Respuesta :
Answer:
[tex]\chi^2 =31.746[/tex]
[tex]p_v =P(\chi^2_{2}>31.746)=1.2x10^{-7}[/tex]
Since our calculated value it's higher than the critical value we have enough evidence to reject the null hypothesis of independence, and there would be a dependence between the gender and the type of automobile.
Step-by-step explanation:
Previous concepts
The Chi-Square test of independence is used "to determine if there is a significant relationship between two nominal (categorical) variables". And is defined with the following statistic:
[tex]\chi^2 =\sum_{i=1}^n \frac{(O-E)^2}{E}[/tex]
Where O represent the observed values and E the expected values.
State the null and alternative hypothesis
Null hypothesis: Styling engineering and fuel economy are independent
Alternative hypothesis: Styling engineering and fuel economy are dependent
The observed values are given by the table given:
STYLING ENGINEERING FUEL ECONOMY TOTAL
MALE 70 130 150 350
FEMALE 30 20 100 150
TOTAL 100 150 250 500
Calculate the expected values
In order to calculate the expected values we can use the following formula for each cell of the table:
[tex]E = \frac{row total* Column total}{Grand total}[/tex]
The following tables represent the xpected values calulated:
STYLING ENGINEERING FUEL ECONOMY TOTAL
MALE 70 105 175 350
FEMALE 30 45 75 150
TOTAL 100 150 250 500
Calculate the statistic
[tex]\chi^2 =\frac{(70-70)^2}{70}+\frac{(130-105)^2}{105}+\frac{(150-175)^2}{175}+\frac{(30-30)^2}{30}+\frac{(20-45)^2}{45}+\frac{(100-75)^2}{75}[/tex]
[tex]\chi^2 =31.746[/tex]
Calculate the critical value
First we need to calculate the degrees of freedom given by:
[tex] df= (rows-1)(columns-1)=(2-1)(3-1)= 2[/tex]
Since the confidence provided is 90% the significance would be [tex]\alpha=1-0.9=0.1[/tex] and we can find the critical value with the following excel code: "=CHISQ.INV(0.9,2)", and our critical value would be [tex]\chi^2_{crit}=4.650[/tex]
We can calculate also the p value:
[tex]p_v =P(\chi^2_{2}>31.746)=1.2x10^{-7}[/tex]
And we got the same decision reject the null hypothesis at 10% of significance.
Conclusion
Since our calculated value it's higher than the critical value we have enough evidence to reject the null hypothesis of independence, and there would be a dependence between the gender and the type of automobile.
