Question

A random sample of monarch butterflies and a random sample of swallowtail butterflies were selected, and the difference in the average flying speed for each sample was calculated. A two-sample t-test for the difference in means was conducted to investigate whether the speed at which monarchs fly, on average, is faster than the speed at which swallowtails fly. All conditions for inference were met, and the p-value was given as 0.072.

Which of the following is a correct interpretation of the p-value?a. The probability that monarchs fly faster than swallowtails is 0.072.b. The probability that monarchs and swallowtails fly at the same speed is 0.072.c. Assuming that monarchs and swallowtails fly at the same speed on average, the probability of observing a difference equal to or greater than the sample difference is 0.072.d. Assuming that monarchs fly faster than swallowtails on average, the probability of observing a difference equal to or greater than the sample difference is 0.072.e. Assuming that monarchs fly faster than swallowtails on average, the probability of the monarchs and swallowtails flying at the same speed is 0.072.

Answer 1

The correct option is;

b. The probability that monarchs and swallowtails fly at the same speed is 0.072

Explanation:

The p- value indicates the probability of which the hypothesis tested, the null hypothesis is true and it is compared against a given confidence level. Whereby the p-value is more than the given confidence level then it indicates that the null hypothesis assumption is correct.

Therefore, whereby the null hypothesis tests proposes the equality of the two means, then at a 5% confidence level or α = 0.05, whereby p-value = 0.072 > α = 0.05 we accept the null hypothesis and state that the probability that monarchs and swallowtails fly at the same speed is 0.072.

Answer 2

Option C is the correct answer

Explanation:

Let μ1 be the average flying speed of monarch butterflies and let μ2 be the average flying speed of swallowtail butterflies.

Also, let n1 be the sample size of monarch butterflies and let n2 be the sample size of swallowtail butterflies.

Thus, defining the hypothesis, we have;

Null hypothesis;H0: μ1 = μ2

Alternative hypothesis; H1: μ1 > μ2

In this case, the p-value = P_H0[t_(n1 + n2 - 2) ≥ t] where t is the test statistic based on tbe differences of sample mean,that follows t distribution with degrees of freedom (n1 + n2 - 2)

With respect to the question, the probability of the monarchs and swallowtails flying at the same speed is 0.072.

Thus,

P_H0[t_(n1 + n2 - 2) ≥ t] = 0.072

We can therefore conclude that in this case, the p-value is the probability of observing a difference equal to or greater than the sample difference under the null hypothesis which states that μ1 = μ2, which is 0.072

Thus, option C is the correct answer