When 40 people used the Weight Watchers diet for one year,...

When 40 people used the Weight Watchers diet for one year, their mean weight loss was 3.0 lb and the standard deviation was 4.9 lb (based on data from “Comparison of the Atkins, Ornish, Weight Watchers, and Zone Diets for Weight Loss and Heart Disease Reduction,” by Dan singer, et al., Journal of the American Medical Association, Vol. 293, No. 1). Use a 0.01 significance level to test the claim that the mean weight loss is greater than 0. Based on these results, does the diet appear to be effective? Does the diet appear to have practical significance?

When 40 people used the Weight Watchers diet for one year, their mean weight loss was 3.0 lb and the standard deviation
was 4.9 lb (based on data from “Comparison of the Atkins, Ornish, Weight Watchers, and Zone Diets for Weight Loss and Heart Disease Reduction,”
by Dan singer, et al., Journal of the American Medical Association, Vol. 293, No. 1). Use a 0.01 significance level to test the claim that the mean weight loss is greater than 0. Based on these results, does the diet appear to be effective? Does the diet appear to have practical significance?

Key Concepts

Statistical vs. Practical Significance

Statistical significance refers to the result of a hypothesis test showing that an effect is unlikely to have occurred by chance, based on a predetermined significance level. Practical significance, on the other hand, considers whether the magnitude of the observed effect is large enough to be of real-world importance or usefulness. It is possible for a result to be statistically significant but have little practical significance if the effect size is very small.

p-value

The p-value is the probability, under the null hypothesis, of obtaining a test statistic as extreme as or more extreme than the value observed in the sample data. It quantifies the strength of the evidence against the null hypothesis. If the p-value is less than the predetermined significance level, the null hypothesis is rejected, suggesting there is statistically significant evidence in favor of the alternative hypothesis.

Significance Level

The significance level, often denoted by alpha (?), is the threshold used in hypothesis testing to decide whether to reject the null hypothesis. It represents the probability of committing a Type I error, which is rejecting the null hypothesis when it is actually true. A lower significance level, such as 0.01, indicates a stricter criterion for evidence, making it harder to claim a statistically significant effect.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about population parameters based on sample data. It involves setting up a null hypothesis (often representing no effect or no difference) and an alternative hypothesis that represents the claim to be tested. The method uses a test statistic calculated from the sample data to determine whether there is enough evidence to reject the null hypothesis at a predetermined significance level.

t-Test

The t-test is a statistical test used to compare the mean of a sample to a known value or to compare the means of two samples, particularly when the sample sizes are relatively small and the population standard deviation is unknown. The test statistic is derived from the sample mean, sample standard deviation, and sample size, and follows a t-distribution under the null hypothesis. This method is generally used in testing claims about population means.

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