Solution to Exercise 14.08.

  I didn't realize it when I first read the problem, but as I set out to solve it, I realized there are two possible interpretations of the classes as they are defined. The first one is two sided, meaning we look on either side of the birthday.

(1)

``Within 7 days'' means 7 days on either side of the birthday, and the birthday itself; there are a total of 15 days in this group.

(2)

``Between 8 and 30 days, inclusive, from the birthday'' means 8 to 30 days after the birthday (23 days) and 8 and 30 days before the birthday (23 more days) for a total of 46 days in this class.

(3)

``Between 31 and 90 days'' before or after would be 120 days total.

(4)

``More than 90 days'' would be all else: 365 - 15 - 46 -120 = 184 days.

The analysis of the data on the basis of this interpretation is given in the following table:

Cate- No. of Prob. Exp. Obs. Z Chi-sq

gory Days Under H₀ Freq. Freq. Terms

1 15 0.041 8.2 11 -0.99 0.941

2 46 0.126 25.2 24 0.26 0.058

3 120 0.329 65.8 69 -0.49 0.160

4 184 0.504 100.8 96 0.68 0.231

Sum 365 1.000 200.0 200 1.389

The third column of the table (``Prob. Under H₀'') is obtained by dividing the second column by 365, i.e. we assume every day is equally likely to be the day the patient was admitted to the hospital. The last column in the table contains the terms $(Obs.Freq. \, - \, Exp.Freq.)^2/Exp.Freq.$, and their sum is the chi-squared test statistic, the lower right entry in the table. The degrees of freedom is 4-1 = 3. The critical value for the $\alpha =0.01$ level of significance is $\chi_{0.01,3}^2$ = 11.344. The $\chi^2$ value of 1.389 is clearly not significant at this level. In fact, it wouldn't be significant at the $10\%$ level of significance since $\chi_{0.10,3}^2$ = 6.251.

The other interpretation that occurs to me is that we are only talking about after the birthday:

(1)

``Within 7 days'' means the birthday and 7 days afterwards for a total of 8 days.

(2)

``Between 8 and 30 days'' would mean 23 days.

(3)

``Between 31 and 90 days'' after would be 60 days.

(4)

``More than 90 days'' would be 365 - 8 - 23 - 60 = 274 .

The corresponding analysis in this situation is given in the following table.

Cate- No. of Prob. Exp. Obs. Z Chi-sq

gory Days Under H₀ Freq. Freq. Terms

1 8 0.022 4.4 11 3.20 9.99

2 23 0.063 12.6 24 3.32 10.31

3 60 0.164 32.9 69 6.89 39.69

4 274 0.751 150.1 96 -8.85 19.52

Sum 365 1.000 200.0 200 79.51

Now the observed chi-squared of 79.51 does exceed the critical value of 11.344, so we do reject H₀ at this level. Looking at the Z values (column 6 of the table), we see that they are all highly significant: the smallest in magnitude 3.20 corresponds to a p-value of 2*.0007 = 0.0014, where 0.0007 is the area to the right of 3.20 (Table A.3) and we multiply by 2 as it is a 2-tailed test. Comparing with a Bonferroni corrected level of significance of $\alpha_{individual}$ = $\alpha_{overall}/(No. \, of \, Comparisons)$ = .01/4 = .0025, we have significance for every one of the categories. Further, we see that the Categories 1, 2, and 3 have Observed values much larger than Expected, and Category 4 has an Observed value much smaller than Expected.

I am sure that the first interpretation (where nothing was significant) is the correct one.