Question

Eight obese persons were placed on a diet for 1 month, and their weights, at the beginning and at the end of the month, were recorded: $$ \begin{array}{ccc} & {\text { Weights }} \\ & & \\ \text { Subjects } & \text { Initial } & \text { Final } \\ \hline 1 & 310 & 263 \\ 2 & 295 & 251 \\ 3 & 287 & 249 \\ 4 & 305 & 259 \\ 5 & 270 & 233 \\ 6 & 323 & 267 \\ 7 & 277 & 242 \\ 8 & 299 & 265 \end{array} $$ Estimate the mean weight loss for obese persons when placed on the diet for a 1 -month period. Use a \(95 \%\) confidence interval and interpret your results. What assumptions must you make so that your inference is valid?

Short answer

Question: Estimate the mean weight loss for obese persons when placed on the diet for 1 month with a 95% confidence interval, given the following data on initial weights and final weights after one month: (310, 263), (295, 251), (287, 249), (305, 259), (270, 233), (323, 267), (277, 242), and (299, 265). Answer: We can estimate with 95% confidence that the mean weight loss for obese persons when placed on the diet for a 1-month period is between 35.07 to 47.18 pounds. To make our inference valid, we must assume the sample is representative, the weight losses are normally distributed, and the weights are independent observations.

Step-by-step solution

Calculate weight loss for each individual

Subtract the final weight from the initial weight for each person to get the weight loss: $$ \begin{array}{ccc} \text { Subjects } & \text { Weight Loss } \\ \hline 1 & 310 - 263 = 47 \\ 2 & 295 - 251 = 44 \\ 3 & 287 - 249 = 38 \\ 4 & 305 - 259 = 46 \\ 5 & 270 - 233 = 37 \\ 6 & 323 - 267 = 56 \\ 7 & 277 - 242 = 35 \\ 8 & 299 - 265 = 34 \\ \end{array} $$

Calculate the sample mean and sample standard deviation.

Add the weight losses together and divide by the number of subjects in the sample (8) to find the sample mean (\(\overline{x}\)) and then use the formula for the unbiased sample standard deviation (s): $$ \overline{x} = \frac{47 + 44 + 38 + 46 + 37 + 56 + 35 + 34}{8} = 41.125 \\ s = \sqrt{\frac{\Sigma (x_i -\overline{x})^2}{(n-1)}} = \sqrt{\frac{(47-41.125)^2 + (44-41.125)^2 + (38-41.125)^2 + (46-41.125)^2 + (37-41.125)^2 + (56-41.125)^2 + (35-41.125)^2 + (34-41.125)^2}{(8 - 1)}} \\ s \approx 7.11 $$

Calculate the 95% confidence interval.

To find the 95% confidence interval, use the t-distribution since the population standard deviation is unknown. In this case, the sample size (n) is 8. So, the degrees of freedom is 7. Using the t-distribution with 7 degrees of freedom, the critical t-value (or t-star) for a two-tailed test at \(\alpha = 0.05\) is 2.365. The margin of error (ME) calculation is: $$ ME = t* \frac{s}{\sqrt{n}} = 2.365 \frac{7.11}{\sqrt{8}} \approx 6.05 $$ The confidence interval (CI) is given by the formula: $$ \text{CI} = \overline{x} \pm ME = 41.125 \pm 6.05 $$ So, the 95% confidence interval is approximately (35.07, 47.18).

Interpretation and Assumptions

We can interpret our results as follows: We are 95% confident that the mean weight loss for obese persons when placed on the diet for a 1-month period is between 35.07 to 47.18 pounds. To make our inference valid, we must assume the sample is representative and that the weight losses are normally distributed. Additionally, we assume that the weights are independent observations.

Key concepts

Weight Loss Study

In this weight loss study, eight participants followed a specific diet for one month, and both their initial and final weights were recorded. The change in their weights, calculated as the difference between the initial and final values, helps us understand the diet's effectiveness. Weight loss is not only about healthier lifestyles but also involves statistical analysis to provide proof of its effectiveness. Here, the focus is on estimating the average weight loss for all obese individuals subjected to the diet. This estimate will help to identify whether the program is working as intended and how much weight participants can expect to lose on average. Clarifying this requires not just the raw data but a deeper analysis including statistical concepts.

Sample Standard Deviation

The `sample standard deviation` (denoted as \(s\)) measures the amount of variability or dispersion in the sample of weight losses recorded. It provides insights into how much the weight losses of the individuals in the study differ from the mean weight loss. A smaller standard deviation would suggest more consistent results across the group, while a larger one would indicate more variability.For this study, we calculate \(s\) using the formula:\[s = \sqrt{\frac{\Sigma (x_i - \overline{x})^2}{(n-1)}}\]Where \(\Sigma (x_i - \overline{x})^2\) represents the sum of the square of differences between each weight loss and the mean weight loss, and \(n-1\) is used for the sample size adjustment. This calculation is crucial as it directly influences the width of our confidence interval later on.

T-Distribution

The `t-distribution` is a vital statistical tool when dealing with small sample sizes, as in this study with eight participants. We use the t-distribution because the population standard deviation is unknown, and instead, we use the sample standard deviation. The t-distribution adjusts for this uncertainty, providing a more accurate reflection of expected variability when the sample size is small. It's more spread out than the normal distribution, accommodating the extra uncertainty. Practically, we refer to it to determine the critical t-value (or t-star), which is essential for constructing the confidence interval. In this study, with 7 degrees of freedom (n-1), the critical t-value for a 95% confidence interval is approximately 2.365. This value is combined with the sample standard deviation to compute the margin of error, which is then used to calculate the confidence interval for the average weight loss.

Statistical Inference

`Statistical inference` involves making conclusions about a population based on a sample. In this exercise, we infer the average weight loss for obese individuals on a specific diet program.The basis of this inference is the confidence interval, which estimates the range within which the true mean weight loss lies with a given level of confidence, in this case, 95%. The confidence interval is given by:\[\text{CI} = \overline{x} \pm ME\]where ME (margin of error) is calculated using the critical t-value and sample standard deviation.However, this inference comes with assumptions:

• The sample should be representative of the entire population of obese individuals.
• The distribution of weight loss should be approximately normal.
• Each observation should be independent.

By adhering to these assumptions, the results will be more likely to reflect the true characteristics of the larger group.