Question

The data below show sport preference and age of participant from a random sample of members of a sports club. Test if sport preference is independent of age at the 0.02 significant level.

H₀: Sport preference is independent of age
Ha: Sport preference is dependent on age

	18-25	26-30	31-40	41 and over
Tennis	40	60	58	44
Swimming	59	76	50	66
Basketball	73	61	67	53

a. Complete the table: Give all answers as decimals rounded to 4 places.

Observed Frequency	Expected Frequency	(O−E)2E(O-E)2E
40
60
58
44
59
76
50
66
73
61
67
53
	Total

(b) What is the chi-square test-statistic for this data?
      Test Statistic: χ2=χ2=  

(d) The p-value is...

• greater than αα
• less than (or equal to) αα

(e) The p-value leads to a decision to...

• reject the null
• fail to reject the null

(f) What is the final conclusion?

• There is not sufficient evidence to conclude sport preference is dependent on age.
• There is sufficient evidence to conclude sport preference is dependent on age.

Answer 1

Given table data is as below MATRIX col1 col2 col3 col4 TOTALS row 1 40 60 58 44 202 row 2 59 76 50 66 251 row 3 73 61 67 53 254 TOTALS 172 197 175 163 N = 707 ------------------------------------------------------------------ calculation formula for E table matrix E-TABLE col1 col2 col3 col4 row 1 row1*col1/N row1*col2/N row1*col3/N row1*col4/N row 2 row2*col1/N row2*col2/N row2*col3/N row2*col4/N row 3 row3*col1/N row3*col2/N row3*col3/N row3*col4/N ------------------------------------------------------------------ expected frequencies calculated by applying E - table matrix formulae E-TABLE col1 col2 col3 col4 row 1 49.143 56.286 50 46.571 row 2 61.064 69.939 62.129 57.868 row 3 61.793 70.775 62.871 58.56 ------------------------------------------------------------------ calculate chisquare test statistic using given observed frequencies, calculated expected frequencies from above Oi Ei Oi-Ei (Oi-Ei)^2 (Oi-Ei)^2/Ei 40 49.143 -9.143 83.594 1.701 60 56.286 3.714 13.794 0.245 58 50 8 64 1.28 44 46.571 -2.571 6.61 0.142 59 61.064 -2.064 4.26 0.07 76 69.939 6.061 36.736 0.525 50 62.129 -12.129 147.113 2.368 66 57.868 8.132 66.129 1.143 73 61.793 11.207 125.597 2.033 61 70.775 -9.775 95.551 1.35 67 62.871 4.129 17.049 0.271 53 58.56 -5.56 30.914 0.528 ᴪ^2 o = 11.656 ------------------------------------------------------------------ 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 = 0.02 from standard normal table, chi square value at right tailed, ᴪ^2 α/2 =15.033 since our test is right tailed,reject Ho when ᴪ^2 o > 15.033 we use test statistic ᴪ^2 o = Σ(Oi-Ei)^2/Ei from the table , ᴪ^2 o = 11.656 critical value the value of |ᴪ^2 α| at los 0.02 with d.f (r-1)(c-1)= ( 3 -1 ) * ( 4 - 1 ) = 2 * 3 = 6 is 15.033 we got | ᴪ^2| =11.656 & | ᴪ^2 α | =15.033 make decision hence value of | ᴪ^2 o | < | ᴪ^2 α | and here we do not reject Ho ᴪ^2 p_value =0.07 ANSWERS --------------- a. 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 b. test statistic: 11.656 c. critical value: 15.033 d. p-value:0.07 e. p value is greater than alpha value decision: do not reject Ho

f. we do not have enough evidence to support the claim that   sport preference is dependent on age.