Question

Insects hovering in flight expend enormous amounts of energy for their size and weight. The data shown here were taken from a much larger body of data collected by T.M. Casey and colleagues. \({ }^{15}\) They show the wing stroke frequencies (in hertz) for two different species of bees, \(n_{1}=4\) Euglossa mandibularis Friese and \(n_{2}=6\) Euglossa imperialis Cockerell. $$ \begin{array}{cc} \text { E. mandibularis Friese } & \text { E. imperialis Cockerell } \\ \hline 235 & 180 \\ 225 & 169 \\ 190 & 180 \\ 188 & 185 \\ & 178 \\ & 182 \end{array} $$ a. Based on the observed ranges, do you think that a difference exists between the two population variances? b. Use an appropriate test to determine whether a difference exists. c. Explain why a Student's \(t\) -test with a pooled estimator \(s^{2}\) is unsuitable for comparing the mean wing stroke frequencies for the two species of bees.

Short answer

Answer: A Student's t-test with a pooled estimator is unsuitable for comparing the mean wing stroke frequencies for the two species of bees because it assumes that both populations have equal variances. In this case, we have already determined that there is a significant difference between the population variances of the two species. Therefore, using a test that assumes equal variances, like the Student's t-test with a pooled estimator, would not provide an accurate comparison of the means. A more appropriate test in this case would be the Welch's t-test, which considers unequal variances when comparing two population means.

Step-by-step solution

Analyze Observed Ranges

First, we need to calculate the ranges for both species of bees. The range is calculated as the difference between the maximum and minimum values. For E. mandibularis Friese: Range = Max - Min = 235 - 188 = 47 For E. imperialis Cockerell: Range = 185 - 169 = 16 Based on the observed ranges of the wing stroke frequencies for both species 47 and 16, we can say that there might be a difference between the two population variances.

Perform an F-test

To determine whether a difference exists between the population variances, we will perform an F-test for equality of variances. First, we need to calculate the sample variance and degrees of freedom for each species. For E. mandibularis Friese: \\ Sample size: \(n_1=4\) \\ Sample variance: \(s_1^2=\frac{1}{n_1-1}((235-209.75)^2+(225-209.75)^2+(190-209.75)^2+(188-209.75)^2)=600.9167\) \\ Degrees of freedom: \(df_1=n_1-1=3\) For E. imperialis Cockerell: \\ Sample size: \(n_2=6\) \\ Sample variance: \(s_2^2=\frac{1}{n_2-1}((180-179)^2+(169-179)^2+(180-179)^2+(185-179)^2+(178-179)^2+(182-179)^2)=32.16\) \\ Degrees of freedom: \(df_2=n_2-1=5\) Now, we can calculate the F-value: \\ \(F=\frac{s_1^2}{s_2^2}= \frac{600.91}{32.16}= 18.68\) Next, we will use an F-table to find the critical value. We need to choose a significance level for our test; let's use \(α=0.05\). The critical value is found at the intersection of the row with \(df_1=3\) and column with \(df_2=5\). For our chosen α, the critical value \(F_{critical}=6.16\). Since the calculated F-value (18.68) is greater than the critical value (6.16), we reject the null hypothesis that there is no difference between the two population variances.

Inappropriateness of Student's t-test with a Pooled Estimator

The Student's t-test with a pooled estimator is used to compare means and assumes that both populations have equal variances. As we have already determined that there is a significant difference between the population variances of the two species, a Student's t-test with a pooled estimator would be unsuitable for comparing the mean wing stroke frequencies for the two species of bees. Instead, a more appropriate test would be the Welch's t-test, which considers unequal variances when comparing two population means.

Key concepts

F-test

The F-test is a statistical procedure used to determine if there are differences between the variances of two populations. Variance measures how much the data in a set is spread out. To perform an F-test, we compare the ratio of the two sample variances. The formula is simple:

• Calculate the variances of both samples, denoted as \(s_1^2\) and \(s_2^2\).
• The F-value is calculated as \(F = \frac{s_1^2}{s_2^2}\).

If this F-value is higher than a critical value from the F-distribution table (for a given significance level \(\alpha\)), it indicates that the variances are significantly different. A crucial point when using the F-test is selecting the appropriate critical value based on the degrees of freedom from each sample. This helps in making an informed decision about the null hypothesis, which posits the variances are equal.

Population variance

Population variance is a measure used to quantify the amount of variation or dispersion in a population dataset. It tells us how much individual values differ from the mean of the dataset. The formula for population variance \( \sigma^2 \) is:\[ \sigma^2 = \frac{1}{N} \sum_{i=1}^{N}(X_i - \mu)^2 \]where:

• \(N\) is the number of data points in the population.
• \(X_i\) represents each data point.
• \(\mu\) is the mean of the population data.

Population variance gives insight into how varied a dataset is and is fundamental in determining if datasets have significantly different spreads. Smaller variance implies that data points tend to be closer to the mean, while larger variance indicates more spread out data.

Student's t-test

A Student's t-test is a statistical test used to compare the means of two groups to see if they are statistically significantly different from each other. There are assumptions when using the t-test:

• Data should be approximately normally distributed.
• The sample sizes should ideally be equal, and the variances should be equal when using a pooled t-test estimator.

The standard formula is:\[ t = \frac{\bar{X}_1 - \bar{X}_2}{s_p\sqrt{(\frac{1}{n_1} + \frac{1}{n_2})}} \]where:

• \(\bar{X}_1\) and \(\bar{X}_2\) are the sample means.
• \(s_p\) is the pooled standard deviation.
• \(n_1\) and \(n_2\) are the sizes of the two samples.

However, if the sample variances are not equal, as illustrated in this exercise, a Student's t-test with a pooled estimator is not suitable. That’s where Welch's t-test comes into play. It is particularly useful when variances depart from equality.

Welch's t-test

Welch's t-test is an adaptation of the Student's t-test which is more robust in situations where the sample sizes and variances are unequal between two groups. Unlike the Student's t-test, it does not assume equal variances. This makes it especially useful when comparing populations with markedly different variations, as was the case in our exercise.The formula for Welch's t-test modifies the degrees of freedom specifically for unequal variances:\[ t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\left(\frac{s^2_1}{n_1}\right) + \left(\frac{s^2_2}{n_2}\right)}} \]where the degrees of freedom \(df\) are calculated using:\[ df = \frac{\left(\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}\right)^2}{\frac{\left(\frac{s^2_1}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s^2_2}{n_2}\right)^2}{n_2 - 1}} \]This flexibility in analyzing datasets with different variances makes Welch's t-test highly valuable in many practical scenarios. While it might seem slightly more complex than a simple t-test, the added accuracy in results is worth it, especially when variabilities are significant.