Minting Quarters Listed below are weights (grams) of quarters ...

Question

Minting Quarters Listed below are weights (grams) of quarters minted after 1964 (based on Data Set 40 “Coin Weights” in Appendix B). Construct a 95% confidence interval estimate of the mean weight of all quarters minted after 1964. Specifications require that the quarters have a weight of 5.670 g. What does the confidence interval suggest about that specification?

Accepted Answer

Hello. In this video, we are told that a researcher collects the following weights in grams of dimes minted after 2000. The values are given as 2.312, 2.284, 2.298, 2.315, 2.289, 2.301, 2.307, 2.294, 2.319, 2.288. And we want to construct a 95% confidence interval estimate of the mean weight of all dimes minted after 2000. The official specification required requires a weight of 2.268 g. Then we want to ask ourselves, does the confident interval suggest the means meets the specification? So, let's just go ahead and list out all the given information. First we are given. The weights of all the gram of the grams of dimes minted after 2000. Notice that we have a total of Of 10 dimes and that means that we have a sample size. Equal to 10 Furthermore, we want to construct a 95% confidence interval, and that means that we are working with a confidence level. of 95%, which translates to 0.95. Furthermore, we are told that the official specification requires a weight to be 2.268 g. So let's go ahead and list that down. OK. So the first thing that we want to do is we want to go ahead and calculate. The mean of the sample and the squared deviation, which will help us find the standard deviation. Now, in order to calculate the mean, what we need to do is we need to take the sum of all the data points that are given to us. We will go ahead and call this mean X bar, and the sum of all the data points given to us is 23.007. Then in order to calculate the mean, we want to divide this sum by the total amount of dimes that we have, which is the sample size, and that is going to be 10. Once we plug this into a calculator, we get the value of 2.3007, and this is going to be the mean weight in grams of the dimes. Now what we want to do is you want to get the square deviation in order to find the standard deviation. In order to take or in order to find the square deviation, we want to take each data point that we have, subtract the value of the mean from it, and square, then we will go ahead and sum up all of these data points. So, let's go ahead and organize our data in the table. OK, so this is going to be our data organized, and we can see that we have all of our data points on the first on the first column. The sum of those data points is 23.007, and that gives us our mean. And what we did with each data point was we took our calculated mean, which is 2.3007, and subtracted it to each data point to produce the second column. Then what we need to do is we need to go ahead and square each of these data points, which produces this third column. And finally, in order to find the square deviation, we will take the sum of all the data points in the third column, and that gives us a square deviation of 0.00133610. Now what we can go ahead and do is we can use this to find our standard deviation of the set. Now, the standard deviation is defined as the following. It is the square root. Of the sum of the square deviation. Divided by our sample size. And Our sample size in this case is going to be N minus 1. So this is going to give us The square root Of 0.001, 33,610, divided by our sample size minus 1, which is 9, and that is going to give us the value of 0.01218. So this is going to be the standard deviation in grams. Now that we have the standard deviation, we want to find the T score with respect to our confidence level. And our T score with respect to a 95% confidence level is going to produce a value of 2.262, based off of our T table. Since we have these values, now we can go ahead and do is we can calculate the standard error. Now, the standard error is going to be our sample standard deviation divided by the square root of our sample size. That is going to be 0.01218, divided by the square root of 10, and in the calculator, this will simplify to be 0.00385. Now that we have the standard error, we can go ahead and use the standard error to calculate our margin of error. Now, the margin of error is defined as our T score with respect to our confidence level, multiplied by our standard error. This is going to give us. 2.262 multiplied by 0.0. 00385. And when plugging this into a calculator, we will get the value of 0.00872. And finally, since we calculated the margin of error, we can now construct the confidence interval, and the end points of the confidence interval is going to be the mean of the set, plus and minus the margin of error. This is going to give us 2.29198 and 2.30942. And this is going to be our confidence interval with 95% confidence level. OK. Now we need to answer the last part of this question. Does the confidence interval suggest the mean meets the specification? Well, since our specified value from earlier, which again is 2.268, is not contained within our confidence interval, then that means that that the confidence interval does not mean. Does not suggest the specification. And so 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

Confidence Interval

Mean Weight

Specification Requirement

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