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 mean weight loss for all such subjects. Does the Atkins program appear to be effective? Does it appear to be practical?

Accepted Answer

Hello. In this video, we are told that a study was conducted on 35 participants who followed the Paleo diet for 10 months. The mean weight loss recorded was 3.4 pounds, with a standard deviation of 5.2 pounds. We want to construct a 90% confidence interval for the mean weight loss for all individuals on the paleo diet, and based on our interval, does the paleo diet show a statistically significant effect? So, let's go ahead and approach the first part of this problem, which is requiring us to find a 95 90% confidence interval. So let's go ahead and list out the given information. First we are told that the sample size. Which we will label as N. is equal to 35. We are told that we have a sample mean. Which we will label as X bar. Equal to 3.4. We are also told that we have a sample standard deviation, which we will label as S, and this sample standard deviation is given to us as 5.2. And finally, we are told to use a confidence level of 90%, which can be represented as a decimal 0.9. Now, in this case here, because the data set is fairly small, we're going to use the T score method. So, in order to approach this problem, the first thing we want to do is we want to determine our degree of freedom. Now, the degree of freedom is defined as N minus 1. We are going to take our sample size and subtract 1 in order to find the degree of freedom. That is going to give us 35 minus 1, which is 34. So we have a degree of freedom of 34. The next thing we want to do is we want to obtain our T score. Now we want to find the area under the endpoints of the normal distribution. That is going to be defined as alpha. Equal to 1 minus the confidence level. And again, our confidence level is equal to 0.9 as a decimal, so this is going to be 1 minus 0.9, which is going to give us 0.1. Now, since there are two tail ends to a normal distribution, we are going to take this area and divide it by 2. So, alpha divided by 2 is equal to 0.1 divided by 2, which gives us a decimal of 0.05. Finally, we are going to take the T score of these areas to determine our critical point. That is going to be T 0.05, and by using the T table in the appendix, that is going to give us an approximation of 1.690 for our critical point. This is going to be our T score, and we are going to set it to the side for now, since it will help us find our margin of error. The next thing we want to do is we want to determine the standard error. Now, the standard error is defined as S E equal to S divided by the square root of N. Now, this is saying that if we want to find the standard error, we are going to take our sample standard deviation and divide it by the square root of the sample size. This is going to give us 5.2, divided by the square root of 35, and this is going to give us an approximate decimal of 0.878. That is going to be our standard error. Now that we have the T score and standard error, we can use these two pieces of data to find the margin of error. The margin of error is defined as our T score multiplied by the standard error. That is going to give us 1.690 multiplied by 0.878, and this is going to give us an approximate decimal of 1.484. Now that we have the margin of error, and we have the sample mean, we can use this to finally find our confidence interval. Now what we are doing specifically here is we are going to find the end points of our confidence interval. The end points of the confidence interval are defined as X bar plus or minus the margin of error. So we are going to take our sample standard, our sample mean, and add and subtract the margin of error to find our confidence interval. That is going to give us the end points of 3.4 minus 1.484, and 3.4 plus 1.484. And this is going to simplify to leave us with 1.916 and 4.884. To put this in the form of an interval, this just tells us that our mean should be between these two values. The mean should be between the values of 1.916 and 4.884 beats per minute. This is going to be the confidence interval for the study. Now, going back to the second part of the problem, the second part of the problem asks us to determine if the paleo diet is statistically significant. Now, in order to determine this, we need to ask ourselves, does the value 0 exist within our confidence interval? In this case, our confidence interval does not include the value of 0. And what this means is that average weight loss. Is greater than 0. And with that being said, the solution to this problem is going to be B. So I hope this video helps you in understanding how to find this, how to find the confidence interval, and I will go ahead and see you all in the next video.

Key Concepts

Confidence Interval

Standard Deviation

Effectiveness and Practicality

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