Answer:
Step-by-step explanation:
a)
[tex]H_0[/tex]: Rows are columns of given table are independent
[tex]H_1[/tex]: Rows and columns of given table are Not independent
(0) Male Female Total
silver 473 296 769
Black 549 309 858
Red 496 373 869
Total 1518 976 2496
We calculate expected frequencies by [tex]E=\frac{(Row\, Total)(Column\, Total)}{Grand\, Total}[/tex]
(E) col-1 col-2
Row-1 [tex]\frac{769+1518}{2496}=467.69[/tex] [tex]\frac{769+978}{2496}=301.31[/tex]
Row-2 [tex]\frac{858+1518}{2496}=521.81[/tex] [tex]\frac{858+978}{2496}=336.19[/tex]
Row-3 [tex]\frac{869+1518}{2496}=528.50[/tex] [tex]\frac{869+978}{2496}=340.50[/tex]
To calculate chi-square test statistic we calculate this formula on each cell:
[tex]\frac{(observed\, count-expected\, count)^2}{expected\, count}[/tex]
observed count(o) expected count(E) [tex]\frac{(O-E)^2}{E}[/tex]
473 467.685 0.0604
296 301.315 0.09375
549 521.812 1.416524
309 336.188 2.198654
496 528.502 1.998867
373 340.498 3.102537
[tex]X^2=\sum \frac{(O-E)^2}{E}=8.870733[/tex]
b)
Critical value [tex]X^2_{0.01}=9.210[/tex]
[tex]X^2_{0.01}[/tex] has an area of 0.01 to its right. To find [tex]X^2_{0.01}[/tex], open any Chi-square table, go across from row [tex]df=v=2[/tex] and down from colum 0.01 and get 9.210
c)
Test statistic X^2=8.871
Significance level = 0.01
Degrees of freedom, [tex]df=(r-1)(c-1)=(3-1)(2-1)=2[/tex]
d1)
To get p-value, we use excel function
CHISQ.DIST.RT(8.870733 , 2) = 0.011850723=0.0119
Since test statistic is not greater than critical value, we fail to reject null hypothesis. At 1% level, there is NOT sufficient evidence that rows and columns are not independent, i.e, it is plausible that rows and columns are independent.
Do not reject [tex]H_0[/tex]: color preference is not dependent on gender
d2)
No
