Question 9.11

For Exercises 9.1 and 9.2, see page 458; for 9.3 and 9.4, see page 461; for 9.5 and 9.6, see page 462; for 9.7 and 9.8, see page 463; for 9.9 and 9.10, see page 465; for 9.11, see page 466; for 9.12 and 9.13, see page 471; for 9.14, see page 473; for 9.15, see page 474; and for 9.16, see page 475.

Question 9.11

9.11 Social media in the supply chain.

Sample proportions from Example 9.1 and the two-way table in Example 9.2 (page 456) report the same information in different ways. We saw in Example 8.9 (pages 442–443) that the $z$ statistic for the hypothesis of equal population proportions is $z = 4.87$ with $P < 0.0004$ .

Find the chi-square statistic $Χ^{2}$ for this two-way table and verify that it is equal (up to roundoff error) to $z^{2}$ .
Verify that the 0.001 critical value for chi-square with $df = 1$ (Table F) is the square of the 0.0005 critical value for the standard Normal distribution (Table D). The 0.0005 critical value corresponds to a P-value of 0.001 for the two-sided $z$ test.
Explain carefully why the two hypotheses
$\begin{array}{l} \begin{array}{l} H_{0} : p_{1} = p_{2} & (z test) \end{array} \\ \begin{array}{l} H_{0} : no relation between company size and social media use & (Χ^{2} test) \end{array} \end{array}$

say the same thing about the population.

9.11

(a) $χ^{2} = 23.74 \approx z^{2} = 23.72$ . (b) ${(z^{*})}^{2} = {3.291}^{2} = 1083 = Χ^{2} *$ . (c) The $z$ test null hypothesis indicates that the proportions are equal or that small and large companies use social media equally; in other words, the size of the company doesn’t matter in determining social media use—i.e., there is no relation between company size and social media use, the null hypothesis for the $Χ^{2}$ test.