Assume the machine shifts and the distribution of the amount o...

Question

Assume the machine shifts and the distribution of the amount of the compound added now has a mean of 9.96 milligrams and a standard deviation of 0.05 milligram. You select one vial and determine how much of the compound was added.

Graph comparing original and shifted distributions of vial masses, showing means and acceptable range limits.

a. What is the probability that you select a vial that is within the acceptable range (in other words, you do not detect that the machine has shifted)? (See figure.)

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 pharmaceutical company fills bottles with a saline solution. After a recent calibration, the mean fill is 50.05 mL, with a standard deviation of 0.08 mL. Bottles are considered acceptable if they contain between 49.90 mL and 50.20 mL. What is the probability that a randomly selected bottle is within the acceptable range? Awesome. So it appears for this particular problem we're ultimately trying to determine what is the probability that a randomly selected bottle lies within this acceptable range, which the acceptable range is 49.90 mL and 50.20 mL. So with that in mind, now that we know that we're ultimately trying to solve for this probability value, let's read off our multiple choice answers to see what our final answer might be. A is 0.1392. B is 0.0304. C is 0.9392. And finally, D is 0.3992. Awesome. So, first off, we need to, in order to solve this problem, we need to recall how to convert bounds to Z scores, and as we should recall, we should be able to write that capital Z subscript 1. will be equal to, for this particular prompt, 49.90 minus 50.05, all divided by 0.08. And if you noticed, I've left out the units. This is just to keep it a little bit more organized, but you need to note that these are all in units and milliliters, so if the prom asks to make sure you write the units, make sure you write the units. But for now we don't have to. So, So now when we perform this calculation for Z1, we will find that it's equal to -1.875 when we round to three decimal places. And similarly for capital Z2, it will be equal to, as we should recall, using the information that's given to us by the prom itself, it's going to be equal to 50.20 minus 50.05. Divided by 0.08, and when we plug that into our calculator, we will find that it's equal to 1.875. Awesome. So now we need to interpolate for Z equals 1.875, and we will use symmetry later to determine Z equals -1.875. Awesome. So now our second step that we need to take is we need to use a Z table and interpolate, so you should have a Z table in your textbook, and from the Z table we will determine that capital Z is equal to 1.87, so 1.87. Which therefore we could state that the probability P, so P of capital Z is less than 1.87. Awesome, will be equal to 0.9693, and also Z is equal to 1.88. We could say that the probability P, so P of capital Z is less than 1.88. Will be equal to 0.9699. Fantastic. So now our next thing that we need to do is we need to interpolate between these values. So for when Z, or specifically capital Z is equal to 1.875. We could say that the fractional distance, which we'll call FD, the fractional distance is equal to 0.5. Fantastic. So we could state that P of capital Z, so P of Capital Z. is less than 1.875, where once again P represents our probability, so we don't forget. So P of capital Z is less than 1.875 will be approximately equal to 0.9693 plus 0.5 multiplied by parentheses 0.9699. Minus 0.9693. Fantastic. So when we plug in that expression into our calculator, we will find that our answer for P of Z is less than 1.875, will be approximately. Or rather, we could just say that it's equal to when we plug it into our calculator and we round to four decial places, it'll be equal to 0.9696. Fantastic. So when we use symmetry, so we're calling and using symmetry, we could go ahead and write that P of Z is less than -1.875. will be equal to 1 minus P of Z. is less than 1.8. 75. So when we plug in our known variables, we will find that it's equal to 1 minus 0.9696. So that will mean that our answer for P of Z is less than -1.875 will be equal to 0.0304. Fantastic. So now, our third and final step that we need to take is we need to perform our final calculation, so we need to recall that P of 49.90 is less than or equal to X and X is less than or equal to 50.20. So that will mean that this is equal to Since we need to recall that this will be equal to for this particular prom, given the conditions set to us by the prom itself, it'll be equal to P of -1. 875, so -1.875 is less than or equal to Z. Awesome. So, So it's gonna be less than or equal to Z. As we determined, and capital Z will be less than or equal to 1.875. Fantastic. So, with that in mind, when we perform that calculation, we will find that our final answer will be equal to So it's going to be when we plug in our uh let's take a moment here to plug in our known variables since we know that it's equal to P of -1.875 is less than or equal to Z and Z Z is less than or equal to 1.875. So that means if you take 0.9. 696 and we subtract 0.0304. We will find that our final answer when we plug it into our calculator will be equal to 0.9392, and that without a doubt is our final answer. Hooray, we did it. So looking at our multiple choice answers, our correct answer without a doubt has to be multiple choice answer letter C, which once again is 0.9392. 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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Key Concepts

Normal Distribution

Mean and Standard Deviation

Probability and Z-scores

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