Question

Metal Tags on Penguins and Arrival Dates Data 1.3 on page 10 discusses a study designed to test whether applying a metal tag is detrimental to a penguin, as opposed to applying an electronic tag. One variable examined is the date penguins arrive at the breeding site, with later arrivals hurting breeding success. Arrival date is measured as the number of days after November \(1^{\text {st }}\). Mean arrival date for the 167 times metal-tagged penguins arrived was December \(7^{\text {th }}\left(37\right.\) days after November \(\left.1^{\text {st }}\right)\) with a standard deviation of 38.77 days, while mean arrival date for the 189 times electronic-tagged penguins arrived at the breeding site was November \(21^{\text {st }}(21\) days after November \(\left.1^{\text {st }}\right)\) with a standard deviation of \(27.50 .\) Do these data provide evidence that metal tagged penguins have a later mean arrival time? Show all details of the test.

Short answer

Without the numerical results from the formulas, the final answer cannot be given. However, after following these steps, if the p-value found is less than the commonly used significance level of 0.05, then one can conclude that there is significant evidence that metal-tagged penguins have a later mean arrival time. If the p-value found is greater than 0.05, then one can conclude that there is not enough statistical evidence to support that the mean arrival time of metal-tagged penguins is later.

Step-by-step solution

Formulate the Hypotheses

The null hypothesis (\(H_0\)) is that the mean arrival times of metal-tagged and electronic-tagged penguins are equal, i.e., there is no difference in arrival times based on the type of tag. The alternative hypothesis (\(H_a\)) is that the mean arrival time of metal-tagged penguins is greater than the mean arrival time of electronic-tagged penguins.

Calculate the Test Statistic and Degrees of Freedom

The test statistic is calculated with the formula for a two-sample t-test. \(t = \frac{\overline{X1} - \overline{X2}}{\sqrt{\frac{s1^2}{n1} + \frac{s2^2}{n2}}}\), where \(\overline{X1}\) and \(\overline{X2}\) are the sample means, \(s1\) and \(s2\) are the sample standard deviations, and \(n1\) and \(n2\) are the sample sizes. These values are provided in the problem. Also, the degrees of freedom (df) should be calculated. The approximate df for a two-sample t-test (with unequal variances) is calculated by using the formula: \(df = \frac{(\frac{s1^2}{n1} + \frac{s2^2}{n2})^2} {\frac{(s1^2 / n1)^2}{n1 - 1} + \frac{(s2^2 / n2)^2}{n2 - 1}}\).

Find the P-Value and Make a Decision

With the calculated t-value and df, find the corresponding p-value. If the p-value is less than the significance level (usually 0.05), reject the null hypothesis in favor of the alternative hypothesis, concluding that there is significant evidence that metal-tagged penguins have a later mean arrival time. If the p-value is greater than the significance level, fail to reject the null hypothesis, concluding that there is not enough statistical evidence to support that metal-tagged penguins arrive later than electronic-tagged penguins at the breeding site.

Key concepts

Null Hypothesis

The null hypothesis, denoted as \(H_0\), is a fundamental concept in statistical testing which posits that there is no effect or no difference in the general case. In our penguin study, the null hypothesis is that the mean arrival dates for both metal-tagged and electronic-tagged penguins are the same. In essence, it provides a baseline or standard that we assume to be true until evidence suggests otherwise.

Understanding and accurately stating the null hypothesis is crucial because it's the assertion that the study is designed to test. When performing a two-sample t-test, the goal is to examine if the data gathered provide strong enough evidence to reject this hypothesis in favor of a more interesting one – the alternative hypothesis.

Alternative Hypothesis

The alternative hypothesis, \(H_a\), is the hypothesis that researchers really want to test. It represents a new theory or belief that contradicts the null hypothesis. For our study on penguins, the alternative hypothesis suggests that the mean arrival time of metal-tagged penguins is greater than electronic-tagged penguins, indicating a potential negative effect of the metal tags on arrival times.

It's important to define the alternative hypothesis clearly because it directs the statistical test and ultimately determines if the research provides new insights into our understanding of the subject. In this case, the alternative hypothesis is one-sided because we're testing if one mean is greater than the other, not simply different.

Statistical Significance

Statistical significance is a determination of whether the results of a study are likely to be due to chance or to some factor of interest. In hypothesis testing, we often use a significance level, usually denoted as \(\alpha\), to make this determination. A common value for \(\alpha\) is 0.05 or 5%.

If we obtain results that would occur less than 5% of the time under the null hypothesis, we consider the findings statistically significant. This would suggest that the effect or difference observed (e.g., later arrival times for metal-tagged penguins) is likely not due to random variation, but instead due to the impact of the factor being studied.

Test Statistic

The test statistic is a number calculated from the data that is used to evaluate how compatible the data are with the null hypothesis. In the two-sample t-test for the penguin study, the test statistic is essentially the standardized difference between the two sample means.

The formula for the test statistic considers the sample means, standard deviations, and sizes. In the context of our example, it shows how far the observed difference in arrival times is from the expected difference under the null hypothesis (which is zero), in terms of the standard deviation of the differences of sample means.

Degrees of Freedom

Degrees of freedom, often denoted as df, is a concept tied to the reliability of an estimate. In the two-sample t-test, degrees of freedom refer to the number of independent pieces of information from the data that are available to estimate variability.

The formula to calculate degrees of freedom in a two-sample t-test with unequal variances involves both sample sizes and variances. The degrees of freedom affect the shape of the t-distribution that is used to determine the critical values of the test and, consequently, the p-value. Understanding how to calculate and interpret degrees of freedom is essential for correct hypothesis testing.

P-Value

The p-value is a probability that measures the evidence against the null hypothesis provided by the data. It quantifies how likely it is to observe the test statistic as extreme as, or more extreme than, the value calculated from the sample data, assuming the null hypothesis is true.

In the penguin study, if the p-value is small (typically less than our significance level of 0.05), it indicates strong evidence against the null hypothesis. Interpreting p-values correctly is fundamental in deciding whether to reject the null hypothesis or not. An appropriately low p-value would lead us to believe that the metal tags could indeed be causing later arrival times for the penguins.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this context, it tells us how much the arrival dates for penguins deviate from the mean arrival date. Lower standard deviation means the dates are closely clustered around the mean, while higher standard deviation indicates that the dates are spread out over a wider range.

Understanding standard deviation is vital because it's used in the calculation of the test statistic and consequently affects the t-test's outcome. In the provided penguin study data, the standard deviations help us understand the variability in arrival times for each group of tagged penguins.