Question

The dataset below contains the average sales for five stores across four different seasons. Which season was the best of all for sales? Explain your answer.
											Store	Fall	Winter	Spring	Summer
1	$            64,866	$             51,382	$              67,480	$                67,282
2	$            55,848	$             70,963	$              68,719	$                64,741
3	$            74,985	$             71,475	$              50,981	$                64,963
4	$            71,215	$             64,265	$              65,634	$                59,487
5	$            67,414	$             74,796	$              57,839	$                74,481

Answer 1

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ₁ = μ₂ = μ₃ = μ₄

Ha: Not all means are equal

The above hypotheses will be tested using an F-ratio for a One-Way ANOVA.

ANOVA

	Group 1	Group 2	Group 3	Group 4
	64866	51382	67480	67282
	55848	70963	68719	64741
	74985	71475	50981	64963
	71215	64265	65634	59487
	67414	74796	57839	74481
Sum =	334328	332881	310653	330954
Average =	66865.6	66576.2	62130.6	66190.8
=∑​Xij2​=	22565570906	22508964759	19528085599	22024578504
St. Dev. =	7254.802	9314.139	7533.729	5442.161
SS =	210528589.2	347012726.8	227028317.2	118468480.8
n =	5	5	5	5

The total sample size is N = 20

The total degrees of freedom are: dftotal​=20−1=19

Also, the between-groups degrees of freedom are dfbetween​=4−1=3, and the within-groups degrees of freedom are:

dfwithin​=dftotal​−dfbetween​=19−3=16

total sum of values and the grand mean.

mean sum of squares

the F-statistic is computed as follows:

Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df1​=3 and df2​=3, therefore, the rejection region for this F-test is R={F:F>Fc​=3.239}.

Decision about the null hypothesis

Since it is observed that F=0.438 ≤ Fc​=3.239, it is then concluded that the null hypothesis is not rejected.

Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that not all 4 population means are equal, at the α=0.05 significance level. Hence we conclude that there is no significant difference between the sales in different seasons