Question

To compare two weight reduction diets \(A\) and \(B, 60\) dieters were randomly selected. One group of 30 dieters was placed on diet \(A\) and the other 30 on diet \(\mathrm{B}\), and their weight losses were recorded over a 30 -day period. The means and standard deviations of the weight-loss measurements for the two groups are shown in the table. Find a \(95 \%\) confidence interval for the difference in mean weight loss for the two diets. Can you conclude that there is a difference in the average weight loss for the two diets? Why or why not?

Short answer

If yes, which diet is more effective? If no, discuss the implications of this result.

Step-by-step solution

Find the difference between the means of the two diets' weight losses

Using given data, calculate the difference between the means of weight losses for diets A and B. Let \(\bar{x}_A\) and \(\bar{x}_B\) denote the mean weight losses for diets A and B, respectively. Then, the difference in means is: $$d = \bar{x}_A - \bar{x}_B$$

Calculate the standard error of the difference

Next, we'll compute the standard error of the difference between the means using the given standard deviations \(s_A\) and \(s_B\) and the number of participants \(n_A = n_B = 30\). The standard error is given by: $$SE(d) = \sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}$$

Find the critical value for a 95% confidence interval

Now, we'll find the critical value \(z^*\) corresponding to a 95% confidence level. Since we want the middle 95% of the distribution, we will have 2.5% in each tail. Therefore, we'll look for the z-score that has a 0.975 probability to the left, which is approximately \(z^* = 1.96\).

Calculate the margin of error

With the critical value \(z^*\) and the standard error \(SE(d)\), we can now calculate the margin of error for the difference in means: $$ME = z^* \cdot SE(d)$$

Find the confidence interval for the difference in means

Using the difference in means \(d\) and the margin of error \(ME\), we can find the 95% confidence interval for the difference in mean weight loss for the two diets: $$(d - ME, d + ME)$$

Analyze the results and conclude if there is a difference in the average weight loss for the two diets

If the confidence interval calculated in Step 5 does not include 0, we can conclude that there is a significant difference in the average weight loss for the two diets at the 95% confidence level. However, if the confidence interval does include 0, we cannot say with 95% certainty that there is a significant difference between the two diets.

Key concepts

Mean Weight Loss Comparison

When we talk about comparing the mean weight loss between two diets, what we are really assessing is whether the difference in average weight loss between the groups is chance or reflects a true difference due to the diets. In the context of our exercise, diet A and diet B were compared by looking at the mean weight loss for 30 dieters on each diet over a 30-day period.

The process involves calculating the difference in the sample means, denoted as \(d = \bar{x}_A - \bar{x}_B\). This value represents the average difference in weight loss between the diets and becomes the centerpiece for further statistical analysis. By itself, the difference in means tells us the average benefit one diet might have over the other but doesn't account for variability or uncertainty in our estimate. To address this, we move onto the steps that involve the standard error and confidence intervals to provide a more robust comparison.

Standard Error Calculation

Within statistics, the standard error is a critical value that helps us understand the variability in our data. Specifically, when comparing means, the standard error of the difference between the means (SE(d)) quantifies how much we expect the difference to vary if we were to repeat the study multiple times.

The formula used in the exercise for the standard error is \(SE(d) = \sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}\), where \(s_A\) and \(s_B\) are the standard deviations for diets A and B, respectively, and \(n_A\) and \(n_B\) are the numbers of participants in each group. A smaller standard error points to a more precise estimate of the mean difference, while a larger standard error suggests greater variability and less precision. After calculating SE(d), we are better equipped to determine the confidence interval for the true mean difference.

Hypothesis Testing

The final step in our comparison is hypothesis testing, which in this context is used to determine if the observed difference in weight loss between diet A and diet B is statistically significant. Hypothesis testing starts with a null hypothesis (often that there is no difference between groups) and an alternative hypothesis (that there is a difference).

The confidence interval calculated from the mean difference and its standard error allows us to perform this test indirectly. If 0 is not within the calculated confidence interval \[ (d - ME, d + ME) \], we have evidence to reject the null hypothesis, implying that the difference in mean weight loss between the two diets is indeed significant. Conversely, if 0 lies within the interval, we do not have enough evidence to conclude a significant difference at the given confidence level, which would typically be 95% in studies like these.