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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.

Step by step solution

01

Formulating Null and Alternative Hypotheses

The null hypothesis (H0) corresponds to the company's claim and its negation forms the alternative hypothesis (H1). Thus: \n H0: \( \mu = 8\), where \( \mu \) is the average weight loss during the first month.\n H1: \( \mu < 8\) - the FTC will file a lawsuit if the average weight loss is less than 8 pounds.

02

Hypotheses Testing and Drawing Conclusions

The sample indicates a loss of 7.9 pounds in the first month with a p-value of 0.006. A commonly used significance level is 0.05. A p-value less than 0.05 suggests strong evidence against the null hypothesis, meaning we reject the null hypothesis in favor of the alternative. Since 0.006 is less than 0.05, this data suggests strong evidence against the null hypothesis.

03

Analyzing Practical and Statistical Significance

Statistical significance is a mathematical tool to inform us whether an outcome occurred by chance. In this case, the results are statistically significant as the p-value is less than 0.05. However, practical significance refers to the actual real-world impact of the result. The difference between 7.9 and 8 pounds may not be practically significant as it's a minor difference unlikely to greatly affect the patrons' satisfaction with the weight-loss program.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

The null hypothesis in statistics is a statement or a default position that there's no effect or no difference, and it serves as a starting point for hypothesis testing. In our weight loss program example, the null hypothesis (\( H_0 \) ) is that the average weight loss is exactly 8 pounds (\( \text{H}_0: \text{average weight loss} = 8 \text{ pounds} \) ), just as the company advertised. It implies that any difference observed in an experiment could be due to chance or random variation rather than a real effect of the program's efficacy.

When testing this hypothesis, we collect data and analyze it to determine if there's enough evidence to 'reject' the null hypothesis. Failing to find this evidence means we 'fail to reject' it, which does not prove the null hypothesis true but rather indicates we don't have enough proof to say otherwise—leaving the company's claim unchallenged.

Alternative Hypothesis

Contrary to the null hypothesis, the alternative hypothesis (\( H_A \text{ or } H_1 \) ) expresses that there is an effect or a difference, and it represents what we aim to support with our data. In the context of the FTC's investigation, the alternative hypothesis is that the actual average weight loss is less than the advertised 8 pounds (\( \text{H}_1: \text{average weight loss} < 8 \text{ pounds} \) ).

This hypothesis is what the FTC would seek to prove in order to substantiate claims of false advertising by the weight loss company. If evidence is strong enough against the null hypothesis, the alternative hypothesis gains credence, which in our case would mean supporting the FTC's suspicion that the company has overstated the effectiveness of its weight loss program.

P-Value Significance

The p-value is a vital concept in hypothesis testing, indicating the probability of observing results as extreme as the ones in your study, assuming the null hypothesis is true. A smaller p-value suggests that the observed data are unlikely under the null hypothesis. In the case of the FTC, a p-value of 0.006 means there is only a 0.6% chance of seeing an average weight loss of 7.9 pounds or less if the true average was 8 pounds, due to random chance alone.

Typically, researchers use a significance level (often 0.05) to determine a threshold for significance. Given that 0.006 is well below 0.05, the FTC's findings are considered statistically significant; hence, there's enough evidence to reject the null hypothesis in favor of the alternative one, suggesting that the advertised claim of an 8-pound weight loss is likely to be false.

Practical Significance vs Statistical Significance

While statistical significance relates to the likelihood that the observed effects occurred by chance, practical significance considers the size of the effect and whether it's meaningful in real-world terms. Statistical significance does not automatically indicate that the result is substantial enough to have real consequences. Thus, even when the FTC finds statistical significance in the weight loss data, it's also essential to ask if the difference between 7.9 and 8 pounds is of practical significance.

For the patrons of the weight loss program, this slight difference might not be substantial or affect their satisfaction with the program. This illustrates the importance of not only stating statistical significance but also taking into account practical significance when considering the implications of results in a tangible context.