reflecting what the sub‐ jects would expect to pay for the product (i.e., their reference price) at the end of the 10‐week period. 0.1 significance

Percent	N u m b e r   o f   P r o m o t i o n s
Discount	Number=1	Number=3	Number=5	Number=7
Discount=10	11.36	11.33	11.15	10.82
	11.76	11.39	11.44	11.17
	11.73	11.51	11.08	11.31
	11.68	11.49	11.35	11.17
	11.82	11.83	11.20	11.37
	11.95	11.59	11.67	10.87
	11.68	11.43	11.40	10.98
	11.43	11.73	11.41	10.95
	11.57	11.86	11.32	11.05
	11.85	11.28	11.16	10.71
Discount=20	10.83	11.46	11.16	10.71
	11.03	11.20	11.03	11.32
	11.16	11.46	11.12	10.61
	11.75	11.14	11.36	10.93
	11.26	11.61	11.36	11.00
	11.92	11.25	11.07	11.06
	11.74	11.27	11.23	11.16
	11.90	11.48	10.93	11.34
	11.57	10.96	11.31	10.78
	11.69	11.74	10.93	11.29
Discount=30	12.20	12.14	11.37	11.15
	11.85	12.06	11.61	11.71
	11.84	11.72	11.43	11.06
	11.74	11.99	11.37	11.41
	11.81	11.22	11.28	11.67
	11.79	11.68	11.67	11.01
	11.85	11.56	11.74	11.24
	11.92	11.94	11.02	11.33
	11.99	11.71	11.92	11.47
	12.50	11.82	11.70	11.49
Discount=40	12.45	12.16	11.57	11.30
	12.14	12.41	11.62	11.48
	12.04	11.94	12.01	11.65
	12.15	12.24	11.88	11.15
	11.95	11.92	11.00	11.52
	12.22	11.72	11.60	11.67
	12.26	11.96	11.78	11.65
	12.19	11.63	11.63	11.78
	12.36	11.95	11.66	11.13
	12.04	12.23	11.78	11.96

Answer 1

1)

The Levene’s Test for Equality of Variances for each combination of promotion and discount is performed in excel by following these steps,

Step 1: Write down the data values for each 16 group. The screenshot is shown below,

Step 2: Calculate the standard deviation for each group using the excel function =STEDEV(). Now, the variance is obtained by taking square of standard deviation.

Step 3: The null and alternative hypothesis are,

Step 4: The F-value is obtained using the formula,

and the corresponding P-value is obtained using the excel function =F.DIST(x,deg_freedom1,deg_freedom2,TRUE)

Step 5: The significance level is 0.01 such that the null hypothesis is rejected if,

P-value < 0.01

	Combination
	Group 1	Group 2	s1^2	s2^2	F=s1^2/s2^2	P-value
1	1	2	0.0341	0.0416	0.8200	0.3862	FALSE
2	1	3	0.0341	0.0310	1.1021	0.5564	FALSE
3	1	4	0.0341	0.0460	0.7428	0.3325	FALSE
4	1	5	0.0341	0.1487	0.2296	0.0196	FALSE
5	1	6	0.0341	0.0550	0.6201	0.2438	FALSE
6	1	7	0.0341	0.0265	1.2865	0.6432	FALSE
7	1	8	0.0341	0.0685	0.4981	0.1569	FALSE
8	1	9	0.0341	0.0543	0.6284	0.2499	FALSE
9	1	10	0.0341	0.0733	0.4657	0.1352	FALSE
10	1	11	0.0341	0.0701	0.4866	0.1491	FALSE
11	1	12	0.0341	0.0580	0.5889	0.2212	FALSE
12	1	13	0.0341	0.0231	1.4770	0.7147	FALSE
13	1	14	0.0341	0.0590	0.5785	0.2137	FALSE
14	1	15	0.0341	0.0721	0.4734	0.1402	FALSE
15	1	16	0.0341	0.0729	0.4685	0.1370	FALSE
16	2	3	0.0416	0.0310	1.3440	0.6666	FALSE
17	2	4	0.0416	0.0460	0.9058	0.4426	FALSE
18	2	5	0.0416	0.1487	0.2799	0.0358	FALSE
19	2	6	0.0416	0.0550	0.7562	0.3420	FALSE
20	2	7	0.0416	0.0265	1.5688	0.7436	FALSE
21	2	8	0.0416	0.0685	0.6074	0.2346	FALSE
22	2	9	0.0416	0.0543	0.7663	0.3491	FALSE
23	2	10	0.0416	0.0733	0.5679	0.2061	FALSE
24	2	11	0.0416	0.0701	0.5935	0.2245	FALSE
25	2	12	0.0416	0.0580	0.7182	0.3149	FALSE
26	2	13	0.0416	0.0231	1.8012	0.8031	FALSE
27	2	14	0.0416	0.0590	0.7055	0.3058	FALSE
28	2	15	0.0416	0.0721	0.5772	0.2128	FALSE
29	2	16	0.0416	0.0729	0.5714	0.2085	FALSE
30	3	4	0.0310	0.0460	0.6740	0.2830	FALSE
31	3	5	0.0310	0.1487	0.2083	0.0143	FALSE
32	3	6	0.0310	0.0550	0.5627	0.2023	FALSE
33	3	7	0.0310	0.0265	1.1673	0.5893	FALSE
34	3	8	0.0310	0.0685	0.4519	0.1262	FALSE
35	3	9	0.0310	0.0543	0.5702	0.2077	FALSE
36	3	10	0.0310	0.0733	0.4226	0.1078	FALSE
37	3	11	0.0310	0.0701	0.4416	0.1196	FALSE
38	3	12	0.0310	0.0580	0.5344	0.1822	FALSE
39	3	13	0.0310	0.0231	1.3402	0.6651	FALSE
40	3	14	0.0310	0.0590	0.5249	0.1755	FALSE
41	3	15	0.0310	0.0721	0.4295	0.1120	FALSE
42	3	16	0.0310	0.0729	0.4251	0.1093	FALSE
43	4	5	0.0460	0.1487	0.3091	0.0476	FALSE
44	4	6	0.0460	0.0550	0.8349	0.3962	FALSE
45	4	7	0.0460	0.0265	1.7320	0.7872	FALSE
46	4	8	0.0460	0.0685	0.6706	0.2805	FALSE
47	4	9	0.0460	0.0543	0.8460	0.4037	FALSE
48	4	10	0.0460	0.0733	0.6270	0.2489	FALSE
49	4	11	0.0460	0.0701	0.6552	0.2694	FALSE
50	4	12	0.0460	0.0580	0.7929	0.3676	FALSE
51	4	13	0.0460	0.0231	1.9885	0.8398	FALSE
52	4	14	0.0460	0.0590	0.7788	0.3579	FALSE
53	4	15	0.0460	0.0721	0.6373	0.2563	FALSE
54	4	16	0.0460	0.0729	0.6308	0.2516	FALSE
55	5	6	0.1487	0.0550	2.7013	0.9225	FALSE
56	5	7	0.1487	0.0265	5.6041	0.9914	FALSE
57	5	8	0.1487	0.0685	2.1697	0.8680	FALSE
58	5	9	0.1487	0.0543	2.7373	0.9252	FALSE
59	5	10	0.1487	0.0733	2.0288	0.8466	FALSE
60	5	11	0.1487	0.0701	2.1199	0.8608	FALSE
61	5	12	0.1487	0.0580	2.5655	0.9117	FALSE
62	5	13	0.1487	0.0231	6.4339	0.9947	FALSE
63	5	14	0.1487	0.0590	2.5201	0.9076	FALSE
64	5	15	0.1487	0.0721	2.0620	0.8520	FALSE
65	5	16	0.1487	0.0729	2.0410	0.8486	FALSE
66	6	7	0.0550	0.0265	2.0746	0.8540	FALSE
67	6	8	0.0550	0.0685	0.8032	0.3747	FALSE
68	6	9	0.0550	0.0543	1.0133	0.5077	FALSE
69	6	10	0.0550	0.0733	0.7510	0.3383	FALSE
70	6	11	0.0550	0.0701	0.7848	0.3620	FALSE
71	6	12	0.0550	0.0580	0.9497	0.4700	FALSE
72	6	13	0.0550	0.0231	2.3818	0.8939	FALSE
73	6	14	0.0550	0.0590	0.9329	0.4596	FALSE
74	6	15	0.0550	0.0721	0.7633	0.3470	FALSE
75	6	16	0.0550	0.0729	0.7556	0.3415	FALSE
76	7	8	0.0265	0.0685	0.3872	0.0868	FALSE
77	7	9	0.0265	0.0543	0.4885	0.1504	FALSE
78	7	10	0.0265	0.0733	0.3620	0.0731	FALSE
79	7	11	0.0265	0.0701	0.3783	0.0819	FALSE
80	7	12	0.0265	0.0580	0.4578	0.1300	FALSE
81	7	13	0.0265	0.0231	1.1481	0.5798	FALSE
82	7	14	0.0265	0.0590	0.4497	0.1248	FALSE
83	7	15	0.0265	0.0721	0.3679	0.0762	FALSE
84	7	16	0.0265	0.0729	0.3642	0.0742	FALSE
85	8	9	0.0685	0.0543	1.2616	0.6326	FALSE
86	8	10	0.0685	0.0733	0.9351	0.4610	FALSE
87	8	11	0.0685	0.0701	0.9770	0.4865	FALSE
88	8	12	0.0685	0.0580	1.1824	0.5965	FALSE
89	8	13	0.0685	0.0231	2.9654	0.9395	FALSE
90	8	14	0.0685	0.0590	1.1615	0.5864	FALSE
91	8	15	0.0685	0.0721	0.9504	0.4704	FALSE
92	8	16	0.0685	0.0729	0.9407	0.4645	FALSE
93	9	10	0.0543	0.0733	0.7411	0.3313	FALSE
94	9	11	0.0543	0.0701	0.7744	0.3548	FALSE
95	9	12	0.0543	0.0580	0.9372	0.4623	FALSE
96	9	13	0.0543	0.0231	2.3504	0.8905	FALSE
97	9	14	0.0543	0.0590	0.9206	0.4520	FALSE
98	9	15	0.0543	0.0721	0.7533	0.3399	FALSE
99	9	16	0.0543	0.0729	0.7456	0.3345	FALSE
100	10	11	0.0733	0.0701	1.0449	0.5256	FALSE
101	10	12	0.0733	0.0580	1.2646	0.6339	FALSE
102	10	13	0.0733	0.0231	3.1713	0.9497	FALSE
103	10	14	0.0733	0.0590	1.2422	0.6240	FALSE
104	10	15	0.0733	0.0721	1.0164	0.5095	FALSE
105	10	16	0.0733	0.0729	1.0060	0.5035	FALSE
106	11	12	0.0701	0.0580	1.2102	0.6096	FALSE
107	11	13	0.0701	0.0231	3.0350	0.9432	FALSE
108	11	14	0.0701	0.0590	1.1888	0.5995	FALSE
109	11	15	0.0701	0.0721	0.9727	0.4839	FALSE
110	11	16	0.0701	0.0729	0.9628	0.4779	FALSE
111	12	13	0.0580	0.0231	2.5079	0.9065	FALSE
112	12	14	0.0580	0.0590	0.9823	0.4896	FALSE
113	12	15	0.0580	0.0721	0.8037	0.3751	FALSE
114	12	16	0.0580	0.0729	0.7956	0.3694	FALSE
115	13	14	0.0231	0.0590	0.3917	0.0894	FALSE
116	13	15	0.0231	0.0721	0.3205	0.0527	FALSE
117	13	16	0.0231	0.0729	0.3172	0.0512	FALSE
118	14	15	0.0590	0.0721	0.8182	0.3850	FALSE
119	14	16	0.0590	0.0729	0.8099	0.3793	FALSE
120	15	16	0.0721	0.0729	0.9898	0.4940	FALSE

We can see that null hypothesis is failed to reject for each combination at 1% significant level. Hence we can conclude that all the variance are equal at 1% significant level.

2)

The two factor ANOVA test is performed in excel by following these steps,

Step 1: Write down the dat avalues in excel. The screenshot is shown below,

Step 2: DATA > Data Analysis > Anova: Two Factor With Replication. The screenshot is shown below,

Step 3: Select the Input Range, Row per sample = 10, Alpha = 0.01 > OK. The screenshot is shown below,

The result is obtained. The screenshot is shown below,

We can see that,

P-value for both the factor is less than 0.01 at 1% significant level. Hence there is significant effect of both the factor. But there is no interaction between two factor (the P-value for interaction is > 0.01)