Question

My second question

A mental health service agency is conducting a study of how clients pay for the counseling services they receive. The agency accepts payment in one of four ways: in person, by mail, by credit card, or by third-party insurance. The agency randomly sampled 400 clients to determine if there is a relationship between the client’s age and the payment method used. The following sample results were obtained:

Counseling Service Payments

Payment Method	Age of Client
	20-30	31-40	41-50	Over 50
In Person	8	12	11	13
By Mail	29	67	72	50
By Credit Card	26	19	5	7
By Third-Party Insurance	23	35	17	6

Based on the sample data, can the agency conclude there is a relationship between the age of the client and the payment method used? Conduct the appropriate test at the α = 0.01 level of significance.

Answer 1

Given table data is as below MATRIX col1 col2 col3 TOTALS TOTALS row 2 8 12 11 13 44 row 3 29 67 72 50 218 row 3 26 19 5 7 57 row 4 23 35 17 6 81 TOTALS 86 133 105 76 400 ------------------------------------------------------------------ calculation formula for E table matrix E-TABLE col1 col2 col3 col4 row 2 row1*col1/N row1*col2/N row1*col3/N row1*col4/N row 3 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 row 4 row4*col1/N row4*col2/N row4*col3/N row4*col4/N ------------------------------------------------------------------ expected frequencies calculated by applying E - table matrix formulae E-TABLE col1 col2 col3 col4 row 2 9.46 14.63 11.55 8.36 row 3 46.87 72.485 57.225 41.42 row 3 12.255 18.953 14.963 10.83 row 4 17.415 26.933 21.263 15.39 ------------------------------------------------------------------ calculate chisquare test statistic using given observed frequencies, calculated expected frequencies from above Oi Ei Oi-Ei (Oi-Ei)^2 (Oi-Ei)^2/Ei 8 9.46 -1.46 2.132 0.225 12 14.63 -2.63 6.917 0.473 11 11.55 -0.55 0.303 0.026 13 8.36 4.64 21.53 2.575 29 46.87 -17.87 319.337 6.813 67 72.485 -5.485 30.085 0.415 72 57.225 14.775 218.301 3.815 50 41.42 8.58 73.616 1.777 26 12.255 13.745 188.925 15.416 19 18.953 0.047 0.002 0 5 14.963 -9.963 99.261 6.634 7 10.83 -3.83 14.669 1.354 23 17.415 5.585 31.192 1.791 35 26.933 8.067 65.076 2.416 17 21.263 -4.263 18.173 0.855 6 15.39 -9.39 88.172 5.729 ᴪ^2 o = 50.314 ------------------------------------------------------------------ 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 =21.666 since our test is right tailed,reject Ho when ᴪ^2 o > 21.666 we use test statistic ᴪ^2 o = Σ(Oi-Ei)^2/Ei from the table , ᴪ^2 o = 50.314 critical value the value of |ᴪ^2 α| at los 0.01 with d.f (r-1)(c-1)= ( 4 -1 ) * ( 4 - 1 ) = 3 * 3 = 9 is 21.666 we got | ᴪ^2| =50.314 & | ᴪ^2 α | =21.666 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: 50.314 critical value: 21.666 p-value:0 decision: reject Ho

we have enough evidence to support the claim that  the agency conclude there is a relationship between the age of the client and the payment method used.