Final answer:
To determine if the three cough remedies are equally effective, a Chi-square test for independence is used. The test compares observed relief frequencies with expected frequencies under the null hypothesis of equal effectiveness. A p-value less than the significance level leads to rejecting the null hypothesis, indicating differences in effectiveness.
Step-by-step explanation:
The question involves conducting a hypothesis test to determine whether three cough remedies are equally effective, which falls under the subject of mathematics, specifically statistics. To test the hypothesis of equal effectiveness, we use a Chi-square test for independence.
With the data provided on the effectiveness of NyQuil, Robitussin, and Triaminic, we compare the observed frequencies of 'no relief', 'some relief', and 'total relief' with the expected frequencies under the null hypothesis that all cough remedies are equally effective.
The test statistic is calculated from the sum of the squared differences between observed and expected frequencies, divided by the expected frequencies. The result is then compared to a Chi-square distribution with degrees of freedom equal to (r-1)*(c-1), where r is the number of levels of relief and c is the number of cough remedies.
The p-value obtained from the test statistic tells us the probability of observing the data we have if the null hypothesis were true. If the p-value is less than the chosen significance level, typically 0.05, we reject the null hypothesis and conclude that there is sufficient evidence to suggest that the cough remedies differ in their effectiveness.
In the student's conclusion, the decision to reject the null hypothesis is made because the p-value is less than the alpha level of 0.05.