Question

The study described in Example 8.1 .1 , involving the compound \(\mathrm{mCPP}\), included a group of men. The men were asked to rate how hungry they were at the end of each 2-week period and differences were computed (hunger rating when taking mCPP minus hunger rating when taking the placebo). The distribution of the differences was not normal. Nonetheless, a sign can be conducted using the following information: Out of eight men who recorded hunger ratings, three reported greater hunger on \(\mathrm{mCPP}\) than on the placebo and five reported lower hunger on \(\mathrm{mCPP}\) than on the placebo. \({ }^{1}\) Conduct a sign test at the \(\alpha=0.10\) level; use a nondirectional alternative.

Short answer

With three '+' signs and five '-' signs, the test statistic is 3. Using a binomial distribution table, the P-value for a test statistic of 3 with a sample size of 8 at a nondirectional hypothesis test is calculated. The P-value is compared to the significance level of 0.10 to determine if the null hypothesis can be rejected.

Step-by-step solution

State the Null Hypothesis

For the sign test, the null hypothesis (H0) asserts that there is no difference in hunger ratings between when the subjects are taking mCPP and when they are taking the placebo. This implies that the probability of a subject reporting greater hunger on either treatment is equal, both having a probability of 0.5.

State the Alternative Hypothesis

The alternative hypothesis (Ha) for a nondirectional (two-tailed) sign test suggests that there is a difference in the hunger ratings between the mCPP and the placebo, but it does not specify the direction of the difference.

Conduct the Sign Test

Calculate the number of '+' signs (mCPP > placebo) and '-' signs (mCPP < placebo). In this case, we have 3 '+' signs and 5 '-' signs.

Determine the Test Statistic

The test statistic for a sign test is the smaller of the number of '+' signs or '-' signs. Here, the smaller number is 3.

Determine the P-Value

For a sample size of 8 and a nondirectional hypothesis test, use the binomial distribution to calculate the P-value. The P-value is the probability of observing the test statistic or something more extreme under the assumption that the null hypothesis is true. Use the binomial probability formula or a binomial table.

Compare P-Value and Alpha

After calculating the P-value, compare it to the significance level \(\alpha = 0.10\). If the P-value is less than or equal to the significance level, reject the null hypothesis.

Conclusion

Based on the comparison between P-value and \(\alpha\), draw a conclusion about the null hypothesis. If null is rejected, this indicates that there is a statistically significant difference in hunger ratings between mCPP and placebo.

Key concepts

null hypothesis

Understanding the null hypothesis is crucial in any statistical test, including the sign test. It's the starting assumption that no effect or difference exists in the population for the specific variable or condition being tested. In the context of our example, which involves hunger ratings on mCPP versus a placebo, the null hypothesis posits that mCPP does not affect hunger levels differently from the placebo; both treatments have the same impact on hunger. This means that the likelihood of reporting greater hunger on either treatment is equally probable.

Students often struggle with conceptualizing the null hypothesis. A helpful analogy could be to think of it as a skeptic's viewpoint, where any observed differences are attributed to random chance rather than a true effect. You maintain this skeptical stance unless the evidence overwhelmingly suggests otherwise.

alternative hypothesis

The alternative hypothesis, on the other hand, directly challenges the null hypothesis. It suggests there is a difference or effect that should be detectable in the data. For our hunger rating study, the alternative hypothesis indicates that there is a difference in the way mCPP and the placebo affect hunger ratings, but it does not specify which treatment increases or decreases hunger; it is nondirectional.

When formulating an alternative hypothesis, clarity and specificity are key so that the statistical test can adequately address the research question. The alternative hypothesis propels our investigation, steering us toward closely examining the data for evidence that contradicts the null hypothesis.

test statistic

The test statistic is a critical component that allows researchers to quantify the strength of the evidence against the null hypothesis. In a sign test, which is non-parametric, the test statistic is the smaller count of positive or negative signs — in this case, indicative of whether hunger is greater or less when taking mCPP compared to the placebo.

Why the Smaller Count?

Focusing on the smaller number of signs maximizes the conservative nature of the test, ensuring we don't too readily discard the null hypothesis unless truly warranted. In our exercise, the smaller count is 3 (indicating greater hunger on mCPP for three men). This number becomes the fulcrum upon which our decision — to reject or not reject the null hypothesis — pivots.

P-value

The P-value is the probability of obtaining a test statistic at least as extreme as the one observed, under the assumption that the null hypothesis is true. In simpler terms, it's an index telling us how surprised we should be by our data, given that the null hypothesis is correct. If this value is extremely low, it means that our observed data is quite unlikely under the null hypothesis, leading us to question the null's validity.

To calculate the P-value in our sign test, we would use the binomial distribution, considering the number of positive and negative signs observed. The resulting P-value is then compared to a pre-decided significance level (alpha); if it's lower, we conclude that what we've observed is sufficiently rare under the null hypothesis and thus significant.

statistical significance

Statistical significance is a determination about the data that suggests an observed effect or difference is unlikely to have occurred due to chance alone. How do we make this call? By setting a threshold before the test — the alpha level. This is the probability of incorrectly rejecting the null hypothesis, known as a Type I error.

For example, our sign test at the alpha level of 0.10 means we're willing to accept a 10% chance of error. If the P-value is less than or equal to 0.10, we reject the null hypothesis. Rejection here indicates a statistically significant difference in hunger ratings between mCPP and placebo. This framework allows researchers to make empirically-based decisions while controlling the likelihood of making incorrect conclusions.