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Numerade Educator



Problem 43 Medium Difficulty

Demonstrates that probability is equal to area under a curve. Given that college students sleep an average of 7 hours per night, with a standard deviation equal to 1.7 hours, use the scroll bar in the applet to find:
a. $\quad P(\text { a student sleeps between } 5 \text { and } 9$ hours)
b. $\quad P(\text { a student sleeps between } 2 \text { and } 4$ hours)
c. $\quad P(\text { a student sleeps between } 8 \text { and } 11$ hours)


a .7606; b .0372; c. .2689


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Video Transcript

okay for this problem were given the context of some sleep information, and we're told it's normally distributed. So we should always sketch out a normal distribution if we're told it's normally distributed the curve. So this problem says, Ah, the average of hours of 97 and the standard deviation equals two 1.7 hours. So I don't make this drawing too busy. But if I had a lot more space, I would kind of dish out and standard deviations and kind of show going above 1.7 and then 1.7 Maurin back 1.7. Um, but for this problem, we're going to show three different answers. So I want to keep some space here. Uh, and the problem asked us to find first for part, A wants us to find the probability of students sleep between five and nine hours. So, um, in between problem. So, um, and it also asked us to use an apple. It So we're over here. My favorite is Ah, stab lit. And we could see all these different categories here. This is a normal distributions and click on that. So to get a good diagram to system where the means standard deviation or we're gonna use the information that gave us. So they told us the meaning of seven anistan. Deviation is 1.7 can see. It populates that for us there is no clear about version. Like I said, 8.7 10.4. That's one standard deviation. Two standard deviations away. Um, because there's three answers. I'm gonna sketch at all three here, and we'll look at each of these. So, um, let's look at our left right boundaries and is in between. Values s really about alone. And so we know over going for 5 to 9 around that, you mean value of seven. Just give us our area s of the area between five and nine is just 0.76 So six. Okay, so just for a little picture there, so five would be a little bit more than once they are deviation below. And nine could be 90.7 a little bit higher. Your teachers are gonna want you to sketch these out. If I had more space, I would sketch three separate ones. But for this problem, that kind of shows what's going on there. So that's for that one. I'm gonna call the 2nd 1 green. I'm not gonna recalculate this cause the same distribution with a mean standard deviation. But for the B of the problem, it asked us to look between two and four hours. So looking between two and four gives us a left and right boundary. We calculate the area, we can see that that is 0.372 So it's right seventh in rotation. The probability that sleep is between normally distributed sleep my research there. So between to for and I just an answer point Stroh 37 to that's that can just to show a little bit direction there. So it would be a little bit little. Also for would be approximately here to be over there. It's a proportionally. It should make sense. Too much smaller amount of sleep context. So that would you that one. And then for my 3rd 1 it asks between, um state in 11 hours. So my channel founded eight and my right better to 11. Since that was the here and will across over years, eight would be around here 11 a little bit higher, roughly about there again, I would shoot on paper and show all three. And I label out a nice sneak label like you see on this, uh, stop. Let you composite can write in these values and then make your boundaries relative to what you see. Um, but for this one, we got that made so between eight and 11 relatively small amount of should be, so looks like it's 0.2689 So we see it. Show the direction we say within inequality for the 3rd 1 For part C, it's Ah, excess between yes, occasional ever probability between eight and 11 hours, if it's normally distributed, is 0.2689 little, almost 27%. Okay, be clear their support. See answer part B. Molo end And we answered kind of way in the middle. There a big chunk A. And that's how we find the values of sleep with an apple. It