Question

The 2009 National Household Survey on drug use and health reported the number (out of 1000) of three age groups: 12-17 years

old, 18-25 years old, and 26 years old, who reported using cannabis (marijuana and hashish). The survey was repeated in 2014, and

the data are presented in the table below. Do a chi-square contingency analysis to see if the reported usage has changed over time or is

reported usage independent of year?

Subjects	12-17	18-25	26+
2009	100	212	130
2014	164	526	461

Ho:

Ha:

test-statistic:

df:

Exact P value for the test-statistic

Conclusion relative to the hypothesis:

Answer 1

The following cross-tabulation have been provided. The row and column total have been calculated and they are shown below:

	12-17	18-25	26+	Total
2009	100	212	130	442
2014	164	526	461	1151
Total	264	738	591	1593

The expected values are computed in terms of row and column totals. In fact, the formula is ​​, where Ri​ corresponds to the total sum of elements in row i, Cj​ corresponds to the total sum of elements in column j, and T is the grand total. The table below shows the calculations to obtain the table with expected values:

Expected Values	12-17	18-25	26+	Total
2009				442
2014				1151
Total	264	738	591	1593

Based on the observed and expected values, the squared distances can be computed according to the following formula:. The table with squared distances is shown below:

Squared Distances	12-17	18-25	26+
2009
2014

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H_0 : The two variables are independent

H_a​: The two variables are dependent

This corresponds to a Chi-Square test of independence.

Rejection Region

Based on the information provided, the significance level is α = 0.05 , the number of degrees of freedom is df = (2 - 1)*(3 - 1) = 2.

Test Statistics

The Chi-Squared statistic is computed as follows:

= 23.619

Decision about the null hypothesis

Since it is observed that = 23.619 > = 5.991, it is then concluded that the null hypothesis is rejected.

Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the two variables are dependent, at the 0.05 significance level.