Question

Low-fat or low-carb? Are low-fat diets or low-carb diets more effective for weight loss? A sample of 81 subjects went on a low-carbohydrate diet for six months. At the end of that time, the sample mean weight loss was 3.1 kilograms with a sample standard deviation of 7.09 kilograms. A second sample of 95 subjects went on a low-fat diet. Their sample mean weight loss was 5.7 kilograms with a standard deviation of 8.65 kilograms. Can you conclude that the mean weight loss of subjects having low-carb diets is less than the mean weight loss of subjects having low-fat diets? Let µ₁ denote the mean weight loss on the low-carb diet. Use the α =0.1 level and the critical value method.

State the appropriate null and alternate hypotheses.

Answer 1

To determine if the mean weight loss of low-carb diets is less than that of low-fat diets, the critical value method is used. The null hypothesis is that the mean weight loss for low-carb diets is equal to or greater than for low-fat diets, while the alternative hypothesis posits that the mean weight loss for low-carb diets is less.

Step-by-step explanation:

To determine whether the mean weight loss of subjects on low-carb diets is less than the mean weight loss of subjects on low-fat diets, we can use the critical value method. The null hypothesis, denoted as μ₁, is that the mean weight loss for low-carb diets is equal to or greater than the mean weight loss for low-fat diets. The alternative hypothesis is that the mean weight loss for low-carb diets is less than the mean weight loss for low-fat diets.

We will use the α = 0.1 level of significance. First, we calculate the test statistic by finding the difference between the sample mean weight losses of the low-carb and low-fat groups: 5.7 - 3.1 = 2.6 kilograms. Next, we calculate the standard error of the difference using the formulas for the standard deviation and sample sizes of each group: sqrt((7.09^2/81) + (8.65^2/95)) = 1.29 kilograms.

The test statistic is calculated as the difference between the sample mean weight losses of the two groups, and the critical value is obtained for the given significance level and degrees of freedom.

Finally, we calculate the critical value for a one-tailed test with α = 0.1 and degrees of freedom equal to the smaller of (81-1) = 80 or (95-1) = 94. From the t-table or using statistical software, the critical value is -1.29. Since the test statistic is less than the critical value, we reject the null hypothesis. Therefore, there is sufficient evidence to conclude that the mean weight loss of subjects on low-carb diets is less than the mean weight loss of subjects on low-fat diets.