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Problem 23 Hard Difficulty

ELECTION PREDICTION One hundred people were asked whether they would vote for Candidate A or Candidate B in an upcoming election. How many said "Candidate A" if the margin of error was 9.6$\% ?$

Answer

$p=64$
$p=36$

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Top Algebra Educators
Anna Marie V.

Campbell University

Alayna H.

McMaster University

Caleb E.

Baylor University

Kristen K.

University of Michigan - Ann Arbor

Video Transcript

another question here that's gonna ask us to use the margin of their formula but not asking us to actually find the margin of error or finding one of the other two variables that we've got here. OK, so it tells us that 100 people were asked whether they could vote for candy Air candidate whether they would vote for Canda Air candidate being an upcoming election. Okay, if 100 people were asked, then that's our sample size meeting they just gave us. And right, if they even set when we say size of the sample for in that means we're talking about how many people got asked survey whatever we're talking about here. So if 100 people got asked, there we go breath After that, they tell us what end is. Okay, that's great. So we continue going and says How? Maney said can't a If the margin of error was 9.6% all right, well, it just told us right there, Then that margin of error was 9.6% so I m e is equal than 9.6%. However we do not. Are you immature on the do not disturb we do not want to leave that as percentage, right? We don't use percentages when we plugged into a formula. So we need to change that to a decimal. The way you change from a percentage to decimal is dividing by 100. So that means we would get 0.96 Okay, so we have two of our three variables, which means we must be trying to solve for P. In this case, on leeway we're gonna be able to do that is take the two pieces of information that we do have plug them into this formula that we have been given and solve it out. So we'd start with 0.96 is equal to two times the square root of P, which we do not know times one minus that same P that we do not know. And all of that is going to be divided by n Okay, so and are sorry. And then we need to plug in 100 friends. I don't know what I'm doing. So to solve this out, we want to try and get P by itself, if at all possible. Okay, so I see that I've got this to outside the square root, since that, too, is outside the squared. I'm gonna focus on that first, because I know I can't mess with anything inside the square until I've gotten rid of square root itself. And we're not supposed to get cancel out square roots unless they've been isolated. Me and they're by themselves on one side of the equation. So, uh, to is being multiplied to the square root eso The opposite is of multiplication is division. So I will be dividing that to the other side. If we take 0.96 and we divide that by two, that would give us 0.0 for eight. Okay. And that is equal to the square root of all that same stuff. Soapy times one minus p. Divided by 100. Now all I have is stuff inside the square on the right side, meaning until I get rid of the square root. I can't keep solving this problem out. So that's my next job is to cancel out the square root. So the question is, what's the opposite of square rooting All the episodes squaring would be square, so I'm gonna have to square both sides here. Okay, if I take 0.0 for eight and square that I get not a nice looking number, but we'll have to deal with it. Which is 0.0 2304 It looks like on the right side nothing changes cause it's just the square in the square cancelling out. So we've got P times One hopes didn't get the color changed pain there. Times one minus p. Divided by 100. That's where we're sitting at right now. Come on now, if we mess with this, we need to be able to get rid of that 100. Or I would say the next best thing of security had 100 movie together side because we want to get p by itself. So we want PT kind of stakes. It's already on that side. We know division bar or kind of blue is we know a fraction bar is the same thing as division. So the opposite of division is multiplication. Meaning I want to multiply by 100 on both sides. If I want to get that 100 moved over. Okay, well, ah 100 times 1000.2304 Would just be 0.230 for And that is now equal to p times the quantity of one minus that exact same peak. Now, the goal here is to try and get be by itself. But I have two different instances of P, so I can't just move one of them to their side. I have to keep them together. What I'm gonna have to do here is distribute right. I have p times a quantity, so I'm gonna have to take p Times one and p times negative p. So I've got 0.2304 p times. One is p p times negative p is negative. P Square's looking at this equation. Now, this is a quadratic because I've got the p squared term, which means I'm gonna have to solve this out using some sort of method for solving quadratic so I can either try and factor. I can use the quadratic formula or I can try and complete the square. Ah, for me, I prefer the quadratic formula. I think that's the one that works the easiest for me. So I'm gonna go for that in order for us to be able to use a quadratic formula. The first thing we have to do is get everything on one side of the equation. Meaning I need to have a equal to zero. Okay, I prefer when I'm doing the quadratic formula to have a positive square term. So I'm going to choose to add p squared to the other side of the equation and also subtract P over here. Okay? Meaning? I have positive P squared minus p plus 0.2304 equals zero. Okay, I am now going to use that line. Quadratic equation. Remember, quadratic equation is X equals negative B plus or minus the square root of B squared minus four a c all over two A. Okay, A represents the coefficient in front of my square terms. That would be this invisible one right here. B represents the coefficient in front of, uh, my variable to the first power. So that would be this negative one right here. And C stay represents the constant, which is whatever number doesn't have any variables with it. Which would be the 0.2304 in this case. So there's my A B and C. I'm gonna toss that into this quadratic formula right over here. Okay. The negative of negative one would be, And in this case, it's really p, not. Exit will say p equals this formula. The negative of negative one would be positive. One plus or minus the square root of RB value. Negative one squared minus four. Times are a value of one. Times are see value of 0.2304 All of that is over. Two times are a value of one. And we're gonna have to go toss this into our calculator because there's no way that squared is gonna come out to a nice number that weaken square in our heads. More than likely at least. Okay, if you type of this in Ah, well, let's take just type the square root part. Okay? So negative one squared is positive one and then take negative four, uh, minus. Uh, our native four times one and times 10.230 for Okay, we'll put it that way. If you have the positive one, and then you take negative four times one times 10.23 or four and actually become vinyl that together you should have one plus or minus the square root of 0.7. Sorry. Zero point 0784 all over to remember There are going to be two answers from this because you need to type in one plus the square root of 10.78 4/2. And then you also need to type in one minus the square root of 10.0 70 Forward to this is definitely for your calculator. Not for you guys to be trying to do in your head. Okay, so if you do one plus the square root of 10.784 divided by two, you should get that P equals 0.64 If you do one minus the square root of zero point 0784 divided by two. You should get P equals 0.36 Remember, these are desperate ALS and we were asked to put stuff in percentage form. Right? We've always been putting all this stuff in percent form. So our answer is either 64% or 36% said they would choose candidate A. We don't know which one. It could be either of them. It just depends. Okay? We don't actually know which one is which. And so we have to answer that. It could be either of these. We don't really have any other options.

University of Central Missouri
Top Algebra Educators
Anna Marie V.

Campbell University

Alayna H.

McMaster University

Caleb E.

Baylor University

Kristen K.

University of Michigan - Ann Arbor