Example 8

EXAMPLE 8 $χ^{2}$ test for independence using the p-value method and technology

youngliving

The National Center for Health Statistics publishes information on the living arrangements of America's young people. Table 9 contains a random sample of 200 young people ages 1-24, indicating their gender and living arrangements. Test whether gender and living arrangement are independent, using the TI-83/84, Minitab, JMP, the p-value method, and level of significance $α = 0.10$ .

Table 11.24: Table 9 Contingency table of living arrangements versus gender

	Living arrangements
Gender	Living with parents	Living with partner	All other arrangements	Total
Female	51	22	28	101
Male	58	14	27	99
Total	109	36	55	200

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Solution

Step 1 State the hypotheses and the rejection rule. Check the conditions.
- $H_{0} : Gender and living arrangements are independent$ .
- $H_{a} : Gender and living arrangements are dependent$ .
Reject $H_{0}$ if the p-value $\leq 0.10$ .

Note that Minitab provides the expected counts (frequencies) below the observed counts. We can then verify that none of the expected frequencies is less than 1, and that none of the expected frequencies has a value less than 5.
Step 2 Calculate $χ_{data}^{2}$ . We use the instructions found in the Step-by-Step Technology Guide at the end of this section. The TI-83/84 results in Figure 17 tell us that $χ_{data}^{2} = 2.225723453$ . The Minitab results in Figure 18 round this to $“Pearson Chi-Square” = χ_{data}^{2} = 2.226$ The JMP results in Figure 19 (“Pearson”) also round this to $“ChiSquare” = χ_{data}^{2} = 2.226$ .

FIGURE 17 TI-83/84 $χ^{2}$ results.

FIGURE 18 Minitab $χ^{2}$ results.

FIGURE 19 JMP $χ^{2}$ results.
Step 3 Find the p-value. From the TI-83/84 results in Figure 17, we have

$p -value = P (χ^{2} > χ_{data}^{2}) = 0.3286172017 \approx 0.329.$
Step 4 State the conclusion and the interpretation. Because p-value ≈ 0.329 is not less than level of significance 0.10, we do not reject $H_{0} .$ There is insufficient evidence that gender and living arrangements are dependent.

NOW YOU CAN DO

Exercises 15–18.