Question

It has been suggusted that the highest priority of retirees is travel. Thus, a study was conducted to investigate the differences in the length of stay of a trip for pre- and post-retirees. A sample of 686686 travelers were asked how long they stayed on a typical trip. The observed results of the study are found below. You may round all answers for this problem to the nearest hundredth.

Number of Nights	Pre-retirement	Post-retirement	Total
4−7	245	163	408
8−13	76	64	140
14−21	34	52	86
22 or more	17	35	52
Total	372	314	686

With this information, construct a table of estimated expected values. Use two digits after the decimal.

Number of Nights	Pre-retirement	Post-retirement
4−7	equation editor Equation Editor	equation editor Equation Editor
8−13	equation editor Equation Editor	equation editor Equation Editor
14−21	equation editor Equation Editor	equation editor Equation Editor
22 or more	equation editor Equation Editor	equation editor Equation Editor

Now, with that information, determine whether the length of stay is independent of retirement using α=0.01

(a) χ2=

Use as many digits after the decimal as possible.

(b) Find the degrees of freedom:

(c) Find the critical value:

Again, use as many digits after the decimal.

(d) The final conclusion is

A. We can reject the null hypothesis that the length of stay is independent of retirement and accept the alternative hypothesis that the two are dependent.
B. There is not sufficient evidence to reject the null hypothesis that the length of stay is independent of retirement.

Answer 1

resince expected value of a cell

Ei=row total*column total/grand total

for (4-7 and Pre cell) , expected value =372*408/686 =221.25

therefore below is expected value table:

Expected	Ei=row total*column total/grand total	Pre	Post
	4-7	221.25	186.75
	8-13	75.92	64.08
	14-21	46.64	39.36
	22 or more	28.20	23.80

a)

Applying chi square test of independence:

chi square    χ2	=(Oi-Ei)2/Ei	Pre	Post	Total
	4-7	2.550	3.021	5.5709
	8-13	0.000	0.000	0.0002
	14-21	3.424	4.056	7.4794
	22 or more	4.447	5.269	9.7157
	total	10.4206	12.3455	22.766
test statistic X2 =		22.77

b)

degree of freedom(df) =(rows-1)*(columns-1)=

3

c)

for 3 df and 0.01 level , critical value       χ2=

11.34 (from excel function: chiinv(0.01,3)

d)

since test statistic is greater than critical value

A. We can reject the null hypothesis that the length of stay is independent of retirement and accept the alternative hypothesis that the two are dependent.