In Exercises 17–24, use the indicated Data Sets from Appendix ...

Question

In Exercises 17–24, use the indicated Data Sets from Appendix B. The complete data sets can be found at www.TriolaStats.com. Assume that the paired sample data are simple random samples and the differences have a distribution that is approximately normal.

Heights of Presidents Repeat Exercise 12 “Heights of Presidents” using all of the sample data from Data Set 22 “Presidents” in Appendix B.

Heights of Presidents Repeat Exercise 12 “Heights of Presidents” using all of the sample data from Data Set 22 “Presidents” in Appendix B.

Accepted Answer

Although, in this problem, we are told that a nutritionist wants to determine if a new diet plan leads to weight loss. The weights in kilograms of 6 participants before and after following the diet for a month are as follows. At the 0.05 significance level, the test test the claim that the mean weight after the diet is less than before and assume that the pair differences are approximately normal distributed. OK. So in order to approach this problem, the first thing we need to do is we need to go ahead and set up our hypothesis. Now, the nutritionist wants to determine if a new diet plan leads to weight loss. So, what this means is that we want our null hypothesis H not. To be that the mean weight loss is less than or equal to zero. So we want to see some negative value for our mean. However, for the opposite, or alternate hypothesis, the mean of the weight loss means that there is no weight loss. In fact, we actually have a growth in the weight loss, so the mean will be greater than 0. So, this is going to be our null hypothesis and our alternate hypothesis, but now what we want to do is we want to go ahead and determine the meaning of the data set. Now, in order to determine the meaning of the data set, we are going to take the values. Of before the weight loss and after the weight loss, and find the differences in the weight loss. So for A, we are going to take the before value which is 92, and subtract the after, which is 90, and that's going to give us a value of 2. And we will repeat this process for B all the way to F. So we have 85 minus 83, which is 2. We have 78 minus 77, which is 1. 88 minus 86, which is 2. 95 minus 92. Which is 3 And finally From dataset F, we have 80. Minus 78, which is equal to 2. Now what we want to do is we want to determine the mean of the averages of the weight loss. So what we are going to do is we are going to take the differences in the before and after weights, sum them all up, and divide by the total amount of data, which is 6. So the mean in this case, which is going to be labeled as X bar, is going to be 2 + 2. 2 + 2 + 1 + 2 + 3 + 2, and we will divide this by the total sample size, which is 6. This is going to result in 12 divided by 6, which is equal to 2. Now what we want to do is we want to determine the sample standard deviation. Now, the sample standard deviation is defined as the following. It is the square root. Of the sum of all the data points, Where we have the differences. Or we take each data point, subtract the mean from it, and square that value, and we will go ahead and divide by the sample size minus 1. So, what we need to do in order to find the squared variation is that we're going to take each of the differences and subtract the mean value from them, and we are going to repeat this process for each data point, and then square each of the data points in the process. So, for example, for A's data point for the squared variance, We are going to take 2 and subtract our mean value, which is 2, and square that value, and that is going to give us 0. For B we have 2 minus 2 squared which is 0. For We have 1 minus 2 squared, which is 1. For D, we have 2 minus 2 squad, which is 0. For E, we have 3 minus 2 squared, which is 1, and for F, we get 2 minus 2 squared, which is 0. Now, if we take the sum of all these data points squared, we get the value of 2. So, for the sample standard deviation, we have the square root. Of 2 divided by 5, and that is going to give us the square root of 0.4, which is equal to 0.6325. This is going to be our sample standard deviation. Now what we need to do is we need to calculate the sample error. Now, the sample error is defined as the following. It is the sample standard deviation divided by the square root of the of the sample size. That is going to be 0.6325 divided by the square root of 6, and this is going to simplify. To give us the following value. We get 0.2582. Now that we have the standard error, let's go ahead and calculate RT statistic for the hypothesis test. Now, for the T statistic, it is defined as the following. We take the mean, which is X bar, subtract the original mean that we are working with. And divide by the standard error. Now for our hypothesis test, recall that we want our null hypothesis to be less than or equal to zero. And so that means that for our null, we are going to plug in 0, and we're going to plug in the mean, which is 2 for the numerator. So we get 2 minus 0. And again, the standard error we calculated was 0.2582. So once we plug this into a calculator, we get 7.746, which is going to be their T statistic. And finally, what we need to do is we need to find the critical value for this T statistic. Now, the critical value is defined as the following. Well, first we need to find the degree of freedom, and the degree of freedom is defined as the sample size minus 1, which is going to be 5. So, our critical error is T 0.05 divided by 2, with respect to the degree of freedom, which is 5. And by using our T table, this is going to be 2.015. Now, what we want to do is we want to go ahead and compare our T statistic to our critical value, but notice that our T statistic, 7.746 is greater than 2.015. This suggests that the mean weight loss is less than before, and that means that we fail. To reject or we completely reject the no hypothesis. So we reject There are no hypothesis, and with that being said, the solution to this problem is going to be B. So, I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Paired Samples

Normal Distribution

Simple Random Sampling

Watch next