Calculation practice: x² contingency analysis. Married couples often split up after one member is diagnosed with a catastrophic disease, such as terminal cancer or a brain tumor. Does the frequency of breakup depend on which member is diagnosed? Glantz et al. (2009) tallied divorces after such serious diagnoses in 515 patients in opposite-sex marriages. Of the 261 couples in which the man was ill, 7 divorced soon after diagnosis. Of 254 couples in which the woman was the patient, 53 divorced. Test for a difference between the two types of couples in the proportion divorcing after diagnosis.

a. Summarize the data in a contingency table and examine the frequencies. Do divorce frequencies appear similar between the two types of patients? If they differ, for which sex does diagnosis seem more often to lead to divorce?

b. State the null and alternate hypotheses for the test.

c. In what proportion of couples was the man the diagnosed patient?

d. What proportion of couples divorced?

e. Under the null hypothesis, what is the expected proportion of each of the four possible combinations of outcomes?

f. What is the total number of observations in the study?

g. Under the null hypothesis, what is the expected number (frequency) of observations in each combination?

h. Examine the expected numbers. Is it legitimate to use a x² contingency test with these data? Why?

i. Calculate the test statistic x² for these data.

j. How many degrees of freedom does the  x² test statistic have?

k. What is the critical value for this test corresponding to significance level α=0.05? 

l. Calculate the P-value  for the test.

m. What is your conclusion? Are sex of the diagnosed patient and divorce independent?