Question

In 2009 U.S. charitable giving fell 3.6 percent to 303.75 billion for the year. Total charitable contributions from American individuals, corporations, and foundations fell to $303.75 billion from $315.08 billion for 2008. The largest share of contributions went to religious organizations, representing 33 percent of total giving. The next largest shares went to educational organizations, receiving an estimated 13 percent of the total, and foundations, which received 10 percent of the total. Suppose a sample of 6 employees is randomly chosen from a large corporation and their charitable contributions in 2008 and 2009 are determined. The following table gives these amounts (in dollars). Assume that the population variances are approximately equal.

Giving in 2008                                                Giving in 2009

232                                                                  215

150                                                                  125

50                                                                    50

400                                                                  350

325                                                                  210

175                                                                  150

Please show all work, including what numbers you plugged in

a) Can we conclude with a=0.01 that the average contribution to charity has decreased in this corporation from 2008 to 2009?

b)Give the assumptions for your test

c)Repeat part a assuming the population variances are not equal

d)compare the results of part a and part c

Answer 1

a)

H0 : mu 1 -mu2 = 0
Ha : mu 1 - mu2 < 0

x1 = 222 , s1 = 125.9921 , n1 = 6

x2 = 183.33 , s2 = 101.7677 , n2 = 6

test statistics:

Sp = sqrt(((n1-1)*s1^2+(n2-1)*s2^2)/(n1+n2-2))
= 114.5222

t = (x1 - x2)/Sp *sqrt(1/n1+1/n2)
= ( 222 - 183.33)/(114.5222 *(1/6 + 1/6))
= 1.013

p value= 0.8325

Do not reject H0

b)

The populations from which the samples have been drawn should be normal - appropriate statistical methods exist for testing this assumption

The standard deviation of the populations should be equal

Samples have to be randomly drawn independent of each other.

c)

x1 = 222 , s1 = 125.9921 , n1 = 6

x2 = 183.33 , s2 = 101.7677 , n2 = 6

t = (x1 - x2)/sqrt(s1^2/n1+s2^2/n2)
= ( 222 - 183.33)/sqrt(125.9921^2/6 + 101.7677^2/6)
= 0.585

SE = sqrt[ (s12/n1) + (s22/n2) ]
(s12/n1)	2645.6682
(s22/n2)	1726.1888
SE	66.1200
DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] }
[ (s12 / n1)2 / (n1 - 1) ]			1399912.056
[ (s22 / n2)2 / (n2 - 1) ]			595945.57
(s12/n1 + s22/n2)2			19113133.87
DF =	9.576

p value = 0.7135

Do not reject H0

d)

Results from A) and c) are same