At a 5% significance, conduct the test of independence of “Response” and “Party

Question

At a 5% significance, conduct the test of independence of “Response” and “Party Affiliation” concerning the first question. Specify the null and alternative hypotheses. Find the value of the test statistic. Make your conclusion using the p-value approach or the critical value approach, and interpret your conclusion.

excel info

Observation Party Pay Cut? Lobbyists? Term Limits? 1 Democrat 1 2 2 2 Democrat 1 1 2 3 Democrat 2 1 2 4 Democrat 1 2 1 5 Democrat 2 2 2 6 Democrat 2 1 1 7 Democrat 1 2 2 8 Democrat 2 1 2 9 Democrat 1 1 2 10 Democrat 1 2 1 11 Democrat 1 2 1 12 Democrat 1 2 1 13 Democrat 2 2 1 14 Democrat 1 2 1 15 Democrat 1 1 1 16 Democrat 1 2 2 17 Democrat 1 1 2 18 Democrat 1 2 1 19 Democrat 1 1 2 20 Democrat 1 1 1 21 Democrat 2 2 1 22 Democrat 1 2 2 23 Democrat 2 1 2 24 Democrat 2 1 1 25 Democrat 2 1 1 26 Democrat 1 1 1 27 Democrat 1 1 2 28 Democrat 2 1 1 29 Democrat 2 2 2 30 Democrat 1 1 2 31 Democrat 1 2 2 32 Democrat 2 1 1 33 Democrat 1 1 2 34 Democrat 2 1 1 35 Democrat 2 1 2 36 Democrat 1 1 2 37 Independent 2 2 1 38 Independent 2 2 1 39 Independent 1 1 1 40 Independent 1 1 2 41 Independent 2 1 2 42 Independent 2 2 1 43 Independent 1 1 2 44 Independent 2 1 1 45 Independent 1 1 1 46 Independent 2 1 2 47 Independent 1 1 1 48 Independent 1 1 1 49 Independent 2 1 2 50 Independent 2 2 1 51 Independent 1 1 2 52 Independent 1 1 1 53 Independent 1 1 2 54 Independent 1 1 2 55 Independent 2 1 2 56 Republican 1 1 1 57 Republican 1 2 1 58 Republican 1 1 2 59 Republican 1 1 1 60 Republican 2 1 2 61 Republican 1 1 2 62 Republican 1 1 2 63 Republican 1 1 1 64 Republican 1 1 1 65 Republican 1 1 1 66 Republican 1 1 2 67 Republican 1 1 2 68 Republican 1 2 1 69 Republican 1 2 1 70 Republican 1 1 1 71 Republican 2 2 1 72 Republican 2 1 1 73 Republican 1 1 1 74 Republican 2 2 1 75 Republican 2 1 1 76 Republican 1 2 1 77 Republican 1 2 1 78 Republican 1 1 2 79 Republican 1 1 1 80 Republican 1 1 1 81 Republican 1 2 1 82 Republican 1 2 2 83 Republican 2 1 2 84 Republican 1 1 1 85 Republican 1 1 2 86 Republican 1 1 1 87 Republican 1 1 2 88 Republican 1 1 2 89 Republican 1 1 1 90 Republican 1 1 1 91 Republican 1 1 1 92 Republican 1 1 1 93 Republican 1 1 1 94 Republican 1 1 1 95 Republican 1 1 1 96 Republican 1 2 1 97 Republican 1 1 1 98 Republican 1 1 1 99 Republican 1 2 2 100 Republican 1 1 1

Explanation / Answer

Here the test of hypothesis is,

H0 :Response and party affilation are independent.

H1 :Response and party affilation are not independent.

Assume alpha = 5% = 0.05

The test statistic is,

X2 = sum(O - E)2 / E

where O is observed frequency and

E is expected frequency.

First we have to complete contingency table.

This we can create by using response yes (1) and no (2) and party affilation are democrat, independent and republican.

The complete table is,

   yes   no   total
democrat   22   14   36
independent   10   9   19
republican   40   5   45
total   72   28   100

First we have to find expected frequency.

Expected frequency (E) = (Row total)*(Column total) / sample size

Expected frequency for 22 :

E = (72*36) / 100 = 25.92

Expected frequency for 14 :

E = (28*36) / 100 = 10.08

Expected frequency for 10 :

E = (72*19) / 100 = 13.68

Expected frequency for 9 :

E = (28*19) / 100 = 5.32

Expected frequency for 40 :

E = (72*45) / 100 = 32.4

Expected frequency for 5 :

E = (28*45) / 100 = 12.6

The complete table of observed and expected frequency is,

O   E   (O-E)^2   (O-E)^2/E
22   25.92   15.3664   0.5928
14   10.08   15.3664   1.5244
10   13.68   13.5424   0.9899
9   5.32   13.5424   2.5456
40   32.40   57.7600   1.7827
5   12.60   57.7600   4.5841
           12.0196

X2 = 12.0196

Now we have to find P-value for taking decision.

P-value synatx in EXCEL for chi-square distribution is,

=CHIDIST(x, deg_freedom)

where x is test statistic value.

deg_freedom = (R-1)*(C-1)

R is number of rows = 3

C is number of columns = 2

deg_freedom = (3-1)*(2-1) = 2

P-value = 0.002

P-value < alpha (0.05)

Reject H0 at 5% level of significance.

Conclusion : Response and party affilation are not independent.

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