SHOPPING According to a recent poll, 33$\%$ of shoppers planned to spend $\$ 1000$ or more during a holiday season. The margin of error was 3$\% .$ How many people were surveyed?
All right, so we're being asked another margin of error question here. That is a little different because unlike the ones we've been doing up to this point, um, it's not asking us to find the margin. Various asked us to find a different part of this, but we're still gonna be using the formula for margin very that you can see here. Remember, we've got three pieces to this formula. Three different things that we want that we need to know to be able to plug everything into this formula. So we've got the margin of error. Of course, P represents the percent of people that respond a certain way. So it's just, you know, whatever it happens to be for that scenario and then end is post represent the size of the sample, the size of the, you know, the number of people that were asking or number of people that were surveying that sort of thing. Okay, so in this scenario tells us that 33% of shoppers plan to spend $1000 or more holiday season. Okay, well, that's our person then, right? That's that's a percentage of people who responded a certain way they're responding that they plan to spend $1000 or more. So 33% is meant to represent RPI value in this case. Now, we don't want to actually use 33%. You know, when we're going to use this in the actual formula, we're gonna want to change that to decimal form percentage. If you're dealing with the percentage. The way you change that to a decimal is you want to divide by 100 right? So 33% divided by 100 gives us 0.33 So that's what we'll be using for P in this formula is 0.33 Okay, then it goes on to tell us that the margin of error is 3%. Well, there's not a lot of debate. There specifically tells you what 3% represents. And that's our margin of air. So slightly darker green. There margin of error is equal to 3%. Same with P. We don't want to leave that as a percentage before. In order to use it are formula. We want to change that to decimal form. 3% divided by 100 would give us 0.3 all right. Make sure you don't forget. It's not 0.3. That be 30% is 300.3 Okay. And then it asks us how many people were surveyed. Well, there's only one variable left that we don't know yet. And that would be in All right. We do not know what end is yet, so we're gonna have to do here. We're gonna take this formula that we've been given. We'll plug in the two variables that we've been given, and we're gonna have to solve it out for in. So looking at this formula, we've got 0.3 for a margin of air is equal. Teoh, too. Times the square root of P, which we said is 0.33 times the quantity of one minus that exact same p. So again, 10.33 And all of that is being divided by what we don't yet know which is n the part that we're trying to find. So let's work towards solving this out. Um, I want to figure out what the numerator of that fraction is going to be. I want to figure out what this part is right here. So I'm going to take points are won minus 0.33 which would give me 0.67 And then I'm gonna multiply 0.33 times, 0.67 And that looks like it should give us just rewrite everything else looks like 0.33 times 0.67 would be 0.22 11 looks like. And then all of that, it's being divided by N Okay, that's a good start. Remember, any time we're solving for a variable, our goal is to get that variable by itself. So I still have a lot of stuff to do because I can't get everything over to the other side of possible. I'm trying to get in by itself is as quick as possible. Um, I see that I have a two here that is outside the radical. Okay, So before I mess with that square root, I want to take care of that, too, because remember, any time you're solving an equation that has a square, root it before you mess with the square root. You want to make sure it is isolated, meaning you want to make sure the square root and Onley the square root is on one side of the equation. OK, so that, too is multiplying to the square root. The opposite of multiplication is division, so I want to divide that to to the other side of the equation. 0.3 divided by two would give me 0.15 And then we've still got the square root of 0.2 to 11 over n on the other side. Okay, so now we've isolated the square root. So now we've isolated weaken, Try and cancel it out. Try and get rid of it. So the question is, what is the opposite of square rooting? Well, the opposite of square rooting would be to square something. Take it to the second power. So I'm going to square both sides of this equation. Okay, uh, on the left side, 0.15 squared would give us 0.0 25 So three zeros and then 25 Sorry. Nope. I read that wrong off my calculator. That would be 0.2 to 5. It's why you should be double checking me with your calculators as well, I guess. Huh? Yeah. Yeah. Now I'm really great. Okay. And then on the other side. We've got that equal to 0.2 to 11 over in. So I'm trying to solve for it we're trying to solve for N. We're not trying to solve for one over end meaning It's a problem that end isn't the denominator. Right now, Before I convinced this problem, I'm gonna need to get in out of the denominator. Which means the next best step would actually be ticket and out of the denominator. And we know fraction Bar means the same thing as division. The opposite division is multiplication. So if I want to get in out of the denominator, I need to multiply by n on both sides. That means that now we've got 0.2 to 5 n equals 0.2 to 11 A lot of decimals. Beautiful stuff here. Finally, to get end by itself, I need to divide by 0.2 to 5 on both sides. If we take point to to 11 and we divide that by 110.0 2 to 5, we get around 982 0.676 repeating okay that we don't want to leave this as a decimal. It would be better for us to go ahead and around this so we will round it up just to be safe. We will say that. 983 people were surveyed.