Atkins Weight Loss Program In a test of weight loss programs, 40 adults used the Atkins weight loss program. After 12 months, their mean weight loss was found to be 2.1 lb, with a standard deviation of 4.8 lb. Construct a 90% confidence interval estimate of the mean weight loss for all such subjects. Does the Atkins program appear to be effective? Does it appear to be practical?

검증된 단계별 안내

Step 1: Identify the key components of the problem. The sample size (n) is 40, the sample mean (x̄) is 2.1 lb, the sample standard deviation (s) is 4.8 lb, and the confidence level is 90%. Since the sample size is greater than 30, we can use the t-distribution to construct the confidence interval.

Step 2: Determine the critical t-value for a 90% confidence level. To do this, calculate the degrees of freedom (df = n - 1 = 40 - 1 = 39) and use a t-distribution table or statistical software to find the t-value corresponding to a 90% confidence level (two-tailed).

Step 3: Calculate the standard error of the mean (SE). The formula for the standard error is:

{SE}_{x̄} = \frac{s}{\sqrt{n}}

, where s is the sample standard deviation and n is the sample size.

Step 4: Construct the confidence interval using the formula:

x̄ ± t_{α/2} × {SE}_{x̄}

. Plug in the values for the sample mean (x̄), the critical t-value, and the standard error (SE) to calculate the lower and upper bounds of the confidence interval.

Step 5: Interpret the results. If the confidence interval includes 0, it suggests that the mean weight loss might not be significantly different from 0, meaning the program may not be effective. Additionally, consider whether the mean weight loss (2.1 lb) is practically significant in the context of weight loss goals.

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주요 개념

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Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. In this case, it estimates the mean weight loss for all adults using the Atkins program. The width of the interval reflects the uncertainty around the estimate, and a 90% confidence level indicates that if the same procedure were repeated multiple times, 90% of the intervals would contain the true mean.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of the Atkins program, a standard deviation of 4.8 lb indicates that individual weight losses varied significantly around the mean of 2.1 lb. A high standard deviation suggests that while some participants may have lost a lot of weight, others may have lost very little or none at all, impacting the overall effectiveness of the program.

Effectiveness and Practicality

Effectiveness refers to how well a program achieves its intended outcome—in this case, weight loss. Practicality considers whether the program can be realistically followed by individuals in their daily lives. Evaluating the Atkins program involves analyzing the mean weight loss and its confidence interval to determine if the results are statistically significant and if the weight loss is substantial enough to be considered practical for long-term adherence.