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Cockroaches tend to rest in groups and prefer dark areas. In a study by Halloy et al. published in Science Magazine in November 2007 , cockroaches were introduced to a brightly lit, enclosed area with two different available shelters, one darker than the other. Each time a group of cockroaches was put into the brightly lit area will be called a trial. When groups of 16 real cockroaches were put in a brightly lit area, in 22 out of 30 trials, all the cockroaches went under the same shelter. In the other 8 trials, some of the cockroaches went under one shelter, and some under the other one. Another group consisted of a mixture of real cockroaches and robot cockroaches ( 4 robots and 12 real cockroaches). The robots did not look like cockroaches but had the odor of male cockroaches, and they were programmed to prefer groups (and brighter shelters). There were 30 trials. In 28 of the trials, all the cockroaches and robots rested under the same shelter, and in 2 of the trials they split up. $$\begin{array}{|l|c|c|}\hline & \text { Cockroaches Only } & \text { Robots Also } \\ \hline \text { One Shelter Used } & 22 & 28 \\\\\hline \text { Both Shelters Used } & 8 & 2 \\ \hline\end{array}$$ Is the inclusion of robots associated with whether they all went under the same shelter? To answer the following questions, assume the cockroaches are a random sample of all cockroaches. a. Use a chi-square test for homogeneity with a significance level of \(0.05\) to see whether the presence of robots is associated with whether roaches went into one shelter or two. b. Repeat the question using Fisher's Exact Test. (If your software will not perform the test for you, search for Fisher's Exact Test on the Internet to do the calculations.) Conduct a two-sided hypothesis test so that the test is consistent with the test in part a. c. Compare the p-values and conclusions from part a and part b. Which statistical test do you think is the better procedure in this case? Why?

Step by step solution

01

Setup null and alternative hypotheses

Let's start by establishing our null hypothesis (H0) and alternative hypothesis (H1). H0 assumes that there is no association between the presence of robots and the preference of roaches for one or two shelters i.e., both scenarios are independent. On the other hand, H1 assumes that there is an association i.e., the two scenarios are not independent.

02

Chi-square test calculation and finding p-value

We apply Chi-square test for homogeneity to check if robots have any effect on choice of roaches. We first calculate the expected frequency for all the entries in the 2x2 table and then we calculate the \( \chi^2 \) test-statistic using the formula \[ \chi^2 = \sum {\frac {(O_i - E_i)^2}{E_i}} \] where O represents observed frequency and E represents expected frequency. After calculating the \( \chi^2 \) value, we find the p-value using the chi-square distribution with 1 degree of freedom.

03

Comparing the p-value with the significance level

We compare the calculated p-value with the significance level (0.05). If the p-value is less than or equal to 0.05, it means there is strong evidence against H0 and we thus reject the null hypothesis confirming that the presence of robots is associated with the shelter preference. If not, we fail to reject H0, indicating that the presence of robots is not associated with the shelter preference.

04

Fisher's Exact Test

We perform Fisher's Exact Test, which is particularly useful when sample sizes are small. Fisher's Exact Test also results in a p-value which we compare with the significance level, in the same way as with the Chi-square test. Note that doing this by hand can be quite difficult and is usually done with a calculator or statistical software.

05

Comparing the results of the two tests

By comparing the p-values from Chi-square test and Fisher's exact test, we can infer about the association between shelter choice of roaches and the presence of robots. While Fisher's Exact Test is more accurate especially for small sample sizes, Chi-square test is good approximation for larger data sets. The better procedure would be based on the p-values and the sample size.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fisher's Exact Test

Fisher's Exact Test is a statistical significance test used to examine the association between two categorical variables from a contingency table when sample sizes are small. Unlike the Chi-square test, Fisher's test does not rely on approximations of the sampling distribution and is, therefore, exact.
It is particularly helpful when the dataset contains cells with expected frequencies less than 5, which is often considered a limitation for the Chi-square test. The test calculates the probability of obtaining the observed data or data more extreme than observed, given that the null hypothesis is true.
In the context of the cockroach and robot study, Fisher's Exact Test calculates the likelihood that the observed distribution of cockroaches across shelters occurred by random chance if the behavior of cockroaches was truly independent of the presence or absence of robot cockroaches.

Calculating Fisher's Exact Test

To perform Fisher's Exact Test, we need to determine all possible combinations of the data that would provide the same marginals as the observed data. Then, we calculate a p-value using these combinations, which represents the probability of observing a result as or more extreme than the current one under the assumption of the null hypothesis. Statistical software is often used for these calculations due to their complexity.

Statistical Significance

Statistical significance is a term used to determine if an observed effect in a study is likely due to something other than random chance. It's often expressed as a p-value, which quantifies the probability of observing data as extreme as the data actually observed, under the assumption that the null hypothesis is true.
The commonly used significance level to determine statistical significance is 0.05, meaning that there is a 5% chance or less that the observed results are due to random variation. In our study with cockroaches, if we obtain a p-value less than 0.05, it indicates a statistically significant association between the presence of robots and cockroach shelter preferences, suggesting that the observation is unlikely to be due to chance alone.
A p-value exceeding this threshold implies that our observations might very well be a result of random fluctuations, and we do not have enough evidence to reject the null hypothesis. It’s important when interpreting statistical significance to also consider the practical significance and context of the results.

Null Hypothesis

The null hypothesis (H0) in a statistical test is a statement of no effect or no difference. It is the assumption that any observations are the result of pure chance. This hypothesis serves as a starting point for statistical testing and is presumed true until evidence suggests otherwise.
The null hypothesis for the cockroach and robot experiment posits that the inclusion of robots has no impact on whether roaches rest under one shelter or split into both. Essentially, it assumes the proportion of trials resulting in either outcome should be the same whether robots are present or not.
Statistical tests, including both the Chi-square test for homogeneity and Fisher's Exact Test, are designed to evaluate the likelihood of the observed data under the assumption that the null hypothesis is true. Only when there is sufficient evidence to suggest a low probability of the data occurring by random chance, statisticians consider rejecting the null hypothesis.

Alternative Hypothesis

The alternative hypothesis (H1 or Ha) directly counters the null hypothesis by stating that there is a statistically significant effect or difference between groups or variables in your study. It represents the outcome that the study is trying to demonstrate.
In the case of the cockroach study, the alternative hypothesis asserts an association between the presence of the robot cockroaches and the shelter choices of the real cockroaches. If the data supports this hypothesis, it suggests that the behavior of the cockroaches is influenced by the robot cockroaches, leading them to prefer one shelter over splitting up.
The aim of our statistical tests is to determine whether the evidence calls for rejecting the null hypothesis in favor of the alternative hypothesis. If our tests result in a low p-value, this indicates that the evidence against the null hypothesis is strong, and we may move to accept the alternative hypothesis.