Out of the 300 people surveyed, a 63said they listen to Spotifyat least one a month. With the level of confidence set at 80%, whatis the margin of error?...

PHYSICIANS In a recent poll, 61$\%$ of the 1010 people surveyed said they considered being a physician to be a very prestigious occupation. What was the margin of error?

Figure out what kind of confidence interval we're looking at in the study that interviewed 1015 adults, had a standard error of 3% and found that 68% of them believe in the specific political viewpoints. Now we're gonna take that The Z score is given by the standard error comes a square root of n over estimates for P and Q hat multiplied together. Our standard error is given by 3% point No. Three. Our end is given by the 1015 adults. R P is given by the 0.68 that agree with the leaning, and the Q is given by the rest of the 0.32 If you multiply all of that together, we got just about 2.5 and that's our Z score. And unfortunately, this doesn't line up with any of the standard ones that we've been looking at. So we're gonna have to go over to our table and find 2.5 by ourselves. Zoom in on the 2.0 row and the 0.5 column And where do they match up? Right here at 0.2 Okay, so 0.0, to what could we do with that? Well, what we're gonna do is we're first going to double it. Let's get points before and then we subtract it from one to get 0.96 0.96 If we convert that to a percentage, we get 96% and that tells us that this is a 96% confidence interval.

So question being asked of us is to find the margin of air. Because of that, I've gone ahead and written out the formula for margin of error. Here. Margin of error is equal to two times the square root of PR percent of people that answered a certain way times the quantity of one minus that exact same p divided by n, which is supposed to be the total size of our sample. Okay, well, we're told here that 83%. So considering that's a percent, I'm pretty sure that's R P right, because only margin of error and PR given in percentages. And so if it is asking for the margin of error than the only other thing that percent could be is P. And then it says there were 1000 and 20 people surveyed, so that's got to be R N. Because that would be the total number of people in our samples of people surveyed right. We have to remember that P cannot be represent as a percentage in the actual formula. We need to change p to a decimal before we can plug it in. Well, the way you change it percent to a decimal is dividing by 100 83. Divided by 100 would give me 1000.83 Okay, that's what we're gonna use for our formula. Now we've established that the rest this is pretty much just plugging it in. Margin of air is going to be equal to to times the square root of P, which we established his 0.83 times one minus the same P value as a quantity all over R N r total sample size, which is 1000 and 20. Make sure you are careful when you plug this into your calculator to make sure that you get the right number. If you plug this into your calculator correctly, you should get 0.2 four because it's 235 and so that firewood around the three up to a four. Now that's great between two changes to a percentage to give our final answer for our margin of air. So to change a decimal two of percent if if a percent gets divided by 100 to go to a decimal than a decibel would get multiplied by 100 to go back to being a percent 0.0 to 4 times 100 would give me 2.4% which means my margin of error would round to roughly approximately 2% because the 20.4 means it would round down.

There are a couple mistakes with this statement that's been need made. One mistake is in sane that 95% of other holes any 5% of other polls would have results. But one mistake is insane that 95% of other polls would have results within 3% points of the results of this survey. And that's really the point that I want to underline. That's kind of the main issue, um, of the results of this survey. That's the big problem. Now what? It should say that other polls would have results within three percentage points of the true proportion. So that's kind of the big change I think that we want to make is ah from of the results of this survey, two of the true proportion. Another mistake is saying at least 19 um, of the 20. So that's another mistake. What? We want to change that, too. Um, what it should say is about 19 so I know it's just a change of, ah, small word, but it kind of has a big implication. So the one mistake is saying that 95% of other polls would have results within 3% points of the results of this survey. So what it really should say is that other polls would have results within 3% points of the true proportion, because that's really what we want to compare it to. Another mistake is saying at least 19 of the 20 surveys would be within three points when it should say about 19%. You can't guarantee the at least, but you can in the about make sense of this.

All right. This question gives us some polling information and wants us to compute the margin of error. So part, eh just wants the margin of error from this sample. So remember that for a proportion, margin of error equals our special Z score times P Hat Times Q hat all over end, which in this case, we're dealing with 95% confidence soar. Special Values 1.96 and P hat and Q hat are both 0.5. And from here we get a margin of error of 0.0 for 42 Now we move on two to them, asking for different margins of error and us competing this sample size. So, for the first example at once margin of error to the equal to 0.4 So we'll use the following equation and all these examples and equals P hat times Q. Hat times The Z score over e quantity squared, which in this case the only thing that's going to change is our margin of error. So it's going to be 0.5 times 0.5 times 1.96 squared over e squared, which weaken group. The point fives together to simplify it, so it looks neater. So all we need to dio is insert are different values of e into this equation. So our first values 0.4 So 0.5 squared times 1.96 squared over 0.4 squared equals and which is 600.25 And for sample size questions with margin of error, we always round up because if we round down, our margin of error would not be small enough. And no, we just repeat the calculation with the different margins of air. So the next nest example it wants is 0.3 So we already have our equation, and now we just plug in point all three. And this time this works out to be, if 1067 0.1, which we round up 10 68 then getting smaller this time our margin of error. His 0.2 So we just plug in point or two for e, and this, for once, actually lands on a whole number 24 point 24 01 no decimal point. Then, for our final answer, our final part of this question Rather it wants a margin of error of 0.1 So using our Formula One last time, this time plugging in 0.1 And this works out to be another hole number 96 04 So, as you can see, decreasing our margin of error keeps making the sample size required going up and up and up. So that's the trade off if you want a smaller, more accurate poll with a small margin where that is.