Question

A researcher surveyed randomly selected Democrats and Republicans asking them what the number one concern should be for the president of the United States. The results of the survey are shown below. Is there evidence to conclude that there is a difference in what Democrats and Republicans think is the most important?

Frequencies of Concerns for Democrats and Republicans

	Economy	Foreign Affairs	Family Values	Environment	Other
Democrats	83	47	81	85	50
Republicans	84	64	94	53	35

What can be concluded at the αα = 0.01 significance level?

1. What is the correct statistical test to use?
   • Paired t-test
   • Goodness-of-Fit
   • Independence
   • Homogeneity
2. What are the null and alternative hypotheses?
   H0:H0:
   • Political affiliation and number one concern are dependent.
   • The distribution of concerns for Democrats is the same as it is for Republicans.
   • The distribution of concerns for Democrats is not the same as it is for Republicans.
   • Political affiliation and number one concern are independent.
   
   
   
   H1:H1:
   • Political affiliation and number one concern are dependent.
   • Political affiliation and number one concern are independent.
   • The distribution of concerns for Democrats is not the same as it is for Republicans.
   • The distribution of concerns for Democrats is the same as it is for Republicans.
3. The test-statistic for this data =  (Please show your answer to three decimal places.)
   
4. The p-value for this sample = (Please show your answer to four decimal places.)  
   
5. The p-value is Select an answer less than (or equal to), greater than   
   
6. Based on this, we should
   • accept the null
   • fail to reject the null
   • reject the null
7. Thus, the final conclusion is...
   • There is sufficient evidence to conclude that the distribution of concerns for Democrats is the same as it is for Republicans.
   • There is sufficient evidence to conclude that the distribution of concerns for Democrats is not the same as it is for Republicans
   • There is insufficient evidence to conclude that political affiliation and number one concern are dependent.
   • There is insufficient evidence to conclude that the distribution of concerns for Democrats is not the same as it is for Republicans.
   • There is sufficient evidence to conclude that political affiliation and number one concern are dependent.

Answer 1

Given table data is as below MATRIX col1 col2 col3 col4 col5 TOTALS row 1 83 47 81 85 50 346 row 2 84 64 94 53 35 330 TOTALS 167 111 175 138 85 N = 676 ------------------------------------------------------------------ calculation formula for E table matrix E-TABLE col1 col2 col3 col4 col5 row 1 row1*col1/N row1*col2/N row1*col3/N row1*col4/N row1*col5/N row 2 row2*col1/N row2*col2/N row2*col3/N row2*col4/N row2*col5/N ------------------------------------------------------------------ expected frequencies calculated by applying E - table matrix formulae E-TABLE col1 col2 col3 col4 col5 row 1 85.476 56.814 89.571 70.633 43.506 row 2 81.524 54.186 85.429 67.367 41.494 ------------------------------------------------------------------ calculate chisquare test statistic using given observed frequencies, calculated expected frequencies from above Oi Ei Oi-Ei (Oi-Ei)^2 (Oi-Ei)^2/Ei 83 85.476 -2.476 6.131 0.072 47 56.814 -9.814 96.315 1.695 81 89.571 -8.571 73.462 0.82 85 70.633 14.367 206.411 2.922 50 43.506 6.494 42.172 0.969 84 81.524 2.476 6.131 0.075 64 54.186 9.814 96.315 1.777 94 85.429 8.571 73.462 0.86 53 67.367 -14.367 206.411 3.064 35 41.494 -6.494 42.172 1.016 ᴪ^2 o = 13.27 ------------------------------------------------------------------ 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 = 0.01 from standard normal table, chi square value at right tailed, ᴪ^2 α/2 =13.277 since our test is right tailed,reject Ho when ᴪ^2 o > 13.277 we use test statistic ᴪ^2 o = Σ(Oi-Ei)^2/Ei from the table , ᴪ^2 o = 13.27 critical value the value of |ᴪ^2 α| at los 0.01 with d.f (r-1)(c-1)= ( 2 -1 ) * ( 5 - 1 ) = 1 * 4 = 4 is 13.277 we got | ᴪ^2| =13.27 & | ᴪ^2 α | =13.277 make decision hence value of | ᴪ^2 o | < | ᴪ^2 α | and here we do not reject Ho ᴪ^2 p_value =0.01 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: 13.27 critical value: 13.277 p-value:0.01 decision: do not reject Ho

we do not have enough evidence to support the claim that  political affiliation and number one concern are dependent.