Question

Metal Tags on Penguins In Exercise 6.148 on page 445 we perform a test for the difference in the proportion of penguins who survive over a ten-year period, between penguins tagged with metal tags and those tagged with electronic tags. We are interested in testing whether the type of tag has an effect on penguin survival rate, this time using a chi-square test. In the study, 10 of the 50 metal-tagged penguins survived while 18 of the 50 electronic-tagged penguins survived. (a) Create a two-way table from the information given. (b) State the null and alternative hypotheses. (c) Give a table with the expected counts for each of the four categories. (d) Calculate the chi-square test statistic. (e) Determine the p-value and state the conclusion using a \(5 \%\) significance level.

Short answer

We fail to reject the null hypothesis. There is no statistically significant evidence to suggest that the type of tag has an effect on penguin survival rate.

Step-by-step solution

Create a Two-way Table

Firstly, create a 2x2 table with rows for 'Metal tag' and 'Electronic tag', and columns for 'Survived' and 'Did not survive'. Our data gives us these figures: 'Metal tag' - Survived: 10, Did not survive: 40; 'Electronic tag' - Survived: 18, Did not survive: 32.

State the Hypotheses

The null hypothesis (\(H_0\)) is that the type of tag does not have an effect on penguin survival. The alternative hypothesis (\(H_A\)) is that the type of tag does have an effect on penguin survival.

Calculate Expected Counts

We calculate the expected count for each cell in the table using the formula \(((\text{row total})(\text{column total}))/(\text{grand total}\). Therefore, the expected counts are: 'Metal tag' - Survived: 14, Did not survive: 36; 'Electronic tag' - Survived: 14, Did not survive: 36.

Calculate the Chi-square Statistic

We then use these expected counts to calculate the chi-square test statistic using the formula: \(\chi^2 = \sum((\text{observed} - \text{expected})^2/\text{expected}\). This gives us: \(\chi^2 = (10-14)^2/14 + (40-36)^2/36 + (18-14)^2/14 + (32-36)^2/36 = 1.143\).

Determine the p-value and Conclusion

Finally, we use a chi-square distribution table to find the p-value corresponding to our chi-square test statistic and 1 degree of freedom (because (2-1)(2-1) = 1). This gives us a p-value around 0.285, which is greater than our significance level of 0.05. Thus, we fail to reject the null hypothesis - we do not have enough evidence to suggest the type of tag has an effect on penguin survival rate.

Key concepts

Penguin Survival Rate

Understanding the factors affecting penguin survival rate is vital for conservation efforts. In our scenario, we compared the survival rates of penguins tagged with different types of tracking devices: metal tags and electronic tags. The survival rate within any study represents the proportion of the subject group—penguins in this case—that survive over a specified period. Such rates are crucial data points for wildlife biologists and environmental scientists who aim to draw conclusions about the effects of external factors, like tagging methods, on an animal population's chances of surviving and reproducing.

In the provided study, survival rates are presented in a clear manner by showing the number of survivors out of the total number tagged for each tag type. This creates a fundamental base to perform statistical analysis, such as the chi-square test, to determine if there's a significant difference between the two tagging methods in relation to their impact on penguin survival.

Statistical Hypothesis Testing

Statistical hypothesis testing is a method for scientists to make decisions about a population based on sample data. In essence, it involves setting up two competing hypotheses—the null hypothesis (\(H_0\)) posits that there is no effect or difference, whereas the alternative hypothesis (\(H_A\)) suggests that there is an effect or difference.

Applied to our exercise, the null hypothesis represents the standpoint that the type of tag, metal or electronic, does not influence penguin survival. Conversely, the alternative hypothesis aligns with the notion that tag type does indeed influence survival rates. Using the chi-square test, researchers can objectively decide whether to reject the null hypothesis or fail to reject it based on the p-value generated from the test statistic and predetermined significance level.

Two-Way Table

Two-way tables are a form of data presentation that helps statisticians organize and disentangle relationships between two categorical variables. For our penguin study, the two-way table separates penguins into groups based on the type of tag and their survival status. With rows and columns designating the different categories, such tables enable researchers to perform the chi-square test by providing a clear visualization of the observed frequencies needed in the calculation.

Moreover, two-way tables are instrumental for summarizing large datasets, allowing for an easy extraction of important information like row totals, column totals, and the grand total, which are necessary for determining expected counts when conducting a chi-square test.

Expected Count Calculation

In a chi-square test, expected counts play a central role; they represent the frequencies we would expect to observe if the null hypothesis were true. These counts are calculated based on the assumption that there’s no association between the categorical variables under consideration. For each cell in a two-way table, the expected count can be calculated using the formula \((\text{row total})(\text{column total}) / (\text{grand total})\).

For instance, in the penguin study with metal and electronic tags, by applying this formula, we establish the expected count for each group had there been no effect of the tag type on survival. This is a fundamental step because the chi-square statistic is ultimately a sum of the squared differences between observed and expected counts, relative to the expected counts.

Significance Level

The significance level is a threshold set by researchers to determine whether to reject the null hypothesis. Commonly denoted by \( \alpha \), it's a measure of how much risk we are willing to take of making a Type I error—rejecting the null hypothesis when it is actually true. A typical significance level is \(5\%\), or 0.05, which indicates a 5% risk of concluding that a difference exists when there is, in fact, no actual difference.

In the scenario given, after calculating the chi-square test statistic, the resulting p-value is compared to this significance level. If the p-value is lower than the significance level, it suggests that the observed data is unlikely under the null hypothesis, leading us to reject the null hypothesis. Conversely, if the p-value is higher, then we fail to reject the null hypothesis, concluding that the data does not provide strong evidence against it—in our case, asserting that tag type does not significantly impact penguin survival.