Noroviruses cause roughly 21 million cases of gastroenteritis ("stomach flu") annually in the United States. A study examined the effectiveness of a new norovirus vaccine. Healthy adult volunteers received, at random, either the vaccine or a placebo. They were then exposed to the norovirus. In all, 14 of the 38 vaccinated subjects suffered from gastroenteritis compared with 27 of the 39 subjects who received a placebo. We want to know if this study provides evidence that an outcome of gastroenteritis depends on the treatment received. Under a null hypothesis of no association between treatment and outcome, what is the contribution to the chi-square statistic (i.e., chi-square component) for the gastroenteritis and vaccine cell of the two-way table?

Question

Quizplus · Accepted Answer

To calculate the contribution to the chi-square statistic for the gastroenteritis and vaccine cell, we need to use the formula:
(observed frequency - expected frequency)^2 / expected frequency

Using the data from the study, we can fill out the following table:

|                | Gastroenteritis | No Gastroenteritis |
|----------------|----------------|-------------------|
| Vaccine        | 14             | 24                |
| Placebo        | 27             | 12                |
|----------------|----------------|-------------------|
| Total          | 41             | 36                |

To calculate the expected frequency for the gastroenteritis and vaccine cell, we use the formula:
(row total * column total) / grand total

Expected frequency = (38 * 41) / 77 = 20.28

So, the contribution to the chi-square statistic for the gastroenteritis and vaccine cell is:

(14 - 20.28)^2 / 20.28 = 1.92

Therefore, the correct answer is B).

Noroviruses Cause Roughly 21 Million Cases of Gastroenteritis ("Stomach Flu")