Mint Specs Listed below are weights (grams) from a simple rand...

Question

Mint Specs Listed below are weights (grams) from a simple random sample of pennies produced after 1983 (from Data Set 40 “Coin Weights” in Appendix B). Construct a 95% confidence interval estimate of for the population of such pennies. What does the confidence interval suggest about the U.S. Mint specifications that now require a standard deviation of 0.0230 g for weights of pennies?

Weights of pennies in grams: 2.5024, 2.5298, 2.4998, 2.4823, 2.5163, 2.5222, 2.4900, 2.4907, 2.5017.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A random sample of eight dimes minted after 2010 has the following weights in grams 2.27, 2.29, 2.28, 2.30, 2.26, 2.28, 2.29, 2.27. Construct a 95% confidence interval for the mean weight of all such dimes. The mint requires a standard deviation of 0.02 00 g for dimes. What does your interval suggest about the mints process? Fantastic. So it appears for this particular problem we're asked to solve for two separate final answers. Our first answer we're trying to solve for is we're asked to construct a 95% confidence interval for the mean weight of all of these particular dimes, so all 8 of our dime samples. And we're also asked to determine what does our interval, since we're trying to solve for an interval value, so what does our interval suggest about the MNS process, and that is our second final answer we're ultimately trying to solve for. So looking at our multiple choice answers, we need to note that our interval, which is written within parentheses, are all in the same units of grams, and then we're given a. Suggestion based off of what our interval states. So A is 2.1661.2.2939. The interval includes 2.268, suggesting the mins production process is accurate and consistent with its specifications. B is. 2.2661.2.2812. The interval includes 2.268, suggesting the min production process is accurate and consistent with its specifications. C is 2.2692, 2.3939. The interval does not include 2.268, suggesting the mins production process is not accurate and consistent with its specifications. And finally, D is 2.2661, 2.2939. The interval includes 2.268, suggesting the mince production process is accurate and consistent with its specifications. Awesome. So let us note that we are told that the sample weights in grams once again is 2.27, 2.29, 2.28, 2.30, 2.26, 2.28, 2.29, and 2.27. And let us also note that we're told that our sample size, which we'll call lowercase n, is equal to 8. We're also told that we have a population standard deviation which will denote this as sigma, and sigma is equal to 0.0200 g, as we're told by the prom itself. We're also told that the confidence interval, which we'll call CI for short, is equal to 95%. Fantastic. And we need to note that since sigma is known, we can recall and use the Z distribution. So, with that said, our first major step to solve this problem is we need to calculate the sample mean. So we need to note that X bar. So X bar is going to be equal to 2.27 plus 2.29 plus 2.28 plus 2.30 plus 2.26 plus 2.28 plus 2.29 plus 2.27. All divided by 8, which will mean that X bar will be equal to 2.28 g. Fantastic. So moving right along here, our next step, our second step is we need to find the Z value for the 95% confidence. So as you should recall, note that for a 95% confidence level, the critical Z level will have to be Z is equal to 1.96. Awesome. Our third step now is we need to calculate the margin of error, which is denoted as E. And E, the margin of error, is equal to Z, multiplied by sigma, divided by the square root of N, which when we plug in our known variables, we will find that it's equal to 1.96 multiplied by. 0.0200 divided by the square root of 8, which will mean that our margin of error E will be approximately equal to 0.0139 when we round to 4 decimal places. Fantastic. And now, Our fourth step that we need to take is we need to construct our confidence interval. So with that said, we need to note that our confidence our confidence interval CI for short, will be equal to X bar plus or minus E, which will be equal to 2.28 plus or minus 0.0139. So that will mean that therefore, as a result, the 95% confidence interval will have to be parentheses 2.2661.2.2939. Grams and that's it. That is our final answer for the interval. Hooray, we did it. Now let's consider our interpretation. So as he should recall a note from the problem itself, the US mint specifies a standard deviation of 0.0200 g and our interval for. The mean weight falls very close to the target dime weight of 2.268 g, which is the official expected weight. So the interval does indeed include 2.268. And suggesting that the mince production process is accurate and consistent with its specifications. So that will mean, without a doubt, when we look at our multiple choice answers, our final answer has to be multiple choice answer letter D. Which once again letter D states. That parentheses 2.2661 2.2939 g. The interval includes 2.268, suggesting the mint's production process is accurate and consistent with its specifications. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Confidence Interval

Standard Deviation

Simple Random Sample

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