Question

It was to be determined whether there was any relation between type of music listened to by 30 year old people and the geographic location of their residence. From the table below, you are requested to test 1% level of significance whether music preference is independent of geographic location.

Region Rock music Romantic music country music classical music

NORTH 150 52 8 20

south 147 49 56 9

west 163 29 6 11

Answer 1

The Chi-Square Test of Independence:

$H_0$ : Music preference is independent of geographic location.

$H_1$ : Music preference is dependent of geographic location.

$\Chi^2=\displaystyle\sum^R_{i=1}\displaystyle\sum^C_{j=1}\frac{(a_{ij}-e_{ij})^2}{e_{ij}}$

$e_{ij}=\frac{row(i)total\cdot col(j)total}{grandtotal}$

$grandtotal=700$

$e_{11}=\frac{230\cdot460}{700}=151.14$ , $e_{12}=\frac{230\cdot130}{700}=42.71$ , $e_{13}=\frac{230\cdot70}{700}=23.00$, $e_{14}=\frac{230\cdot40}{700}=13.14$

$e_{21}=\frac{261\cdot460}{700}=171.51$ , $e_{22}=\frac{261\cdot130}{700}=48.47$ , $e_{23}=\frac{261\cdot70}{700}=26.10$, $e_{24}=\frac{261\cdot40}{700}=14.91$

$e_{31}=\frac{209\cdot460}{700}=137.34$ , $e_{32}=\frac{209\cdot130}{700}=38.81$ , $e_{33}=\frac{209\cdot70}{700}=20.90$, $e_{34}=\frac{209\cdot40}{700}=11.94$

$\Chi^2=0.01+2.02+9.78+3.58+3.50+0.01+34.25+2.34+$

$+4.79+2.50+10.62+0.07=73.47$

Degrees of freedom:

$df=$ (number of columns – 1)(number of rows – 1)

$df=3\cdot2=6$

p-value$=P(\Chi^>73.47)=0$

$\alpha=0.01$ , $\alpha>$ p-value

Since α > p-value, reject H₀. This means that the factors are not independent.