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Most salamanders of the species \(P\). cinereus are red striped, but some individuals are all red. The all-red form is thought to be a mimic of the salamander \(N\). viridescens, which is toxic to birds. In order to test whether the mimic form actually survives more successfully, 163 striped and 41 red individuals of \(P\). cinereus were exposed to predation by a natural bird population. After 2 hours, 65 of the striped and 23 of the red individuals were still alive. \({ }^{4}\) Use a chi-square test to assess the evidence that the mimic form survives more successfully. Use a directional alternative and let \(\alpha=0.05 .\) (a) State the null hypothesis in words. (b) State the null hypothesis in symbols. (c) Compute the sample survival proportions for each group and display the values in a table similar to Table 10.1 .2 (d) Find the value of the test statistic and the \(P\) -value. (e) State the conclusion of the test in the context of this setting.

Short Answer

Expert verified
The null hypothesis H0: p1 = p2 is not supported, as a directional chi-square test may show a P-value below 0.05, suggesting that the red form does indeed have a survival advantage over the striped form when exposed to predation by birds.

Step by step solution

01

- Stating the Null Hypothesis in Words

The null hypothesis in words is that there is no difference in survival rates between the striped and red forms of the salamander species P. cinereus when exposed to natural bird predation.
02

- Stating the Null Hypothesis in Symbols

The null hypothesis in symbols can be represented as: H0: p1 = p2, where p1 is the survival rate of the striped form and p2 is the survival rate of the red form of P. cinereus.
03

- Computing Sample Survival Proportions

To compute the sample survival proportions, divide the number of surviving individuals by the total number of individuals for each group.Striped survival proportion = Number of surviving striped / Total striped = 65 / 163Red survival proportion = Number of surviving red / Total red = 23 / 41.
04

- Display Survival Proportions in a Table

Create a table to display the survival proportions for each group.| Form | Survivors | Total | Proportion ||------------|-----------|-------|------------|| Striped | 65 | 163 | 65/163 || Red | 23 | 41 | 23/41 |
05

- Calculate Expected Counts

Calculate the expected number of survivors for each form if the null hypothesis were true, using the combined survival rate and the total numbers in each group.Combined survival rate = (Total survivors) / (Total salamanders) = (65 + 23) / (163 + 41)Expected striped survivors = Combined survival rate * Total stripedExpected red survivors = Combined survival rate * Total red
06

- Calculate Chi-Square Test Statistic

Use the formula for the chi-square test statistic: X^2 = Σ [(Observed - Expected)^2 / Expected], separately for each group and sum the values. Since this is a test for two proportions, the test statistic can also be calculated using a chi-square test for independence or homogeneity with appropriate adjustments for directionality.
07

- Find the P-Value

Use the calculated chi-square test statistic to find the P-value by comparing it to the chi-square distribution with one degree of freedom, taking into account the direction of the alternative hypothesis which is looking for the survival advantage of the red form.
08

- Conclusion of the Test

Based on the comparison of the P-value to the significance level α = 0.05, determine whether to reject or fail to reject the null hypothesis. If P-value ≤ α, reject the null hypothesis and conclude that there is evidence to suggest the red form has a survival advantage. If P-value > α, fail to reject the null hypothesis, suggesting no evidence of a survival difference between forms in the context of predation by birds.

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Key Concepts

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

Null Hypothesis
In any statistical analysis, understanding the null hypothesis is critical. It serves as the starting assumption for statistical tests, including the chi-square test mentioned in our salamander example. The null hypothesis posits that there is no effect or no difference between groups; in this case, no difference in survival rates between the striped and red forms of the salamander species P. cinereus when exposed to bird predation. Symbolically, it's expressed as H0: p1 = p2, where p1 and p2 represent the survival rates of the striped and red salamanders, respectively.

To assess this null hypothesis, researchers use actual data from experiments or observations. If the evidence suggests that the differences observed are too great to have occurred merely by chance, the null hypothesis can be rejected. However, rejecting the null hypothesis doesn't prove the existence of an effect; rather, it suggests that our data are inconsistent with the assumption of no effect and that further investigation may be warranted.
Survival Rate Comparison
In studies like the one involving the salamander species P. cinereus, a survival rate comparison is essential for evaluating the success of different forms or treatments. Such comparisons typically involve calculating the proportion of individuals surviving in each group. For our salamanders, this meant dividing the number of survivors by the total number of salamanders in both the striped and red groups (e.g., 65/163 for striped and 23/41 for red).

Once we determine these proportions, we express them in a table to clearly present our findings. Comparing these rates helps us understand whether the mimicry exhibited by the red salamanders gives them a survival advantage. If the observed survival rates significantly differ, researchers may attribute this discrepancy to the mimicry's effectiveness against predation. However, such conclusions hinge on the statistical significance of the observed differences, which is where the chi-square test plays a crucial role.
Statistical Significance
When researchers talk about statistical significance, they refer to the likelihood that the observed difference between groups is not due to random chance. This concept is intimately tied to the p-value, which in our salamander study, comes from the chi-square test. The p-value represents the probability of observing results as extreme as, or more extreme than, the ones obtained if the null hypothesis were true.

In our example, if the p-value is less than or equal to 0.05 (the chosen significance level α), we reject the null hypothesis, suggesting that the survival advantage for the red salamanders is statistically significant—it's unlikely that such results would have occurred if there were no real difference in survival rates. Conversely, a p-value greater than 0.05 implies that the observed difference could likely be due to random chance, providing insufficient evidence to support the claim that the mimic form survives more successfully.

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