Are the events "happy-level" and "grade" independent?

a. Fill in the blanks. Write your answers in 2 decimals.

Here is the frequency table (f_ij)

	A	B	C	D	F	Total
Very happy	21	14	16	5	2	58
Happy	12	7	21	9	1	50
Sad	9	8	18	16	4	55
Very sad	15	12	12	8	5	52
Total	57	41	67	38	12	215

Here is the expected frequency table (e_ij)

	A	B	C	D	F
Very happy	15.38	11.06	18.07	10.25	3.24
Happy	13.26	9.53		8.84	2.79
Sad	14.58	10.49	17.14	9.72	3.07
Very sad	13.79	9.92	16.20	9.19	2.90

Here is the table for(fijeij)2eij

	A	B	C	D	F
Very happy	2.06	0.78	0.24	2.69	0.47
Happy	0.12	0.67		0.00	1.15
Sad	2.14	0.59	0.04	4.06	0.28
Very sad	0.11	0.44	1.09	0.15	1.52

b. The chi-square test statistic is 20.48, find the p -value.

Answer: p-value is between and . Write your answers in 2 decimals from small to large.

c. Can we say the events are independent?

Answer: . Write 1 for YES and 2 for NO.