Question

1. Does including a gift with a request for a donation influence the proportion who will make a donation? This question was investigated in a recent study. Letters were sent to a large number of potential donors in Japan. The letter requested a donation for funding schools in Mali. The letter recipients were assigned at random to one of three groups. Those in the first group received a letter with no gift. Those in the second group received a letter that included a small gift (a postcard), and those in the third group received a letter with a larger gift (four postcards). The response of interest was whether or not the letter resulted in a donation.
   

   	Donation	No Donation
   No Gift	36	435
   Small Gift	78	502
   Large Gift	72	399

   
   Perform a test of independence. What is the critical value for the study at a 5% level of significance?

   	a.	5.99
   	b.	1.645
   	c.	11.07
   	d.	7.81
   	e.	3.84

Answer 1

Given table data is as below MATRIX col1 col2 TOTALS row 1 36 435 471 row 2 78 502 580 row 3 72 399 471 TOTALS 186 1336 N = 1522 ------------------------------------------------------------------ calculation formula for E table matrix E-TABLE col1 col2 row 1 row1*col1/N row1*col2/N row 2 row2*col1/N row2*col2/N row 3 row3*col1/N row3*col2/N ------------------------------------------------------------------ expected frequecies calculated by applying E - table matrix formulae E-TABLE col1 col2 row 1 57.56 413.44 row 2 70.88 509.12 row 3 57.56 413.44 ------------------------------------------------------------------ calculate chisquare test statistic using given observed frequencies, calculated expected frequencies from above Oi Ei Oi-Ei (Oi-Ei)^2 (Oi-Ei)^2/Ei 36 57.56 -21.56 464.83 8.08 435 413.44 21.56 464.83 1.12 78 70.88 7.12 50.69 0.72 502 509.12 -7.12 50.69 0.1 72 57.56 14.44 208.51 3.62 399 413.44 -14.44 208.51 0.5 ᴪ^2 o = 14.14 ------------------------------------------------------------------ set up null vs alternative as null, Ho: no relation b/w X and Y OR X and Y are independent alternative, H1: exists a relation b/w X and Y OR X and Y are dependent level of significance, α = 0.05 from standard normal table, chi square value at right tailed, ᴪ^2 α/2 =5.99 since our test is right tailed,reject Ho when ᴪ^2 o > 5.99 we use test statistic ᴪ^2 o = Σ(Oi-Ei)^2/Ei from the table , ᴪ^2 o = 14.14 critical value the value of |ᴪ^2 α| at los 0.05 with d.f (r-1)(c-1)= ( 3 -1 ) * ( 2 - 1 ) = 2 * 1 = 2 is 5.99 we got | ᴪ^2| =14.14 & | ᴪ^2 α | =5.99 make decision hence value of | ᴪ^2 o | > | ᴪ^2 α| and here we reject Ho ᴪ^2 p_value =0 ANSWERS --------------- null, Ho: no relation b/w X and Y OR X and Y are independent alternative, H1: exists a relation b/w X and Y OR X and Y are dependent test statistic: 14.14 critical value: 5.99 option:A p-value:0 decision: reject Ho

we have enough evidence to support the claim that there is association of The letter requested a donation for funding schools in Mali. The letter recipients were assigned at random to one of three groups