Suppose a large-scale randomized experiment were conducted to compare two weight loss plans. At the end of the experiment, the weight loss for each subject (in pounds) was recorded. The p-value turned out to be .004, and a 99% confidence interval for the difference in means turned out to be (0.26, 0.84).
a. Do the data provide strong evidence that the mean weight loss differs between the two plans? Explain briefly.
b. Do the data provide strong evidence that the mean weight loss differs considerably (i.e., a lot) between the two plans? Explain briefly.
c. Describe briefly what a Type II error would mean in this context.
d. Describe briefly what a Type I error would mean in this context.
a. In order to determine if the data provide strong evidence that the mean weight loss differs between the two plans, we need to look at the p-value and the confidence interval. The p-value is a measure of the strength of evidence against the null hypothesis, which states that there is no difference in mean weight loss between the two plans. A p-value of .004 suggests that there is strong evidence against the null hypothesis, indicating that the mean weight loss does differ between the two plans.
b. The confidence interval provides an estimate of the range within which the true difference in means is likely to fall. In this case, the 99% confidence interval for the difference in means is (0.26, 0.84). Since this interval does not include zero, we can conclude that there is strong evidence that the mean weight loss differs considerably between the two plans. The fact that the upper bound of the interval is as high as 0.84 suggests a significant difference.
c. A Type II error in this context would mean failing to reject the null hypothesis when it is actually false. In other words, it would mean concluding that there is no difference in mean weight loss between the two plans, when in reality there is a difference. This would be a missed opportunity to detect and acknowledge the effectiveness of one plan over the other.
d. A Type I error in this context would mean rejecting the null hypothesis when it is actually true. In other words, it would mean concluding that there is a difference in mean weight loss between the two plans, when in reality there is no difference. This could lead to erroneously favoring one plan over the other based on the study results, potentially resulting in wasted resources and misguided recommendations.