11.81 Normal and chi-squared with df • 1 When df = 1, the P-value from the chi-squared test of independence is the same as the P-value for the two-sided test comparing two proportions with the z test statistic. This is because of a direct connection between the standard normal distribution and the chi-squared distribution with df = 1: Squaring a z-score yields a chi-squared value with df = 1 having chisquared right-tail probability equal to the two-tail normal probability for the z-score.

a. Illustrate this with z = 1.96, the z-score with a two-tail probability of 0.05. Using the chi-squared table or software, show that the square of 1.96 is the chi-squared score for df = 1 with a P-value of 0.05.

b. Show the connection between the normal and chi-squared values with P@value = 0.01.