Quarters Assume that weights of quarters minted after 1964 are...

Question

Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).

a. Find the probability that a randomly selected quarter weighs between 5.600 g and 5.700 g..

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 machine produces metal rods with lengths that are normally distributed. Has a mean of 50 centimeters and a standard deviation of 2 centimeters. What is the probability that a randomly selected rod has a length between 48 centimeters and 53 centimeters? Awesome. So it appears for this particular problem we're ultimately trying to figure out what is the probability that a randomly selected rod. Has a length between 48 centimeters and 53 centimeters. So with that said, now that we know that we're ultimately trying to figure out this probability value, that's our final answer we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is 0.7475. B is 0.7547. C is 0.7745 and D is 0.5774. Awesome. So first off, In order to help us solve this problem more effectively and efficiently, let's take a moment to write down all of our known variables or the variables that are given to us by the prom itself. We're told that the mean, which will denote as mu, the mean, is equal to 50 centimeters. We're also told that the standard deviation, which we'll call sigma, is equal to 2 centimeters. And we need to note that we're ultimately trying to solve for the probability P of 48 is less than X and X is less than 53. So our first step in order to solve this problem is we need to convert to Z scores. So as we should recall, the Z score equation states that Z, the Z score is equal to X minus mu divided by sigma. So now we need to convert 48 centimeters to our Z value, and then we will do 53 centimeters for that specific Z score. So with that said, we will find that Z1 is equal to 48 minus 50 divided by 2, which is going to be equal to -1 and Z2 is going to be equal to, once again, since we're plugging in our secondary value is going to be equal to 53. -50 divided by 2, which will be equal to when we plug that into our calculator, 1.5. So now, our second step is we need to use the standard normal table or use our calculator, and your standard normal table, there should be one of those in your textbook. And with that said, we will find that the probability P of -1 is less than Z and Z is less than 1.5, is going to be equal to, once again, where P is the probability of Z is less than 1.5 minus P. Of Z is less than -1, so using the Z table, we will find that P of Z is less than 1.5 is going to be approximately equal to 0.9332, and P of Z is less than -1. It's going to be approximately equal to 0.15. 87, and once again, we got both of these values by using our standard normal table, which you can find in your textbook or you can simply use your calculator to solve. So that will mean that solving for our final answer, we will take the probability P of -1 is less than Z and Z is less than 1.5. is equal to 0.9332 minus 0.1587, which is going to be equal to we plug into our calculator, 0.7745, and that is our final answer. Hooray, we did it. So looking at our multiple choice answers, our final answer without a doubt has to be multiple choice answer letter C, which once again is 0.7745. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).
a. Find the probability that a randomly selected quarter weighs between 5.600 g and 5.700 g..

Key Concepts

Normal Distribution

Z-Score

Probability Calculation

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Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).a. Find the probability that a randomly selected quarter weighs between 5.600 g and 5.700 g..