Question

11) The staff at a hospital emergency room are convinced that the number of patients they see during the midnight shift is affected by the phases of the moon. They decide to test their theory by recording the number of patients they see during the midnight shift for each phase of the moon over the course of a month. The results are summarized in the table below. At the 0.05 significance level, test the claim that the number of patients is affected the phase of the moon. Phase of Moon New Moon 1st Quarter Full Moon Last Quarter Number of Patients 85 66 97 68

Answer 1

As we are testing here whether the number of patients is affected by the phase of the moon, therefore we are testing here whether the frequency is equally distributed in the 4 categories here.

Moon Phase	New Moon	1st Quarter	Full Moon	Last Quarter
# of Patients (O_i)	85	66	97	68	316
Expected Frequency(E_i)	79	79	79	79	316
Chi square_i	0.4557	2.1392	4.1013	1.5316	8.2278

The expected frequency for each of the 4 categories here is computed as:
= (Total Frequency) / 4 = 316/4 = 79

The chi square test statistic here is computed as:

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} = 8.2278$

For k - 1 = 3 degrees of freedom, the p-value here is obtained from the chi square distribution tables here as:

$p = P(\chi^2_3 > 8.2278) = 0.0415$

As the p-value here is 0.0415 < 0.05 which is the level of significance, therefore the test is significant here and we can reject the null hypothesis here. Therefore we have sufficient evidence here that the number of patients is affected by the phase of the moon