Question

The eating habits of 12 bats were examined in the article "Foraging Behavior of the Indian False Vampire Bat" (Biotropica [1991]: \(63-67)\). These bats consume insects and frogs. For these 12 bats, the mean time to consume a frog was \(\bar{x}=21.9\) minutes. Suppose that the standard deviation was \(s=7.7\) minutes. Is there convincing evidence that the mean supper time of a vampire bat whose meal consists of a frog is greater than 20 minutes? What assumptions must be reasonable for the one-sample \(t\) test to be appropriate?

Short answer

First, calculate the test statistic from the sample mean, sample standard deviation and present hypotheses. Then, compute the P-value, keeping in mind that it is a one-tailed test. If the P-value is less than or equal to 0.05, there is convincing evidence that the mean supper time of a vampire bat is greater than 20 minutes. If the P-value is greater than 0.05, there is not enough evidence to reject the null hypothesis, and it may not be longer than 20 minutes. Values for the test statistic and P-value are not provided, but should be calculated based on the given steps and information.

Step-by-step solution

State the Hypotheses

The null hypothesis \(H_0\) is that the mean supper time of the bats is 20 minutes: \(H_0: \mu = 20\). The alternative hypothesis \(H_a\) is that the mean supper time is greater than 20 minutes: \(H_a: \mu > 20\).

Compute the Test Statistic

The test statistic for the one-sample \(t\) test is given by \(T = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(s\) is the sample standard deviation, \(n\) is the number of observations and \(\mu_0\) is the presumed population mean under the null hypothesis. Here, \(\bar{x} = 21.9, s = 7.7, n = 12, \mu_0 = 20\). So, \(T = \dfrac{21.9 - 20}{7.7/\sqrt{12}}\).

Calculate the P-value

The \(P\)-value is the probability of observing a test statistic as extreme as the one calculated, under the null hypothesis. In this case, as the alternative hypothesis is that the mean supper time is greater than 20 minutes, the P-value is the probability of observing a \(t\) value as extreme as calculated or more, in the right tail of the t-distribution with \(n-1\) degrees of freedom. We calculate \(P(T > t)\) where \(t\) is the calculated test statistic value.

Interpret Results

If the \(P\)-value is less than or equal to a significance level (often 0.05), then we conclude that there is convincing evidence to reject the null hypothesis in favor of the alternative; thus concludes that the mean supper time of a vampire bat whose meal consists of a frog is greater than 20 minutes. If the \(P\)-value is greater than 0.05, there isn't enough evidence to reject the null hypothesis, hence we do not have convincing evidence that the mean supper time is over 20 minutes.