Question

In the following data set, the columns indicate young adults’ smoking habit, while the rows indicate their exercise status. Please conduct a hypothesis to determine whether smoking habit and exercise status are associated. Choose α = 0.05. (Please make sure to check assumptions, if assumptions are not met, you may stop).

this q is for a biostatistical subject.

	Smoking Habit
Exercise Status	Frequent	Some	None	Total
Never	98	86	35	219
Occasion	29	47	23	99
Regular	17	9	17	43
Heavy	9	7	19	35
Total	153	149	94	396

Answer 1

To check whether there smoking habit and exercise status are associated, we will use two way ANOVA process. Since there are two factors of variation due to rows and columns, the hypothesis are as follows ;

NULL HYPOTHESIS:

H_0R : there is no significant difference between the exercise statuses.

^{H_0C : there is no significant difference between the smoking habits.}

^{ALTERNATIVE HYPOTHESIS :}

^{H_1R : at least two of the exercise statuses differs significantly.}

^{H_1C : at least two of the smoking habits differs significantly.}

		Smoking habit
Exercise status	frequent	some	none	Total	ti.^2
never	98	86	35	219	47961
Occasion	29	47	23	99	9801
Regular	17	9	17	43	1849
Heavy	9	7	19	35	1225
Total	153	149	94	grand total =396	total = 60836
t.j^2	23409	22201	8836	total= 54446

N = 4*3 = 12
Raw sum of squares = 22954
Correction Factor = G^2 / N = 13068
Total sum of Squares = RSS - CF = 9886
Row sum of Squares = (sum of ti.^2/ 3) - CF = 7210.667
Column Sum of Squares = (sum of t.j^2/ 4) - CF = 543.5
Error sum of squares = TSS - RSS - CSS = 2131.833

		ANOVA TABLE
Sources of Variation	S.S.(1)	d.f.(2)	M.S.S.    (3) = (1)/(2)	Variance Ratio (4)
Between Columns	543.5	3-1 =2	271.75	0.764834769
Between rows	7210.667	4-1= 3	2403.555667	6.76475784
Error	2131.83	2*3 = 6	355.3055
Total	9886

Given alpha level of significance is 0.05

now, tabulated F_0.05(2,6) is 5.14 and the calculated value is 0.765 ,which is much less than the tabulated value, it is not significant and we fail to reject H0R at 5% level of significance. Hence there is no significant difference between the smoking habits.

Again,the tabulated F_0.05 (3,6) is 4.76 and the calculated value is 6.765 ,which is greater than the tabulated value , so we will reject the null hypothesis at 5% level of significance, and conclude that there is significant difference between the exercise statuses.