Question

Scientists studying lion attacks on humans in Tanzania \(^{32}\) found that 95 lion attacks happened between \(6 \mathrm{pm}\) and \(10 \mathrm{pm}\) within either five days before a full moon or five days after a full moon. Of these, 71 happened during the five days after the full moon while the other 24 happened during the five days before the full moon. Does this sample of lion attacks provide evidence that attacks are more likely after a full moon? In other words, is there evidence that attacks are not equally split between the two five-day periods? Use StatKey or other technology to find the p-value, and be sure to show all details of the test. (Note that this is a test for a single proportion since the data come from one sample.)

Short answer

The short answer will be based on whether the null hypothesis was rejected or failed to reject after comparing the p-value with the significance level. This will determine whether there is statistical evidence to claim that lion attacks are more likely to occur after a full moon.

Step-by-step solution

Formulate the hypothesis

We start off by stating our null hypothesis (H0) and alternative hypothesis (Ha). For this exercise, the null hypothesis is that the lion attacks are equally likely before and after the full moon that is, \( p = 0.5 \) the alternative hypothesis is that the lion attacks are more likely after the full moon, which could be represented as, \( p > 0.5 \). Here, 'p' denotes the probability of an attack after the full moon.

Calculate the observed proportion

Next, compute the observed proportion which is the proportion of attacks after a full moon. This calculation can be done by dividing the attacks after the full moon by the total attacks. There were 71 attacks after full moon out of 95 attacks in total, therefore the observed proportion, \( p^* \), is \( p^* = \frac{71}{95} \).

Calculate the test statistic

We need to calculate the test statistic for the null hypothesis. The z-score for this test is given by the formula \( Z = \frac{p^* - p }{ \sqrt{ \frac{p (1-p)}{n} } } \). Here, 'n' is the total number of attacks, 'p' is the assumed probability of an attack after a full moon under the null hypothesis, and \( p^* \) is the observed probability. Substituting the given values you find the calculated test statistic.

Compute the p-value

Once the test statistic is calculated, use tables or a calculator to find out the p-value.

Interpret the results

Compare the calculated p-value with the significance level (generally 0.05). If the p-value is less than the significance level, then reject the null hypothesis. If the p-value is greater than the significance level, then there is insufficient evidence to reject the null hypothesis. Provide your interpretation based on the result.