Atkins Weight Loss Program In a test of weight loss programs, ...

Question

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 standard deviation of the weight loss for all such subjects. Does the confidence interval give us information about the effectiveness of the diet?

Accepted Answer

Hello everyone. Let's take a look at this question together. A group of 50 volunteers participated in a mindfulness program. After 8 weeks, their mean reduction in stress score was 4.6 points. With a sample standard deviation of 5.3 points, construct a 99% confidence interval estimate of the standard deviation of the reduction in stress scores for all such participants. Is it answer choice A 4.19 to 7.11? Answer choice B, 5.55 to 7.26? Answer choice C 4.55 to 7.26, or answer choice D 3.55 to 6.26. So, in order to solve this question, we have to recall how to construct a 99% confidence interval estimate so that we can construct one of the standard deviation of the reduction in stress scores for all of the participants. And so looking at the data that we are given from the question, we know that we have a sample size represented by N of 50, a sample standard deviation represented as S of 5.3, and a confidence level of 90%. And our first step is to use chi square distribution. In which the formula for the confidence interval of the population variant sigma squared is calculated using the following formula. And then to get the confidence interval for the standard deviation, which is sigma, we take the square root of sigma squared. And in order to determine the chi square critical values, we start with the degrees of freedom, which is calculated as N minus 1, so the degrees of freedom is 49. And using a 90% confidence level. Our alpha is 0.01, our lower tail area is alpha divided by 2, which is equal to 0.005. And our upper tail area is 1 minus alpha divided by 2, which is 0.995. And using a chi square table or a calculator, our chi square value for the lower tail area is 78.231, and our chi square value for the upper tail area is 27.249. And so using these values, we plug them into the formula starting with the lower bound or variants we plug in our lower tail values which results in 17.5941762217 as the lower bound for variants and the upper bound for variance is 50.5123123784. And since the confidence interval for the standard deviation sigma is equal to the square root of sigma squared. Which is the value for our variants, we take the square roots to get the bounds for the standard deviation. Starting with the lower bound for the standard deviation, we get 4.19 as the value, and for the upper bound for the standard deviation, we get 7.11. So the values for our 90% confidence interval are 4.19 and 7.11. Which, looking at our answer choices, we can identify that answer choice A is the correct answer, as it contains the correct confidence interval estimate of the standard deviation of the reduction in stress scores for all participants, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

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Key Concepts

Confidence Interval

Standard Deviation

Effectiveness of a Diet

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