Question

Question No 3. Tests Using Contingency Tables:

A researcher selected sample of customs from 4 companies and asked them if the companies care give warranty on sold items or not. Assume observed values from your own (fill the below table by assuming any values of your choice) and test the claim that the proportion of customers of each company who got warranty is the same for each company by choosing alpha of your own choice.  (Marks 2.5)

	Company A	Company B	Company C	Company D
Warranty: Yes	34	86	56	73
Warranty: No	14	25	31	26

Answer 1

Step -1: State the hypothesis:

Null hypotheisi: H0: The proportion of customers of each company who got warranty is not the same for each company
Alternative hypothesis: H1: the proportion of customers of each company who got warranty is the same for each company


Step-2: Observed Frequencies:

	Company A	Company B	Company C	Company D	Total
Warranty: Yes	34	86	56	73	249
Warranty: No	14	25	31	26	96
Total	48	111	87	99	345

Step-3:Expected Frequencies:

Fe11 = ( 249 x 48 ) / 345 = 34.64

Fe12 = ( 249 x 111 ) / 345 = 80.11

Fe13 = ( 249 x 87 ) / 345 = 62.79

Fe14 = ( 249 x 99 ) / 345   = 71.45

Fe21 = ( 96 x 48 ) / 345     = 13.35

Fe22 =( 96 x 111 ) / 345   = 30.88

Fe23 =( 96 x 87 ) / 345      = 24.20

Fe24 = ( 96 x 99 ) / 345    = 27.54

	Company A	Company B	Company C	Company D	Total
Warranty: Yes	34.64	80.11	62.79	71.45	249
Warranty: No	13.35	30.88	24.20	27.54	96
Total	48	111	87	99	345

Step-4: Compute Chi-square:

χ2 = ∑ [ (Oi - Ei)² / Ei ]

= (34 – 34.64)² / 34.64 +    (86 – 80.11)² /80.11      +        (56 – 62.79)² / 62.79                                                                         (73 – 71.45)² /71.45      +    ( 14 – 13.35)² / 13.35       +       (25 – 30.88)² /30.88      +      (31 - 24.20)² /24.20    +    ( 26 – 27.54)² / 27.54

= 0.012 + 0.4326 + 0.7345 + 0.0335 + 0.031 + 1.122 + 1.9052 + 0.087

χ2 = 4.3578 --------------------------------( Calculated value)

Step-5: Compute the degrees of freedom (df):
df = (2 - 1)⋅(4 - 1)

df = 3
Consider level of significance = 5% = 0.05

χ2 = 7.815 --------------------------------( Tabulated value)

Step-6: Decision:

Calculated value < Tabulated value

4.3578   < 7.815

Step-7: Conclusion:

Null hypothesis is accepted.

H0: The proportion of customers of each company who got warranty is not the same for each company