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. Yes, the data provide strong evidence that the mean weight loss differs between the two plans. This conclusion is based on the given p-value of 0.004. In statistical hypothesis testing, the p-value is the probability of observing a result as extreme as the one obtained, assuming that the null hypothesis (which states that there is no difference between the two plans) is true. A p-value of 0.004 indicates that such extreme or more extreme results would occur only 0.4% of the time if the null hypothesis were true. Typically, a p-value below a chosen significance level (commonly 0.05) is considered statistically significant. In this case, the p-value is much smaller than 0.05, providing strong evidence against the null hypothesis and in favor of the alternative hypothesis (which states that there is a difference in mean weight loss between the two plans).
b. The data do provide strong evidence that the mean weight loss differs between the two plans, as explained in part (a). However, the confidence interval provides additional information about the magnitude of the difference. The 99% confidence interval for the difference in means, (0.26, 0.84), suggests that the true difference in mean weight loss between the two plans falls within this range with 99% confidence. Since this interval does not include zero, it indicates that the mean weight loss in one plan is likely considerably higher than the mean weight loss in the other plan.
c. In this context, a Type II error would mean failing to reject the null hypothesis when it is actually false. Specifically, it would occur if the experiment did not find significant evidence of a difference in mean weight loss between the two plans, even though such a difference does exist in the population. In other words, the experiment fails to detect the effect of the weight loss plans, possibly due to a small sample size or other factors. This would be a missed opportunity to identify the effectiveness of one plan over the other.
d. In this context, a Type I error would mean rejecting the null hypothesis when it is actually true. More specifically, it would occur if the experiment found significant evidence of a difference in mean weight loss between the two plans, even though there is no actual difference in the population. It implies concluding that one weight loss plan is superior when there is no real difference, potentially leading to incorrect decisions or recommendations.