Question

Weight Loss Program Suppose that a weight loss company advertises that people using its program lose an average of 8 pounds the first month, and that the Federal Trade Commission (the main government agency responsible for truth in advertising) is gathering evidence to see if this advertising claim is accurate. If the FTC finds evidence that the average is less than 8 pounds, the agency will file a lawsuit against the company for false advertising. (a) What are the null and alternative hypotheses the FTC should use? (b) Suppose that the FTC gathers information from a very large random sample of patrons and finds that the average weight loss during the first month in the program is \(\bar{x}=7.9\) pounds with a p-value for this result of \(0.006 .\) What is the conclusion of the test? Are the results statistically significant? (c) Do you think the results of the test are practically significant? In other words, do you think patrons of the weight loss program will care that the average is 7.9 pounds lost rather than 8.0 pounds lost? Discuss the difference between practical significance and statistical significance in this context.

Short answer

The null and alternative hypotheses are \(H_0: \mu = 8\) and \(H_a: \mu < 8\) respectively. Since the p-value (0.006) is less than the common significance level (0.05), the result of the test is considered statistically significant and we reject the null hypothesis. That is, the average weight loss is less than 8 pounds. Despite this, the small difference (7.9 vs 8 pounds) may not be practically significant as most clients would not consider the 0.1-pound difference as significant.

Step-by-step solution

Formulate Hypotheses

In this scenario, the null hypothesis (denoted by \(H_0\)) is that the average weight loss of a given population is equal to 8 pounds. Therefore, we have \(H_0: \mu = 8\), where \(\mu\) represents the population mean. The alternative hypothesis (denoted by \(H_a\)) should be that the average weight loss is less than 8 pounds. Therefore, we have \(H_a: \mu < 8\).

Interpret the p-value

The p-value of 0.006 obtained in the task suggests that assuming the null hypothesis is true (i.e. mean weight loss is 8 pounds), there is a 0.006 (or 0.6%) chance of obtaining the measured sample mean weight loss of 7.9 or an even lower amount. When p-value is less than or equal to the significance level, usually denoted by 0.05 or 5%, due to its low probability, we reject the null hypothesis. Therefore, since 0.006 is less than 0.05, we reject the null hypothesis. The results are statistically significant.

Practical Significance vs Statistical Significance

Practical significance refers to the measure of the real-world (practical) impact of a result. Although the test results are statistically significant, one may argue that the difference between 7.9 pounds and 8 pounds weight loss is not practically significant because the difference is very small. Customers who sign up for the program may not consider the 0.1-pound difference in weight loss as a major factor in their decision, hence showing a less than 8-pound weight loss may not practically matter to them. On the other hand, statistical significance shows that the observed difference is not just due to chance and that there is a statistically significant difference, even if the difference is slight in numerical value.

Key concepts

Null Hypothesis (H0)

The null hypothesis, symbolized as H₀, is a fundamental concept in hypothesis testing and serves as the starting point for statistical analysis. It represents the default or original claim, which we assume to be true until evidence suggests otherwise. In the context of the weight loss program, the null hypothesis is the company's claim that their program results in an average weight loss of 8 pounds in the first month. Formally, this is expressed as H₀: \[ \mu = 8 \], with \mu representing the population mean.

This hypothesis is known as 'null' because it typically states that there is no effect, no difference, or no association when comparing two groups or treatments – in this case, that there is no difference from the advertised weight of 8 pounds lost. An understanding of the null hypothesis is crucial because it lays the foundation for the direction and nature of the statistical test to be performed.

Alternative Hypothesis (Ha)

In contrast to the null hypothesis, the alternative hypothesis, denoted by H_a, represents what the researcher is trying to prove, it suggests that there is an effect, a difference, or an association. For the weight loss program example, the alternative hypothesis posits that the actual average weight loss is less than the advertised 8 pounds, or H_a: \( \mu < 8 \).

Whether the alternative hypothesis is directional (as in this case, where it states 'less than') or non-directional (stating 'not equal to') depends on the context of the research question and can greatly affect the interpretation of the test results. In establishing the alternative hypothesis, researchers build the framework for what would constitute evidence sufficient enough to reject the null hypothesis and favor this competing claim.

p-value

The p-value is a pivotal measure in hypothesis testing that helps to determine the significance of the results. It quantifies the probability of observing the sample results, or something more extreme, if the null hypothesis is actually true. For instance, a p-value of 0.006 in the weight loss study indicates there is a 0.6% chance that a sample would show an average weight loss of 7.9 pounds (or less) if the true average weight loss is 8 pounds as the null hypothesis states.

The lower the p-value, the stronger the evidence against the null hypothesis. Conventionally, a p-value less than 0.05 is considered statistically significant and leads to the rejection of the null hypothesis. This threshold is somewhat arbitrary and chosen as a balance between Type I and Type II errors, but it has been widely accepted in many scientific fields.

Statistical Significance

Statistical significance is a term used to state that the observed difference or relationship in the data is not likely due to sampling variability; in other words, the findings are not happening by mere chance. It often hinges on the p-value; when the p-value falls below the pre-set alpha level (commonly 0.05), we consider the result statistically significant.

In the case of the weight loss program, since the p-value of 0.006 is below the commonly used threshold, we would declare the findings statistically significant. This means the FTC has sufficient statistical evidence to reject the company's claim and potentially take legal action. Statistical significance is critical in research to draw conclusions from data and separate meaningful results from those that are random fluctuations.

Practical Significance

While statistical significance relates to the likelihood that the observed results are real and not due to random chance, practical significance refers to the real-world relevance or importance of the effect size. It answers the question: even if the result is statistically significant, does it matter in a practical, substantive way?

Considering the weight loss example, a difference in weight loss of 0.1 pounds (from 8 to 7.9) might be statistically significant due to a very large sample size, yet it may not be clinically or practically significant. In other words, individuals looking to lose weight might not find this slight difference to impact their choice of a weight loss program. It's crucial to differentiate between statistical and practical significance, as a result can be statistically meaningful but not necessarily important or impactful in a practical context.