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Problem 7 Medium Difficulty

MEDIA For Exercises 6 and 7 , use the following information. A survey found that 5$\%$ of consumers said they will not have any debt from holiday spending. Suppose the survey had a margin of error of 3$\%$ .
How many people were surveyed?

Answer

1089

Discussion

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Top Algebra Educators
GH
Grace H.

Numerade Educator

Catherine R.

Missouri State University

Anna Marie V.

Campbell University

Michael J.

Idaho State University

Video Transcript

So on this one, we have been told that we have a margin very 3%. We've been told that RPR percentage as 57%. That's how many people responded a certain way on this survey which happened to be about debt. Doesn't matter, though. We're being asked to figure out how many people were surveyed, which would be in in this case, that's asking us the size of our sample. So I've got the formula for margin of error written here. We're gonna pluck in the two things that we do no margin of error and percent, and then we're just gonna have to solve this equation out for it. Okay, so we start with our margin of air. It is 3%. But remember, we don't use percentages and formulas. We write them is decibels. So remember, 3% is really 30.3 Because percent is a term that means divide by 100 a time. You want change a percent to a decimal divide by 100. So we're really dealing with 0.3 That's gonna be equal to two times the square root of our percent R p, which is not 57 because again, that's a percent. We need to divide that by 100 to change it to a decimal, which would be 0.57 And then that is being multiplied by the quantity of one minus that exact same p of 10.57 again. And all of that is being divided by what we're actually here for, which is, and it's the one thing that we do not know right now. Okay, so we need to work our way towards solving this stuff out. Um, I know at some point I may need to get rid of that square root because the end is inside the square root. And until I deal with the square root, I can't mess with anything inside it. But I don't want to mess with the square right now until it is isolated. Meaning my number one issue is that, too, that is outside the square root, but still on the same side of the equation as it It is two times that square root. So the opposite of multiplication would be division, meaning I'm going to divide to over to the other side. Okay, so if we take 0.3 and divide that by two that would give us 0.15 and then that is equal to what we've got left over here, which is the square root of 0.57 times. Let's go ahead and do one minus 10.57 one minus 10.57 would be 0.43 Okay, and then that's divided by what we don't know. Which is it? All right. So now the square root is isolated. Now I'm ready to actually take care of it. Well, the question then, is what is the opposite of Square Rudy? The opposite of square rooting is squaring, meaning, taking something to the second power. So I am going to square both sides of this equation. If I take 0.15 and square it, I will get 0.0 00 2 to 5. Okay, then over here I am going to go ahead and take 0.57 times 0.43 because the square in the square cancelling out. I guess I should have said that, right? That's cancelling out. That's the whole point of this. So 0.0.57 times 0.43 is 0.2451 and I still am dividing that by end. What? I actually what we actually came here for? Okay, Well, and isn't the denominator I need to get that up to the numerator because I'm trying to get in by itself. I'm solving for N. I'm not solving for one over in, right. Well, we know a fraction bar is the same thing. Is division the office? Oh, division is multiplication. So I'm going to multiply n on both sides of the equation that will cancel out here, And I will have 0.2 to 5 times N is equal to 0.2451 which means final step to get our answer here, I have that decimal multiplying to end. So the opposite of multiplication is a division, meaning I am going to take 0.2 to 5 and divide that on both sides of the equation. And whatever I get should be your answer. So if I take 0.2451 and divided by 0.2 to 5, looks like we get around 1000 and 89. Yes, there is a decibel. Their bullets. Remember what we were asked. We were asked how many people were surveyed. You can't really survey 1/3 of a person, so we're gonna ignore that 890.3 repeating. Our answer is just 1089.

University of Central Missouri
Top Algebra Educators
GH
Grace H.

Numerade Educator

Catherine R.

Missouri State University

Anna Marie V.

Campbell University

Michael J.

Idaho State University