Answer to Question #195555 in Statistics and Probability for Rin

Question

Suppose that PTPTN would like to investigate if the repayment score and income level of an

individual are independent of one another. PTPTN selected a random sample of 164 employed

adults and asked them to report their repayment score and their income range. The repayment

score is measured based on the overdue of student loan repayment in months and the

following contingency table presents these results:

 

Annual Income Range (RM)

Repayment Score

Poor

(Over 3 months)

Average

(2 – 3 months)

Good

(Below 2 months)

Below RM 40 000

30

27

10

RM 40 000 – RM 80 000

17

20

15

Above RM 80 000

11

16

18

 

a. Set up the competing hypotheses to determine if the repayment score and income level

are dependent. (2 Marks)

b. What is the degree of freedom for the test? (1 Mark)

c. Calculate the value of the test statistic (𝜒2). (4 Marks)

d. Compute the p-value. (2 Marks)

e. Does the evidence suggest the repayment score and income level are dependent at the

1% significance level? (1 Mark)

Answer 1

The given data is-

The Expected value of the Reypayment score is calculated by the formula-

$E=\dfrac{R_i\times C_i}{T}$

Where, $C_1=58,C_2=63,C_3=43, R_1=67,R_2=52,R_3=45$ and $T=164$

The table for chi-square calculation is-

(a) Null Hypothesis

$H_o:$ There is no association between the repayment score and income level.(i.e. They are independent)

Alternate Hypothesis

$H_a$ : There is an association between the reypayment score and income level.(i.e. Theya are not independent.)

(b) Degree of freedom for this test,

$df=(\text{ Total row }-1)\times (\text{ Total colum}-1)$

    $=(3-1)\times (3-1)=2\times 2=4$

(c) Chi square test-

$\chi^2=\sum\dfrac{(O-E)^2}{E}$

    $=1.67+0.11+1.51+0.07+0+0.10+3.28+0.14+3.26$

     $=10.143$

Therefore chi-square test statistics is $\chi^2=10.14$

(d) The calculated value of chi- square (10.14) with 4 degree of freedom at 1% level of significance level

The p-value is p=0.0381

(e) Conclusion: Asp-value<0.01. Thus the Null hypothesis is rejected. We conclude that the repayment score and income level are independent.