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Problem 34 Hard Difficulty

A set of 250 data values is normally distributed with a mean of 50 and a standard deviation of 5.5
What percent of the data lies between 39 and 61$?$

Answer

95.44$\%$

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Heather Z.

Oregon State University

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University of Michigan - Ann Arbor

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Idaho State University

Video Transcript

That's a This problem is meant to be a review over stuff those covered in section 12.7 so or 12-7. So if you haven't done 12-7. Yeah, you probably should be backing up in taking a look at that. But into old F seven talked about normal distributions, which is what I've got drawn here up at the top of the screen is a generic normal distribution. Um, so the idea is with normal distribution things. They're distributed evenly. Um, and we've got the mean the average dead center in the middle of the graph, and then it just kind of goes out from there. Um, you can see we've got percentages off to the right there. That is called the Empirical rule. And those are some general assumptions you can make that 60% of your data is kept within one standard deviation of the mean. 95% of the data is kept within two standard deviations of the mean and 99% of the data is kept within three standard deviations of the mean. Okay, so then we can take this generic knowledge and we can apply it to specific problems like the one we're gonna do here. So in this case, were given a specific problem that says we've got 250 data values that's normally distribute with a mean of 50 and a standard deviation of 5.5. So down here, I've got a blank normal distribution drawn out here. So the mean is meant to be the center, that black line. So if it told us that our meanness 50 will go ahead and label that there's our mean of 50. Okay, then tells us that our standard deviation is 5.5. So what that means is I'm going to take 50 and I'm going toe Add 5.5 and I'm gonna take 50 and I'm going to subtract 5.5. So if 50 added to 5.5 is 55.5, that is my first positive standard deviation. I could also then take 50 minus 5.5, which gives me 44.5. And there's one negative standard deviation. Okay, so I have not, uh, can finish this. I still have two and three standard deviations, so I'm going to go and I'm going to go another standard deviation. So I'm gonna add another 5.5, which would give me 61 right here. All right, then I'm gonna take go and subtract another 5.5, which would give me 39. There's my two standard deviations. There's my my two second standard deviations so that I'm gonna go again. I will add another 5.5 61 plus 5.5 is 66.5. That all go and I will subtract another 5.5 39 minus 5.5 is 33 0.5. That's how you build yourself out. A normal distribution. OK, so now we look at what We're actually being asked for this specific question, which is what person of the data lies between 39 61. Here's 39. Here's 61. So it wants to know what percentage of the data is this distance covering right here. Okay, well, that's both the red lines which, looking at my generic standard deviation appear. The red line represents a second standard deviation, right? Two standard deviations. So, judging off the empirical rule right over here, two standard deviations holds 95% of my data. So if I'm on to see if I'm within two standard deviations than the answer to this question would be 95% of the data.

University of Central Missouri
Top Algebra Educators
Heather Z.

Oregon State University

Alayna H.

McMaster University

Kristen K.

University of Michigan - Ann Arbor

Michael J.

Idaho State University