Question

Can People Delay Death? A study indicates that elderly people are able to postpone death for a short time to reach an important occasion. The researchers \({ }^{10}\) studied deaths from natural causes among 1200 elderly people of Chinese descent in California during six months before and after the Harbor Moon Festival. Thirty-three deaths occurred in the week before the Chinese festival, compared with an estimated 50.82 deaths expected in that period. In the week following the festival, 70 deaths occurred, compared with an estimated 52. "The numbers are so significant that it would be unlikely to occur by chance," said one of the researchers. (a) Given the information in the problem, is the \(\chi^{2}\) statistic likely to be relatively large or relatively small? (b) Is the p-value likely to be relatively large or relatively small? (c) In the week before the festival, which is higher: the observed count or the expected count? What does this tell us about the ability of elderly people to delay death? (d) What is the contribution to the \(\chi^{2}\) -statistic for the week before the festival? (e) In the week after the festival, which is higher: the observed count or the expected count? What does this tell us about the ability of elderly people to delay death? (f) What is the contribution to the \(\chi^{2}\) -statistic for the week after the festival? (g) The researchers tell us that in a control group of elderly people in California who are not of Chinese descent, the same effect was not seen. Why did the researchers also include a control group?

Short answer

Calculations are as follows; for part (d) the chi-square contribution is \(\frac{(33 - 50.82)^2}{50.82}\) and for part (f) it is \(\frac{(70 - 52)^2}{52}\). P-value is likely to be small, indicating that observed data did not occur by chance. Regarding the ability to delay death, before the festival, fewer deaths were observed than expected, after the festival, the count of deaths was significantly higher than expected.

Step-by-step solution

Chi-Square Statistic

A chi-square (\( \chi^{2}\)) statistic measures the difference between observed data and the data we would expect to get by chance alone. If the observed data fit the expectation perfectly, the chi-square statistic is 0. The larger the chi-square statistic, the greater the discrepancy between the observed data and the data expected by chance alone. So for question (a), because the observed count is significantly different from the expected count, the \( \chi^{2}\) statistics would likely be relatively large.

The P-value

A p-value is the probability that the results of the observed data occurred by chance. If the p-value is small, it indicates that it is unlikely that the observed data occurred by chance. Therefore, for question (b), since the observed count differs significantly from the expected count, it’s not likely due to chance, and therefore the p-value would likely be relatively small.

Observed Count vs Expected Count Pre-festival

When comparing the observed and expected count in the week before the festival, it's seen that the observed count (33 deaths) is lower than the expected count (50.82 deaths). This suggests that elderly people might be able to postpone death to reach an important occasion such as the festival.

Pre-festival Chi-square Contribution

The contribution to the chi-square statistic for the week before the festival can be calculated using the formula \(\frac{(observed-expected)^2}{expected}\). Substituting the given values gets \(\frac{(33 - 50.82)^2}{50.82}\). Calculate this for (d).

Observed Count vs Expected Count Post-festival

In contrast, in the week following the festival, the observed count (70 deaths) is higher than the expected count (52 deaths). This could indicate that there may have been a delay in deaths leading up to the festival, followed by a higher count afterward.

Post-festival Chi-square Contribution

For the contribution to the chi-square statistic for the week after the festival use the formula \(\frac{(observed-expected)^2}{expected}\) again. This time plug the post-festival observed and expected counts, which are 70 and 52, respectively. Calculate this for (f).

The Significance of Control Group

The purpose of the control group is to provide a baseline against which to compare the observed effects in the test group. By noting that the effect was not seen in a control group of elderly non-Chinese, they can be more confident that the effect they observed in the Chinese group was indeed significant, and not due to other factors, validating the effect of the Festival for the Chinese group in (g).

Key concepts

P-value

The concept of a p-value is integral to statistical hypothesis testing. It’s a measure that helps researchers determine the significance of their results. The p-value represents the probability of observing the data—or something more extreme—assuming that the null hypothesis is true. In the context of the study on whether elderly people can delay death, a small p-value would suggest a low probability that the observed decrease in deaths before the Harbor Moon Festival, and the subsequent increase after, occurred by just chance.

This has practical implications for interpreting the study. For example, if researchers find a p-value of 0.03, this means there’s only a 3% chance that the pattern observed was a result of random variation in natural death rates. When the p-value is lower than a predetermined significance level, typically 0.05, the result is considered statistically significant, leading to the rejection of the null hypothesis—indicating that there is something other than chance at play concerning the festival's influence on mortality rates among the study participants.

Observed vs Expected Count

When working with chi-square tests, the terms 'observed count' and 'expected count' are foundational. The observed count is the actual number collected from experimental data—in the study’s case, the number of deaths that occurred. The expected count, however, is what statisticians anticipate in terms of frequency under the null hypothesis, based on probability distributions or historical data.

By comparing the observed and expected counts, statisticians can infer whether deviations might be due to random fluctuations or indicate an underlying effect. Referring to the Harbor Moon Festival study, there were fewer deaths observed before the festival than expected, and more deaths afterward. This pattern suggests that participants indeed managed to postpone their deaths until after the festival. This difference between the observed and expected counts is crucial to the calculation of the chi-square statistic and the resulting p-value, which will quantify the statistical significance of the findings.

Control Group in Research

The control group forms the cornerstone of experimental research design. By providing a baseline, researchers can isolate the factor or treatment of interest—the variable they believe might cause a change. In studies like the one examining death rates around the Harbor Moon Festival, the control group consists of a similar population of elderly people who are not of Chinese descent and do not celebrate the festival.

This group is essential for comparison because if both the tested and control groups show no significant difference, then it is likely that the festival itself has no real impact on mortality rates. However, if the effect is seen in the test group (Chinese descent celebrating the festival) but not in the control group, it corroborates the hypothesis that the festival may have an influence. Control groups help ensure that the results observed are due to the independent variable (the festival) and not other extraneous factors.