Archeology Archeologists have studied sizes of Egyptian skulls...

Question

Archeology Archeologists have studied sizes of Egyptian skulls in an attempt to determine whether breeding occurred between different cultures. Listed below are the widths (mm) of skulls from 150 A.D. (based on data from Ancient Races of the Thebaid by Thomson and Randall-Maciver). Construct a 99% confidence interval estimate of the mean skull width.

Accepted Answer

Hello. In this video, we are told that a team of anthropologists measured the femur lengths in millimeters of ancient skeletons from a burial site dated to 200 BC. The recorded femur lengths are 410, 425, 418, 432, 440, 427, 415, and 422. We want to construct a 99% confidence interval estimate for the mean femur length of this population. So, let's just go ahead and list out the given information. First, we are given various femur estimations, and there are a total of 8 measurements, which means that we have a sample size, which we will call N. To be 8 Furthermore, since we want to construct a 99% confidence interval, that means that we are working with a confidence level. Of 99%, and this can be translated to be 0.99. Now, in order to approach this problem, let's go ahead and start by determining the sample mean and the sample square deviation. In order to determine the sample mean, which is X bar, we need to take the sum of all the femur measurements and divide by the sample size. The sum of all the femur measurements will be 3,389, and we are going to divide this by the sample size, which is 8. This will simplify to be 423.625. Now, what we want to do is we want to go ahead and find the square deviation. Now, in order to determine the square deviation, we are going to take each of our measurements, which we will call X. Subtract The mean from them and square this value. And we're going to have to repeat this process for all 8 of the femur measurements. So let's go ahead and organize our data in a table. So what I provided here is a table that gives us the square deviation. Notice that this lists out all the measurements of the femurs. Now this second column here are the values of each data point while subtracting the mean from it. So we took each data point, like 410, subtracted our mean, which is 423.625, and that gives us the second column. Then what we do is we take each of these data points and we square them, and that is going to be the resulting column. And our square deviation is going to be the sum of all these data points, and the sum of all these data points ends up being 645.877. Now, the reason why we did this is because we need this value, 645.877, in order to find the sample standard deviation. The sample standard deviation. Which we will label as S, is defined as the following. It is the square root. Of the square deviation, so the sum of all those data points. Divided by the sample size. Now we have all this information in order to determine the sample standard deviation. We know that the sum of the squared deviation is 645.877, so we are taking the square root of 645.877, and dividing this by our sample size, which is going to be 8. And if we plug this into a calculator, this estimates to be 9.61 millimeters. Now that we have our sample standard deviation, and we have our mean, let's go ahead and determine the T critical value. Now, we are going to use T scores, since our sample size is less than 30. So we want to find the T score. The T score of the complement of the given uh the given. Confidence level and that's going to be alpha divided by 2 with respect to our degree of freedom. Now degree of freedom is equal to the sample size minus 1, which is 7. So we are going to take alpha divided by 2 with respect to degree of freedom 7, and this is going to give us a value of 3.449 when we use the T score table. So now that we have the T score, we can now calculate the sample standard error. Now, for the standard error, standard error is defined. As our sample standard deviation divided by the square root of our sample size. This is going to be 9.61, divided by the square root of 8, and this will simplify to be 3.40. Finally, now that we have the standard error and the T score, we can calculate the Margin of error Now margin of error is defined as our T score. Multiplied by our standard error, and this is going to be 3.499, multiplied by 9.61. Apologies, not 9.61. It'll be 3.40. And once we multiply these values together, we get 11.90. So now that we have the margin of error, we can go ahead and create the confidence interval. Now the confidence interval. Is defined as our mean of the femur measurements plus and minus the margin of error. Now, this is going to give us the following confidence interval. 411.73,435.53, and this is going to be the confidence interval that we are looking for. And with that being said, the solution to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem, and I will go ahead and see you all in the next video.

Key Concepts

Confidence Interval

Mean

Standard Deviation

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