Question

Do workers prefer to buy lunch rather than pack their own lunch? A survey of employed Americans found that 75% of the 18 to 24 year-olds, 77% of the 25 to 34 year-olds, 72% of the 35 to 44 year-olds, 58% of the 45 to 54 year-olds, 57% of the 55 to 64 year-olds, and 55% of the 65 + year-olds buy lunch throughout the workweek. Suppose the survey was based on 200 employed Americans in each of six age groups.

a. At the 0.05 level of significance, is there evidence of a difference among the age groups in the preference for buying lunch?

b. Determine the p-value in (a) and interpret its meaning.

Answer 1

a) Null hypothesis:Ho: preference for buying lunch is similar among different age groups

Alternate hypothesis:Ho: preference for buying lunch is different among different age groups

degree of freedom(df) =(rows-1)*(columns-1)=		5
for 5 df and 0.05 level of signifcance critical region       χ2=			11.070
Applying chi square test of independence:

Expected	Ei=row total*column total/grand total	18-24	25-34	35-44	45-54	55-64	65+	Total
	buy	131.3333	131.3333	131.3333	131.3333	131.3333	131.3333	788
	not buy	68.6667	68.6667	68.6667	68.6667	68.6667	68.6667	412
	total	200	200	200	200	200	200	1200
chi square    χ2	=(Oi-Ei)2/Ei	18-24	25-34	35-44	45-54	55-64	65+	Total
	buy	2.6531	3.9120	1.2217	1.7902	2.2876	3.4653	15.3299
	not buy	5.0744	7.4822	2.3366	3.4239	4.3754	6.6278	29.3204
	total	7.7276	11.3942	3.5582	5.2141	6.6631	10.0931	44.6503
test statistic X2 =		44.650

since test statistic falls in rejection region we reject null hypothesis

we have sufficient evidence to conclude that preference for buying lunch is different among different age groups

b)

for above test statistic and 5 df, p value <0.0001

this is the probability of having above or more extreme sample if preference for buying lunch is similar among different age groups