In: Statistics and Probability
Reading Nook Bookstore has 750 retail outlets across the country. The sales director wanted to see if Christmas music affects book sales in December. She randomly assigned some of the outlets to pipe in music and others not to. Then sales records for the month of December were kept. The results are shown in below. is there convincing evidence that there is an association between sales and christmas music?
<10,000 10-20,0000 >20,000
with music 5 18 7
without music 10 7 3
Type of test
Ho
Ha
Requirements
test statistic
Pvalue
Reject/fail to reject
Solution:
Here, we have to use chi square test for independence of two categorical variables.
Null hypothesis: H0: there is no association between sales and Christmas music.
Alternative hypothesis: Ha: there is an association between sales and Christmas music.
We assume/given level of significance = α = 0.05
Test statistic formula is given as below:
Chi square = ∑[(O – E)^2/E]
Where, O is observed frequencies and E is expected frequencies.
E = row total * column total / Grand total
We are given
Number of rows = r = 2
Number of columns = c = 3
Degrees of freedom = df = (r – 1)*(c – 1) = 1*2 = 2
α = 0.05
Critical value = 5.991465
(by using Chi square table or excel)
Calculation tables for test statistic are given as below:
Observed Frequencies |
||||
Column variable |
||||
Row variable |
<10 |
10 to 20 |
>20 |
Total |
With music |
5 |
18 |
7 |
30 |
Without music |
10 |
7 |
3 |
20 |
Total |
15 |
25 |
10 |
50 |
Expected Frequencies |
||||
Column variable |
||||
Row variable |
<10 |
10 to 20 |
>20 |
Total |
With music |
9 |
15 |
6 |
30 |
Without music |
6 |
10 |
4 |
20 |
Total |
15 |
25 |
10 |
50 |
Calculations |
||
(O - E) |
||
-4 |
3 |
1 |
4 |
-3 |
-1 |
(O - E)^2/E |
||
1.777778 |
0.6 |
0.166667 |
2.666667 |
0.9 |
0.25 |
Test Statistic = Chi square = ∑[(O – E)^2/E] = 6.361111
χ2 statistic = 6.361111
P-value = 0.041563
(By using Chi square table or excel)
P-value < α = 0.05
So, we reject the null hypothesis
There is sufficient evidence to conclude that there is an association between sales and Christmas music.