91Ó°ÊÓ

Refer to the description in exercise 10.71. There were 22 trials with only cockroaches (no robots) that went under one shelter. In 16 of these 22 trials, the group chose the darker shelter, and in 6 of the 22 the group chose the lighter shelter. There were 28 trials with a mixture of real cockroaches and robots that all went under one shelter. In 11 of these trials, the group chose the darker shelter, and in 17 the group chose the lighter shelter. The robot cockroaches were programmed to choose the lighter shelter (as well as preferring groups; Halloy et al. 2007 ) $$ \begin{array}{lcc} & \text { All under One Shelter } \\ \hline & \text { Cockroaches Only } & \text { Robots Also } \\ \hline \text { Darker } & 16 & 11 \\ \text { Lighter } & 6 & 17 \\ \hline \end{array} $$ Is the introduction of robot cockroaches associated with the type of shelter when the group went under one shelter? Assume cockroaches were randomly sampled from some meaningful population of cockroaches. a. Use the chi-square test to see whether the presence or absence of robots is associated with whether they went under the darker or the brighter shelter. Use a significance level of \(0.05\) b. Do Fisher's Exact Test with the data. If your software does not do Fisher's Exact Test, search the Internet for a Fisher's Exact Test calculator and use it. Report the p-value and your conclusion. c. Compare the \(\mathrm{p}\) -values for parts a and \(\mathrm{b}\). Which do you think is the more accurate procedure? The p-values that result from the two methods in this question are closer than the p-values in the previous question. Why do you think that is?

Step by step solution

01

Setting Up the Hypotheses

The null hypothesis H0: Cockroach preference for shelter is independent of the presence or absence of robot cockroaches. The alternative hypothesis H1: Cockroach preference for shelter is dependent on the presence or absence of robot cockroaches.

02

Compute Expected Frequencies

Under the null hypothesis, the expected frequencies for each cell can be calculated using formula, \[Expected Frequency = (Row Total * Column Total) / Grand Total \]

03

Compute Chi-Square Test Statistic

The chi-square statistic is calculated using the formula: \[X^2 = \sum \frac{(Observed Frequency - Expected Frequency)^2}{Expected Frequency}\]. This step results in a test statistic that must be compared with a chi-square distribution with 1 degree of freedom (because (2-1) * (2-1)). The p-value is obtained and compared with the significance level of 0.05.

04

Fisher's Exact Test

The Fisher's exact test is performed. Fisher's exact test is used in the analysis of contingency tables when sample sizes are small. The test is usually performed through a statistical software or Fisher's exact test calculator available online.

05

Compare p-values and Conclude

The findings from chi-square test and Fisher's exact test are compared to draw a conclusion about the association of the type of cockroaches and preference of shelter type.

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Key Concepts

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

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing, where it serves as the default assumption that there is no difference or no association between two measured phenomena. In the context of our exercise, the null hypothesis (H_0) asserts that the real cockroaches' shelter choice is independent of whether robot cockroaches are present or not. This means that the observed difference in the preference for darker or lighter shelters with and without the robots could be due to random chance rather than a real effect of the robot cockroaches.

To test this hypothesis, one performs a statistical test to examine the observed data against what we would expect to happen if the null hypothesis were true. If the test shows a low probability that the observed outcomes are due to chance (a result deemed statistically significant), we may reject the null hypothesis in favor of the alternative hypothesis (H_1), which suggests that there is indeed an association.

Expected Frequencies

In the realm of chi-square tests, expected frequencies play a vital role. They represent the theoretical frequency of occurrences in each category if the null hypothesis were true - that is, if no difference exists between groups or no association is present between variables. The calculation involves taking the product of the corresponding row and column totals and then dividing by the grand total of all observations.

For instance, to calculate the expected frequency of cockroaches choosing a darker shelter when only cockroaches are present, one would multiply the total number of trials with only cockroaches by the total number of times any group chose the darker shelter, then divide by the overall number of trials. This figure is crucial because in the chi-square test, the observed frequencies are compared to these expected frequencies to understand if the deviations are likely due to random variation or if they point to a meaningful phenomenon.

Fisher's Exact Test

Fisher's exact test is an alternative to the chi-square test, particularly useful when dealing with small sample sizes or when the expected frequencies in a contingency table are very low, typically less than 5. Unlike the chi-square test, it doesn't rely on approximations and instead calculates an exact p-value based on the hypergeometric distribution.

The test works well for 2x2 contingency tables and considers all possible distributions of the data that are as extreme as the observed distribution in terms of probability. While it can be computationally intensive for larger tables, for small datasets, it provides a more accurate assessment regarding the null hypothesis. With the increasing availability of computational resources and software, applying Fisher's exact test has become a convenient and reliable method to assess statistical significance in scenarios where the chi-square test might not be appropriate.

Statistical Significance

Statistical significance is a determination about the non-randomness of the observed result. It is evaluated using a p-value, which measures the probability of obtaining an observation at least as extreme as the one observed if the null hypothesis is true. A p-value lower than the chosen significance threshold (commonly set at 0.05) signals that the observed data are unlikely under the null hypothesis, and thus we may consider our results statistically significant.

In our exercise, we compare the p-value from the chi-square test to the significance level of 0.05. If the p-value is lower, we have grounds to reject the null hypothesis, which would indicate an association between the presence of robot cockroaches and the choice of shelter by the cockroaches. It is vital, however, to interpret p-values carefully since they do not measure the size or importance of an effect, only how likely the effect is under the null hypothesis.

Contingency Table Analysis

Contingency table analysis is a method to analyze the relationship between two categorical variables. Our exercise involves a 2x2 contingency table, showing frequencies of shelter choices (darker or lighter) with two different conditions (presence or absence of robot cockroaches).

In such an analysis, one tabulates the observations into a table format, which organizes the data to highlight discrepancies between observed and expected frequencies under the null hypothesis. For each cell in the contingency table, the expected frequency is calculated based on the total observations of rows and columns, as stated previously. Analyzing these tables with chi-square tests or Fisher's exact test allows researchers to probe the evidence for independence or association between the variables, providing a visual and statistical way to understand and interpret complex data sets.