A survey cross-tabulates happiness (not too happy, pretty happy, very happy) against family income (above average, average, below average), with row counts 58/278/117 out of 453, 59/285/101 out of 445, and 66/294/140 out of 500. Identify the explanatory and response variables, then find and interpret the conditional proportions of happiness at each income level.

Step-by-step solution

1. 1

   Decide which variable explains which. The stated concern is 'whether happiness varies depending on income level' — happiness is what might change, income is what might cause the change. So income is the explanatory variable and happiness is the response variable. This choice is not arbitrary bookkeeping: it dictates the direction in which the percentages must be computed.

2. 2

   Condition in the direction the explanatory variable dictates. Because income is explanatory and it labels the rows, divide each count by its row total. That answers 'among people at this income level, what fraction are very happy?' Dividing by column totals would answer a different, less relevant question.

3. 3

   Compute the row for above-average income (total 453).

   $\frac{58}{453}=12.8\%,\qquad \frac{278}{453}=61.4\%,\qquad \frac{117}{453}=25.8\%.$

   These sum to $100\%$, which is the check that the conditioning was done correctly.

4. 4

   Compute the other two rows. Average income (total $445$):

   $\frac{59}{445}=13.3\%,\qquad \frac{285}{445}=64.0\%,\qquad \frac{101}{445}=22.7\%.$

   Below-average income (total $500$):

   $\frac{66}{500}=13.2\%,\qquad \frac{294}{500}=58.8\%,\qquad \frac{140}{500}=28.0\%.$

5. 5

   Compare the rows to judge the association. Read down each column. 'Not too happy' barely moves: $12.8\%$, $13.3\%$, $13.2\%$. 'Very happy' varies a little more: $25.8\%$, $22.7\%$, $28.0\%$ — and, against the usual expectation, the below-average income group has the highest share.

6. 6

   State the interpretation. If income and happiness were independent, every row would match the overall split of $13.1\%$ / $61.3\%$ / $25.6\%$. The rows differ from that by only a few percentage points and not in a consistent direction, so the association between income level and happiness in this sample is weak. Differences this small in a sample of $1398$ are the kind that a formal chi-square test would need to confirm before being called real.