Question

We have presented a contingency table that gives a cross-classification of a random sample of values for two variables x and y, of a population.

Perform the following tasks

a. Find the expected frequencies Note: You will first need to compute the row totals, column totals, and grand total.

b. Determine the value of the chi-square statistic

c. Decide at the 5% significance level whether the data provide sufficient evidence to conclude that the two variables are associated.

Short answer

The data do not provide sufficient evidence to conclude that the two variables are associated at the 5% significance level.

Step-by-step solution

Step 1. Given

Step 2. Solution a). Find the Expected frequencies using MINITAB

MINITAB procedure:

Step 1: Choose Stat > Tables > Chi-Square test (Two-Way Table in Worksheet).

Step 2: In Columns containing the table, enter the columns of AandB.

Step 3: Click OK.

Step 3. MINITAB output

Step 4. Row total, column total and grand total

y	A	B	Total
a	5	35	40
b	20	80	100
c	35	85	110
Total	50	200	250

Step 5. Expected Frequencies

y	A	B	Total
a	8	32	40
b	20	80	100
c	22	88	110
Total	50	200	250

Step 6. Solution b)

Determine the value of chi-squared statistic

From the MINITAB output, the value of chi-squared statistic is$\boxed{1.918}$

Step 7. Solution c) 

Check whether or not the data provide sufficient evidence to conclude that the two variables are associated at the 5% significance level

The hypotheses are given below

Null hypothesis:

: The two variables are not associated

Alternative hypothesis:

: The two variables are associated

Step 8. Conclusion for 5% significance level

From the output, the value of test statistic is $\boxed{1.918}$and the p-value is $\boxed{0.383}$.

Here, the p-value is lesser than the level of significance

That is, $p-value(=0.383)>\alpha (=0.05)$.

Therefore, the null hypothesis is not rejected at 5% level

Thus, the data do not provide sufficient evidence to conclude that the two variables are associated at the 5% significance level.