Problem

Determine whether the data in each table appear t…

04:38

Problem 8 Medium Difficulty

For Exercises $5-8,$ use the following information. The useful life of a certain car battery is normally distributed with a mean of $100,000$ miles and a standard deviation of $10,000$ miles. The company makes $20,000$ batteries a month.
What is the probability that if you buy a car battery at random, it will last between $80,000$ and $110,000$ miles?

Answer

0.815

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Intro Stats / AP Statistics

Algebra

Chapter 12

Probability and Statistics

Section 7

The Normal Distribution

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Video Transcript

As you can see, I've got to normal distributions drawn out here. 1st 1 up top is generic. Meaning it's just any time you were told that you have a data set that is normally distributed. This is the type of scenario that you're looking at. Okay, We know from the empirical rule that 60% of our data is kept within one standard deviation. 95% of our data is kept within two standard deviations, and 99% of our data is kept within three standard deviations. I've gone ahead and split those out because we know normally distributed means that are mean or average is smack dab in the middle. So it's splitting everything in half. You can see that 34 on 34 68. So on so forth. I am assuming that you guys understand a basic normal distribution before we're going through with this problem. If all those percentages means nothing to you and you have not been taught H and Eric normal distribution, then you're not going to be able to understand how to apply it to a specific situation like this problem. So before you continue with this video, make sure you understand how I've got those generic percentages before we start trying to apply it here? Okay, Now, assuming that you do understand those looking at this second normal distribution, this represents the actual problem we're doing. We've been told we're talking about the life of car batteries. We were told that the mean is 100,000 miles and that the standard deviation is 10,000. So I've gone ahead and marked out our three standard deviations. I added 10,003 times to the right, subtracted 10,003 times to the left to get my 1st 2nd and third standard deviation away from Army and of 100,000. Okay, Now the question here is asking us what is the probability that you will have a car that you can get a car battery that lasts between 80,000 and 100 and 10,000? Well, okay, let's mark what they're talking about. Then here's 80,000. Here's 100 and 10,000. So it's asking us about this portion this interval of our chart of our data of our graph. Okay, if you understand the generic normal distribution and it's percentages, this problem really isn't that tough. All you have to do is add up each of those slivers each of those portions of the graph. So on the generic one, here's the second standard deviation cause noticed. The 80,000 is on the red. It's the second standard deviation. Here's my positive first standard deviation because the 110,000 is the blue. It's the first standard deviation. So we're talking about this portion, meaning we simply just need to add all three of these percentages together. So that means we're taking 13.5% from the red, plus 34% plus another 34% right. That's the portion of the graph that we're looking at here. If you take 13.5 plus 34 plus 34 that will give you 81.5% or remembering that percent means to divide by 100 81.5 divided by 100 meaning in decimal form, that would give us 0.815 That is actually all we have to do for this one, because it did just ask us for a probability we don't care about the fact that the company makes 20,000 batteries a month because it didn't ask how many batteries this time, it just asked, was the probability. So we're done.

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