Question

Equivalent Tests A\({\chi ^2}\)test involving a 2\( \times \)2 table is equivalent to the test for the differencebetween two proportions, as described in Section 9-1. Using the claim and table inExercise 9 “Four Quarters the Same as $1?” verify that the\({\chi ^2}\)test statistic and the zteststatistic (found from the test of equality of two proportions) are related as follows:\({z^2}\)=\({\chi ^2}\).

Also show that the critical values have that same relationship.

Short answer

The critical values and the test statistic of \({\chi ^2}\;{\rm{and}}\;{z^2}\) shows the same relationship; that is \({\chi ^2} = {z^2}\).

Step-by-step solution

Given information

The data for the students, whether they purchased gum or kept the money,is provided.

Compute the test statistic

Referring to Exercise 9 of section 11-2,

The value of the chi-square test statistic is 12.162.

From Table A-4, the critical value for the row correspondsto 1 degree of freedom and at 0.05 level of significance 3.841.

Therefore, the critical value is 3.841.

Compute the proportions and z test statistic

Let\({\hat p_1}\)representthe sample proportion of students who purchased the gum and students given four quarters.

Let\({\hat p_2}\)representthe sample proportion of students who purchased the gum and students given a $1 Bill.

The proportions are computed as,

\(\begin{aligned}{c}{{\hat p}_1} = \frac{{27}}{{27 + 16}}\\ = 0.628\end{aligned}\)

Similarly,

\(\begin{aligned}{c}{{\hat p}_2} = \frac{{12}}{{12 + 34}}\\ = 0.261\end{aligned}\)

The value of the pooled sample proportion is computed as follows:

\(\begin{aligned}{c}\bar p = \frac{{{x_1} + {x_2}}}{{{n_1} + {n_2}}}\\ = \frac{{12 + 27}}{{46 + 43}}\\ = 0.438\end{aligned}\)

\(\begin{aligned}{c}\bar q = 1 - \bar p\\ = 1 - 0.438\\ = 0.562\end{aligned}\)

The value of the test statistic is computed below:

\(\begin{aligned}{c}z = \frac{{\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - \left( {{p_1} - {p_2}} \right)}}{{\sqrt {\frac{{\bar p\bar q}}{{{n_1}}} + \frac{{\bar p\bar q}}{{{n_2}}}} }}\;\;\;\;{\rm{where}}\left( {{p_1} - {p_2}} \right) = 0\\ = \frac{{\left( {0.261 - 0.628} \right) - 0}}{{\sqrt {\frac{{\left( {0.438} \right)\left( {0.562} \right)}}{{46}} + \frac{{\left( {0.438} \right)\left( {0.562} \right)}}{{43}}} }}\\ = - 3.487395274\end{aligned}\)

Thus, the value of z test statistic is -3.487395274.

The critical value of z corresponding to \(\alpha = 0.05\) for a two-tailed test is equal to \( \pm \)1.96.

Show the relationship

The calculations are as follows,

For test statistic:

\(\begin{aligned}{c}{z^2} = {\left( { - 3.487395274} \right)^2}\\ = 12.162\end{aligned}\)

Thus,\({\chi ^2} = {z^2}\)

For critical values:

\(\begin{aligned}{c}{z^2} = {\left( {1.96} \right)^2}\\ = 3.841\end{aligned}\)

Thus,\({\chi ^2} = {z^2}\)

Therefore, the critical value of chi-square and square of z critical value is approximately the same.