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.

Measured and Reported Weights Repeat Example 1 using all of the 2784 measured and reported weights of males listed in Data Set 4 “Measured and Reported” in Appendix B. Did the larger data set have much of an effect on the results?

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says in a study of 2000 patients, researchers recorded both the actual and estimated number of hours left per night. For each patient, the difference, the actual minus the estimated was computed. The sample mean of the differences is -0.3 hours, and the sample standard deviation is 1.2 hours. Test whether there is a significant difference between the actual and estimated sleep hours at the 0.05 significance level. What is the P-value for this hypothesis test? And we'll give 4 possible choices as our answers. For choice A, we have P is equal to 0.05, and there's not a significant difference between the actual and estimated sleep hours. For choice B, we have P is equal to 0.001, and there is a significant difference between actual and estimated sleep hours. For choice C we have P is equal to 0, and there is a significant difference between actual and estimated sleep hours and for choice D, we have P is equal to 0.10, and there is no significant difference between actual and estimated sleep hours. So, to answer this problem, the first thing we need to do is write down our hypotheses. So, for our null hypothesis, H dot. We're gonna have that mu is going to be equal to 0. Where me here is the mean difference. For our alternative hypothesis, HA. We're gonna have that U is not equal to 0. Now, since we're dealing with a large sample, since we have over 2000 points in our sample, we'll be using a Z test for the hypothesis test. So, we call your formula for calculating our Z square. So we'll have Z is equal to the quantity of D minus you not in quantity, divided by the quantity of S divided by the square root of N in quantity. Where D-bar is the sample mean of the differences. You not is the hypothesized mean difference, S is going to be our sample standard deviation, and N is our sample size. And we're actually given all those values in our problem. So substituting in the values, we see that Z is going to be equal to. The quantity of ford bar, we have minus 0.3. Then we'll have minus muk not, we're testing that mu is equal to 0, so we'll set that equal to 0 in quantity, and that's going to be divided by the quantity of S, which is our sample standard deviation, and we were given a value of 1.2 in our problem, and that gets divided by the square root of N, wheren here is going to be 2000. And when we evaluate this expression, this is could be approximately equal to -11.18. So we can look up the P value for the Z score in a standard normal distribution table or using a statistical tool. However, a Z value of -11.18 is extremely far in the tails of the distribution. So the P value is going to be essentially zero. So we'll just have our P-value equal to 0, and that's going to be the answer for the first part of our question. So now we need to compare our P-value to our significance level. In our case, our significance level alpha. Was given to be equal to 0.05. In our case, P is going to be less than alpha. That's 0.05. is actually greater than 0. And so since P is less than alpha, that means that we will reject our no hypothesis, so we'll reject. Each night So what that means is that there is there is a significant difference between the actual and estimated sleep hours. And so based on that conclusion, if we look at our answer choices, we see that the correct answer choice actually corresponds to answer choice C. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Paired Samples

Normal Distribution

Effect of Sample Size

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