demonstrates-that-probability-is-equal-to-area-under-a-curve-given-that-college-students-sleep-an-av

Question

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. $\quad P(	ext { a student sleeps between } 5 	ext { and } 9$ hours)
b. $\quad P(	ext { a student sleeps between } 2 	ext { and } 4$ hours)
c. $\quad P(	ext { a student sleeps between } 8 	ext { and } 11$ hours)

Gus Steppen · Accepted Answer

okay for this problem were given the context of some sleep information, and we&#x27;re told it&#x27;s normally distributed. So we should always sketch out a normal distribution if we&#x27;re told it&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s one standard deviation. Two standard deviations away. Um, because there&#x27;s three answers. I&#x27;m gonna sketch at all three here, and we&#x27;ll look at each of these. So, um, let&#x27;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&#x27;s going on there. So that&#x27;s for that one. I&#x27;m gonna call the 2nd 1 green. I&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s 0.2689 So we see it. Show the direction we say within inequality for the 3rd 1 For part C, it&#x27;s Ah, excess between yes, occasional ever probability between eight and 11 hours, if it&#x27;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&#x27;s how we find the values of sleep with an apple. It

Andrew Kim · Answer

all right. According to the Sleep Foundation, the mean number of hours asleep people get it&#x27;s 6.8 hours. And standard deviation for that we&#x27;re assuming is 0.6 hours. But x b the random variable. That is the number of hours of sleep one gets right. So Part asks us to find the probability that somebody gets more than eight hours of sleep. All right, what? Z score eight against thes statistics. Uh, and that is, uh, two standard. You just two point. Oh, and that gives us according to our table 0.9772 So up here, the secrets, the probability that Z is greater than 2.0 So we&#x27;re gonna take one, minus the probability that Z, it&#x27;s less than or equal to 2.0 So that&#x27;s one minus 0.9772 which is 0.228 All right, part B. We need to find the probability that someone get six hours of sleep or less, which is not healthy. But, hey, it happens. All right? What&#x27;s the score? Six. All right, this is gonna be well, this up here is gonna be negative. 0.8 over. That&#x27;s negative. Forth. Uh, 0.6 which is negative for third. So this is 1.33 We checked this with our table. We get that this is equal to 0.0 918 This is the probability and sees list some 1.33 I&#x27;m sorry. Negative. 1.33 So this is 0.918 All right, so we transcribe this information just so it&#x27;s in front of me. Ah, for part c were given that doctors recommend between seven and nine hours of sleep every night. So it asked us to find the proportion of people who actually get that amount of sleep. All right, let&#x27;s find our Z lower and the upper So Z lower is gonna be seven minus 6.8. Also, overs zero points expired away. I&#x27;d like to point out the lower bound of this already greater than our mean. I don&#x27;t think this looks good already. So this is 0.2 of reserve when six. That&#x27;s 1/3 so 0.33 from the probability that Z is less than or equal to 0.33 is that we check our table. Um what? See three 0.62 93 right here we have I&#x27;m minus 6.8 all over 0.6, and ah, well, this is 3.67 because this is greater than our Z value of 3.50 Where is it going to say that the probability 0.9999 So anyway, this is gonna be the probability that ah, sorry Z is between, uh 0.33 and 3.867 And that is the probability that Z is less than or equal to 3.67 minus the probability that Z is less than or equal to 0.33 That&#x27;s 0.9999 minus 0.6293 Ah, doing some mental subtraction. That 0.37 06 So 37.6 people percent of people are getting a proper amount of sleep. That&#x27;s not great. But hey, such a CZ life

Heather Zimmers · Answer

okay, We&#x27;re using a graphing utility for this problem. So here&#x27;s my graphing calculator and we go into the y equals menu and you can see that I&#x27;ve typed in the equation that were given in the problem. This is a galaxy and model, so we&#x27;re expecting to see a bell curve, then for window dimensions. I used the X values given in the problem. It said X is between four and seven. So I use those is my X Men and my ex Max and then for the UAE maximum of 0.9. I took a cue from the equation and we see in the equation we have a 0.7979 so I wanted to go slightly larger than that. So here&#x27;s what the graph looks like. And for Part B. We&#x27;re going to estimate the average number of hours per week that the student uses the tutoring center and that would be bound at that maximum right there. We can find the maximum. If we go into the Calculate menu, which is second trace, choose number four maximum. Put the cursor to the left of the maximum and press enter into the right of the maximum press enter and then in between, and we end up with approximately 5.4 as our X value. So that&#x27;s telling us that the average student spends or that students spend an average number of 5.4 hours in the tutoring center.

Maurizia De Palma · Answer

So the question asked us to You find the round sweet between 30 and 16. And the problem gives us these parameters Here, this first function, if we are in grams, sleep greater or equal to zero minutes and the last nickel to 40 minutes and this second function if were greater than 40 minutes but also best than or equal t 80 minutes and zero if it does not fulfill any of these two time requirements. Great. So her a is asking us to find the room sleep between 30 to 60 minutes. And with that, we need to use both functions that were provided above because together both functions with Phil this 32. 16 in that time period. So to start off, the are going to have our definite integral with their limits 30 to 40 has given in the first function above with one over 1600 t d t. Don&#x27;t forget your d t. That is what helps you actually and agree. And our second portion would be 40 to 60 minutes with the function 1/20 minus one over 1600 t Bt Wonderful. So we have our in a rolls and our limits set up. Do you make this a little easier? And since this is a continent or a constant, we&#x27;re going to pull out this one over 16 hundreds that will have one over 1600 and then we can actually integrate. That&#x27;s he right above. And remember, this is technically t to the first. So integrated. We want to add one to the exponents to give us tea scored and then divide by whatever this new experiment is. So it happens to meet, too, and we want to keep her limits 30 2 40 And then for our second portion, we have one over 20 T minus one over 1600 similar to the first portion t squared. We were too under limits, 40 to 60. So that&#x27;s what we&#x27;re working with. We are now done with a calculus portion, and what I like to consider the hard portion is the algebra, because this is where you can start to make all of these mistakes. Um, like plugging in an order of operations. So we just want to keep that in mind as we are doing problem. So to start off will keep this one over 1600 and we&#x27;re going to just straight substitute in. We have 40 squared over to my nest, 30 squared over to. And if this is a little confusing to see, this is the same thing is 1/2 over T square just were in in a different way. So that&#x27;s our first question. And our second question is going to be 1/20 times 60 minus one over 1600 times 60 squared. Well, Rick, too. Oh, this deceits, uh, minus one. Over 20 trains, 40 minus one over 1600 times 40 squared over to really long and complicated re, and you can see why algebra can get a little confusing, a little messy. We&#x27;re gonna break this all down. So when we simplify this first portion here we are going to get wait to one, eat no. 218 75 And then when you simplify Oh, this will get plus 1.875 That&#x27;s what you get when we simplify the green minus 1.5. And that&#x27;s what he checked out when we simplified blue. So now we&#x27;re street. Just, uh, adding and subtracting. So looking at 0.21875 Let&#x27;s 0.375 and that&#x27;ll give us Wait, My nine, 375 Remember, this is a study, and when you give someone decimals a little hard to understand, so for practicality, senses will multiply this by 100%. And we&#x27;ll get a final answer when we round 2 59 Wait for person. And that&#x27;s the answer Report. A heart beats an ass Us. Do you find the me amount of friends Week? And to do that, we want to add Hall of our non zero parameters, meaning that first in section election and in my medical senses mean his reign as Mu and we want to add all our non zero portions rate. So they give it to infinity to infinity. The T deserve ill variable and lift functions. And for tea DT mathematically, I is what is written. Hey, logistically, it doesn&#x27;t make too much sense, right? So we&#x27;re gonna start putting in our functions. Teoh hopefully and hopefully that I don&#x27;t need a little more sense. So Oliver, non zero parameters rates So that first function gave us limits of 0 to 40 t as oh, that&#x27;s a variable. We&#x27;re trying to find times one over 1600 t day 80 plus you want to add our second function, which had the limits 42 et and T That&#x27;s variable that we&#x27;re trying to find. And then the function of they gave us the sunrise. That was 1/20 minus one over 1600 t And of course, Haouari 18 and again. So we have this constant here so we can pull this constant out. So that&#x27;s what Would you have One wish? 1600. And now, um, way can multiply this tea in this team together which will give us t squared. I can t well, this our 40 t 80 and then we can distribute this t tow both terms, so we&#x27;ll have 40 to 80 in our limits of 1/20 T minus one over 1600 t squared Don t which we look at, It can look a little like some hard. Um, remember, we&#x27;re gonna break this time piece by piece, sailor. First step is to an agreed. We&#x27;re going to keep our one over 1600 on the outside and against integrate, we want to add one to her exponents and then defined by that new exponents, so t cute over three over here. Zero t 40 plus our second inter goal. Well, our second integration, which would be 1/20 but t squared over to minus one over 1600 times t cute over three. 46 42. You all right? And hopefully this starts to make a little more sense. So we&#x27;re gonna keep her one over 1600 on the outside. Appear we can put our 40 cubes over three minus zero. Keen over three. And that&#x27;s just zero. Plus our 1/20 times 80 squared over two minus one over 1600 times. Any cute over three minus. We went over 20 chains 40 squared over two minus one over 1600 times. 40 kids over three. And again, this is where algebra, uh, can get a little hard. Little tricky if not careful if you move too fast by with that first, this first funeral here, this simplifies to 13.33 roughly. And then this second portion here. Ah, when we do Oh, we&#x27;ll get it. Plus 53 made 33 minus 26 6 seven. And again, we want to subtract this and then at that and then antis 13.33 So when we simplify that, we get about 39.96 minutes or to round this up and make it nice and me about 40 minutes.

Bryan Meares · Answer

Petey at Florida State University, has 14 statistics. Class is scheduled for a summer 2013 terms. One class of space available for 30 students. Eight classes have space for 60 students. One class have space for 70 students, and four classes have space for 100 students. The question A What is the average class size? Assuming each class is filled to capacity? Because in this problem were interested in the class size, we&#x27;re gonna let our random variable X be represented with 30 60 7100 to represent the students in each class. Because we have 14 classes, we can represent the probabilities of each of those students. Class size is to be one at a 14 for 30 ate at a 14 for 60 one out of 14 for 70 and four out of 14 for 100. So the question is saying, What is the problem? What does the average class size no average mor mu is found by adding together, each x value multiplied with probability. So we have our 30 multiplied by its one and a 14 60 multiplied by a 14 70 multiplied. Both won at a 14 and 100 times four out of 14. We can add together these four products and get an average class size 0 70 be spaces available for 980 students. Suppose that each class is filled to capacity and select a statistic student at random. Let the random variable hex equal the size of the student&#x27;s class to find the pdf Breck. So because we&#x27;ve already found our X values and our probabilities of those X values, we&#x27;re just gonna put those in the table to create their probability distribution we never expect. Used for 30 60 70 and 100. The probability of 30. It was one of 14. The 60 814 70 was one of 14 and 100. It was for out of port. You know that this is legitimate probability distribution is each probability 01 and then some of the four probabilities is one. See, find the mean of X, the mean of X. It&#x27;s the same exact thing is you&#x27;re expected value. So for this problem, we&#x27;re doing the same thing We didn&#x27;t, eh? We know that the means is 70 deep. Find the standard deviation. Our standard deviation is found. It was a pretty long formula, but it&#x27;s not that complex square root of the summation of the X value, minus the mean squared times. The probability because we have four X values. What this is gonna look like. It&#x27;s 30 modest, 70 squared multiplied by the probability of 30 which was one of 14 because that 70 it&#x27;s the same mean for every single one of these at 70 LaRue be repeated in all four for the next 60 minus 70 squared. Stop the probability of 60 touches 8 14 at that to 79 70. Weird, I&#x27;m the one about a 14. Lastly, at 100 minus 70 cleared four out of 14. Then we have to remember to include a radical drama radical at the end. Make sure it&#x27;s long enough. You can type this in. We can find that are standard deviation. 20 0.7

Lily An · Answer

Okay, so this one says the circle graphs above show the distribution of students at two universities based on how much time, on average and minutes they studied each day. If one student has chosen at random from all the students at universities A and B, what is the probability to the nearest ten thousands that he or she studies, on average at least three hours per day? Okay, so if we want to know for both universities A and B and we have to combine them, we have to do the total students of A and B. That&#x27;s what we&#x27;re pulling out of, which is going to be the fifteen thousand from a and then plus a twenty five thousand from B, which should give us something over already thousand and then on top. It should be the one hundred eighty plus minutes of a plus. The students that study one hundred eighty plus minutes from B. So, in order to get the number of students that study one hundred eighty plus minutes from A, we know that out of the fifteen thousand students policy on ly ten percent study more than one hundred and eighty minutes. So then we can multiply fifteen thousand by point one on which got one thousand five hundred students. We&#x27;re gonna put that here and done for us. University, be out of the twenty five thousand students. We know that the number of people that students that studied ah, hundred and eighty minutes or more is just eight percent, which is point oh eight. So if we do that, it should be two thousand students. So then we can add two thousand students here. So then, if we did the two thousand students plus the one thousand five hundred students, that gives us three thousand five hundred and then from here we just have to divide the three thousand five hundred students by the forty thousand total students and that should give us zero point zero eight seven five.

Problem

Demonstrates the effects that the mean and standa…

Problem 43 Medium Difficulty

Answer

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

Related Courses

Elementary Statistics

Related Topics

Discussion

Top Intro Stats / AP Statistics Educators

Srikar K.

Catherine R.

Michael J.

Joseph L.

Intro Stats / AP Statistics Courses

Data And Variables

Data And Variables - Example 1

Recommended Videos

Watch More Solved Questions in Chapter 6

Video Transcript

Robert Johnson, Patricia Kuby

Elementary Statistics

Srikar K.

Catherine R.

Michael J.

Joseph L.

Data And Variables

Data And Variables - Example 1

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

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

Elementary Statistics

Srikar K.

Catherine R.

Michael J.

Joseph L.

Data And Variables

Data And Variables - Example 1

Robert Johnson, Patricia KubyElementary Statistics

Srikar K.

Catherine R.

Michael J.

Joseph L.

Robert Johnson, Patricia Kuby

Elementary Statistics