Question
In: Statistics and Probability
The dataset below contains the average sales for five stores across four different seasons. Which season...
| 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 | ||||||
Solutions
Expert Solution
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
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