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 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 for association.

Step 2. In Columns containing the table, enter the column of A, B and C

Step 3. In Rows, select y

Step 4: Under Statistics, select Chi-square test, Display counts in each cell, Display marginal counts and expected cell counts.

Step 5: Click OK.

Step 3. MINITAB output

Step 4. Row total, column total and grand total

y	A	B	C	Total
a	10	15	75	100
b	0	25	75	100
Total	10	40	150	200

Step 5. Expected Frequencies

y	A	B	C	Total
a	5	20	75	100
b	5	20	75	100
Total	10	40	150	200

Step 6. Solution b)

Determine the value of chi-squared statistic

From the MINITAB output, the value of chi-squared statistic is $\boxed{12.5}$.

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:

$H_0$: The two variables are not associated

Alternative hypothesis:

$H_1$: The two variables are associated

Step 8. Conclusion for 5% significance level

From the output, the value of test statistic is $\boxed{12.5}$and the p-value is $\boxed{0.002}$.

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

That is,$p-value(=0.002)<\alpha (=0.05)$.

Therefore, the null hypothesis is rejected at 5% level

Thus, the data provide sufficient evidence to conclude that the two variables are associated at

the 5% significance level