Join our free STEM summer bootcamps taught by experts. Space is limited.Register Here 🏕

Like

Report

Numerade Educator

Like

Report

Problem 67 Hard Difficulty

The extraction force required to remove a cork from a bottle of wine has a normal distribution with a mean of 310 Newtons and a standard deviation of 36 Newtons.
a. The specs for this variable, given in Applied Example $6.13,$ were "$300\mathrm{N}+100 \mathrm{N} /-150 \mathrm{N}$" Express these specs as an interval.
b. What percentage of the corks is expected to fall within the specs?
c. What percentage of the tested corks will have an extraction force of more than 250 Newtons?
d. What percentage of the tested corks will have an extraction force within 50 Newtons of $310 ?$

Answer

a. from $150 \mathrm{N}$ to $400 \mathrm{N}$
b. 0.99377 or $99.4 \%$
c. 0.9525 or $95.3 \%$
d. 0.8354 or $83.5 \%$

Discussion

You must be signed in to discuss.

Video Transcript

this problem is talking about the removal of quirks from wine bottles being normally distributed and the force it takes has a mean of 310 Newtons and a standard deviation of 36 Newtons. So for part A here it gave us some notation and it says the specs for the very for this variable were 300 Newtons plus 100 Newtons. Then it used disease, division, sign or a slash and it goes minus 1 50 Newtons, or what this really means is that we're going to do 300 Newtons plus 100 Newtons to get the top end and we're gonna do 300 Newtons minus of 1 50 Newtons to get the bottom. And so the interval of force is going to be from 150 Newtons up to 400 newness and often times that is expressed as a parenthetical notation of 1 50 come a 400. So let's approach part B in part B. We're looking for a percentage of corks that fell in within those specs, So we're really going to start with what's the probability that we are between 1 54 100 so in order to calculate the probability we will need to set up our bell shaped curve. We're gonna put the average in the center and our average was 310. So in this case, we're trying to find the probability between 150 and 400 so we will need a Z score. So to refresh your memory, the formula for Z scores are Z equals X minus mu over sigma. So the Z score associated with 1 50 would be 150 minus 310 divided by 36 which is negative 4.44 So that would go right here and the Z score associated with 400 would be 400 minus 310 over 330 over 36 which would be approximately 2.5. So we've got 2.5 here. So when we're solving this problem, we're talking about being in between here. So if we are in between 154 100 it's no different than the Z score being between negative 4.44 and 2.5. So we will have to find the probability that Z is less than 2.5 will subtract the probability that Z is less than negative 4.44 and we will utilize our standard normal table. So you're going to go to the back. Your book in the area to the left of 2.5 would be 0.9938 and we're going to subtract zero, because for negative 4.44 is off the chart. There's nothing there. Um, so as soon as you get to that bottom end, the probability is nothing. It's not going to happen. So therefore, the probability that the extraction, um, force for the cork is between 150 Newtons and 400 Newtons would be 4000.9938 and this is what percentage of the corks. So we've got a transition that into percentages. So it's 99.38% of the corks will have an extraction force between 154 100 Newtons. When we move to Part C in Part C, it's asking what percentage of tested corks will have. An extraction force of more than 2 50 says X is greater than 2 50 so we will set up our bell curve And remember, our center is that 310. And for visual purposes, I like to use that curve. 250 would be over here, so we'll need a Z score. Z equals to 50 minus 310 divided by the standard deviation of 36. So our Z score in part C turns out to be negative one 0.67 So we're gonna put a negative 1.67 here. So being greater than 250 is the same as Z being greater than negative 1.67 And since our table in the back of the book is set up to go to the left and not to the right, we're going to have to say one minus the probability that Z is less than negative 1.67 So we're gonna use the back of the book and find the area to the left of a negative 1.67 and you're gonna find 0.475 And that results in an answer of 0.95 to 5. But part c asked for percentage, not a probability. So we're then going to transition to a percentage and we could say that 95.25% of the quirks that air tested are going to have to have an extraction force of more than 250. And then finally, let's do Part D and in Part D. We're asking for a percentage of tested corks having a force within 50 Newtons of 310. So that's saying we have 310. Will 50 Newtons above or 50 Newtons below is going to put us between 3 60 and 260. So this question is really asking What's the probability that the extraction force is between 263 160? So here's our bill shaped curve again. We've got 310 in the front in the center, 260 would be back here and 360 be right here, and we're going to need a Z score for each so Z equals 2 60 minus 310 divided by our standard deviation, which was 36. And that gives us a Z score of negative 1.39 and then we do a Z score for 360 so 3 60 minus 310 over 36 which gets us a Z score of positive 1.39 so we can put those appear on our bell. So being between an extraction force between 2 63 60 is no different than a Z score being between negative 1.39 and positive 1.39 So to solve that, we're gonna have to do the easy being less than 1.39 minus the probability that Z is less than negative 1.39 You will use your standard normal table from the back of the book. In the area to the left of positive, 1.39 is 0.9177 in the area to the left of negative 1.39 is 0.8 to 3, resulting in a probability of 0.8354 Or we could say that 83.54% of the tested corks will have an extraction force within 50 Newtons. Of the 310 which was average