A x² test involving a 2 × 2 table is equivalent to the test for the difference between two proportions, as described in Section 9-1. Using the claim and table in Exercise 9 “Four Quarters the Same as $1?” verify that the x² test statistic and the z test statistic (found from the test of equality of two proportions) are related as follows: z² = x². Also show that the critical values have that same relationship.

Data From Exercise 9:

In a study of the “denomination effect,” 43 college students were each given one dollar in the form of four quarters, while 46 other college students were each given one dollar in the form of a dollar bill. All of the students were then given two choices: (1) keep the money; (2) spend the money on gum. The results are given in the accompanying table (based on “The Denomination Effect,” by Priya Raghubir and Joydeep Srivastava, Journal of Consumer Research, Vol. 36.) Use a 0.05 significance level to test the claim that whether students purchased gum or kept the money is independent of whether the  were given four quarters or a $1 bill. Is there a “denomination effect”?