Question

Female primates visibly display their fertile window, often with red or pink coloration. Do humans also do this? A study \(^{18}\) looked at whether human females are more likely to wear red or pink during their fertile window (days \(6-14\) of their cycle \()\). They collected data on 24 female undergraduates at the University of British Columbia, and asked each how many days it had been since her last period, and observed the color of her shirt. Of the 10 females in their fertile window, 4 were wearing red or pink shirts. Of the 14 females not in their fertile window, only 1 was wearing a red or pink shirt. (a) State the null and alternative hypotheses. (b) Calculate the relevant sample statistic, \(\hat{p}_{f}-\hat{p}_{n f}\), for the difference in proportion wearing a pink or red shirt between the fertile and not fertile groups. (c) For the 1000 statistics obtained from the simulated randomization samples, only 6 different values of the statistic \(\hat{p}_{f}-\hat{p}_{n f}\) are possible. Table 4.7 shows the number of times each difference occurred among the 1000 randomizations. Calculate the p-value.

Short answer

The null hypothesis is that the proportions are equal, and the alternative hypothesis is that the proportion is greater in the fertile group. The sample statistic for the difference in proportions is 0.329. The exact p-value cannot currently be calculated without the distribution of simulated statistics, but would be the proportion of simulated statistics greater than or equal to 0.329.

Step-by-step solution

State the Null and Alternative Hypotheses

The null hypothesis (H0) is that the probability of wearing red or pink is the same whether a woman is in her fertile window or not. In mathematical terms, \( H_{0}: p_{f} - p_{nf} = 0 \), where \( p_{f} \) is the proportion of women in their fertile window wearing red or pink, and \( p_{nf} \) is the proportion of non-fertile women wearing red or pink. The alternative hypothesis (Ha) is that the probabilities are not the same, or in other words the probability of wearing red or pink is higher during the fertile window. Mathematically, \( H_{a}: p_{f} - p_{nf} > 0 \).

Calculate the Relevant Sample Statistic

To calculate the sample statistic, we first calculate the sample proportions. For the fertile group, \( \hat{p}_{f} = \frac{4}{10} = 0.4 \). For the non-fertile group, \( \hat{p}_{nf} = \frac{1}{14} ≈ 0.071 \). The sample statistic is then the difference between these two proportions: \( \hat{p}_{f}-\hat{p}_{nf} = 0.4 - 0.071 = 0.329 \).

Calculate the p-value

The p-value is derived from the simulated randomization samples. From the provided information, it's given that only 6 different values are possible. For a valid p-value calculation, it's necessary to know the observed frequency of each possible value. Without this information, the exact p-value cannot be computed. However, if the observed frequency distribution was provided, to calculate the p-value, the proportion of simulated statistics greater than or equal to the observed statistic 0.329 would be found. That proportion represents the p-value.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is crucial when conducting statistical hypothesis testing. The null hypothesis, symbolized as H₀, represents a statement of no effect or no difference. In the context of the exercise, it asserts that the likelihood of females wearing red or pink during their fertile window is the same as during other times, mathematically denoted as H₀: p_f - p_nf = 0.

The alternative hypothesis, represented by H_a, is the hypothesis that researchers want to test for. It indicates there is an effect or a difference. In this case, it proposes that females are more likely to wear red or pink during their fertile window compared to other times, expressed as H_a: p_f - p_nf > 0. Testing these hypotheses helps determine whether there's sufficient evidence to reject the null hypothesis in favor of the alternative.

Sample Statistic Calculation

The sample statistic is a numerical value calculated from the data that provides insight into the hypotheses being tested. It's a point estimate that reflects a population parameter.

In our exercise, we calculate the sample proportions first. For females in their fertile window (fertile group), the proportion who wore red or pink is \( \hat{p}_{f} = \frac{4}{10} = 0.4 \). For females not in their fertile window (non-fertile group), the proportion is \( \hat{p}_{nf} = \frac{1}{14} ≈ 0.071 \). To assess the difference hypothesized, we compute the difference between these two proportions: \( \hat{p}_{f}-\hat{p}_{nf} = 0.4 - 0.071 = 0.329 \).

This sample statistic represents the observed difference between the two sample proportions and serves as the basis for our hypothesis test.

P-Value Calculation

The p-value is a crucial component in hypothesis testing, as it helps us to determine the strength of the evidence against the null hypothesis. It is the probability of observing a sample statistic as extreme as the test statistic, assuming the null hypothesis is true.

In the exercise given, the p-value would be calculated from the simulated randomization samples provided by the study. Unfortunately, specific values from the simulation were not supplied, but generally, one would tally the number of statistics from the simulation that are grater than or equal to the observed sample statistic, in this case 0.329. This count is then divided by the total number of simulations, 1000, to obtain the p-value.

This p-value tells us how likely it is to observe a difference in proportions as large or larger than the one we calculated, given that there is actually no difference (i.e., under the null hypothesis). A low p-value indicates that such an observation would be very unlikely, thus providing evidence against the null hypothesis. A common threshold for 'low' is 0.05, but this can vary depending on the field of study and specific circumstances.