Margin of error 2 Finding the 95% confidence interval for the proportion of a population voting for a candidate.
Margin of error 2
- Where we left off in the last video I kind
- of gave you a question.
- Find an interval so that we're reasonably confident-- we'll
- talk a little bit more about why I have to give this kind
- of vague wording right here-- reasonably confident that
- there's a 95% chance that the true population mean, which is
- p, which is the same thing as the mean of the sampling
- distribution of the sampling mean.
- So there's a 95% chance that the true mean-- and
- let me put this here.
- This is also the same thing as the mean of the sampling
- distribution of the sampling mean is in that interval.
- And to do that let me just throw out a few ideas.
- What is the probability that if I take a sample and I were
- to take a mean of that sample, so the probability that a
- random sample mean is within two standard deviations of the
- sampling mean, of our sample mean?
- So what is this probability right over here?
- Let's just look at our actual distribution.
- So this is our distribution, this right here is our
- sampling mean.
- Maybe I should do it in blue because that's
- the color up here.
- This is our sampling mean.
- And so what is the probability that a random sampling mean is
- going to be two standard deviations?
- Well a random sampling is a sample from this distribution.
- It is a sample from the sampling distribution of the
- sample mean.
- So it's literally what is the probability of finding a
- sample within two standard deviations of the mean?
- That's one standard deviation, that's another standard
- deviation right over there.
- In general, if you haven't committed this to memory
- already, it's not a bad thing to commit to memory, is that
- if you have a normal distribution the probability
- of taking a sample within two standard deviations is 95--
- and if you want to get a little bit more
- accurate it's 95.4%.
- But you could say it's roughly-- or maybe I could
- write it like this-- it's roughly 95%.
- And really that's all that matters because we have this
- little funny language here called reasonably confident,
- and we have to estimate the standard deviation anyway.
- In fact, we could say if we want, I could say that it's
- going to be exactly equal to 95.4%.
- But in general, two standard deviations, 95%, that's what
- people equate with each other.
- Now this statement is the exact same thing as the
- probability that the sample mean, that the sampling mean--
- not the sample mean, the probability of the mean of the
- sampling distribution is within two standard deviations
- of the sampling distribution of x is also going to be the
- same number, is also going to be equal to 95.4%.
- These are the exact same statements.
- If x is within two standard deviations of this, then this,
- then the mean, is within two standard deviations of x.
- These are just two ways of phrasing the same thing.
- Now we know that the mean of the sampling distribution, the
- same thing as a mean of the population distribution, which
- is the same thing as the parameter p-- the proportion
- of people or the proportion of the population that is a 1.
- So this right here is the same thing as the population mean.
- So this statement right here we can switch this with p.
- So the probability that p is within two standard deviations
- of the sampling distribution of x is 95.4%.
- Now we don't know what this number right here is.
- But we have estimated it.
- Remember, our best estimate of this is the true standard, or
- it is the true standard deviation of the population
- divided by 10.
- We can estimate the true standard deviation of the
- population with our sampling standard deviation, which was
- 0.5, 0.5 divided by 10.
- Our best estimate of the standard deviation of the
- sampling distribution of the sample mean is 0.05.
- So now we can say-- and I'll switch colors-- the
- probability that the parameter p, the proportion of the
- population saying 1, is within two times-- remember, our best
- estimate of this right here is 0.05 of a sample mean that we
- take is equal to 95.4%.
- And so we could say the probability that p is within 2
- times 0.05 is going to be equal to-- 2.0 is going to be
- 0.10 of our mean is equal to 95-- and actually let me be a
- little careful here.
- I can't say the equal now, because over here if we knew
- this, if we knew this parameter of the sampling
- distribution of the sample mean, we could
- say that it is 95.4%.
- We don't know it.
- We are just trying to find our best estimator for it.
- So actually what I'm going to do here is actually just say
- is roughly-- and just to show that we don't even have that
- level of accuracy, I'm going to say roughly 95%.
- We're reasonably confident that it's about 95% because
- we're using this estimator that came out of our sample,
- and if the sample is really skewed this is going to be a
- really weird number.
- So this is why we just have to be a little bit more exact
- about what we're doing.
- But this is the tool for at least saying
- how good is our result.
- So this is going to be about 95%.
- Or we could say that the probability that p is within
- 0.10 of our sample mean that we actually got.
- So what was the sample mean that we actually got?
- It was 0.43.
- So if we're within 0.1 of 0.43, that means we are within
- 0.43 plus or minus 0.1 is also, roughly, we're
- reasonably confident it's about 95%.
- And I want to be very clear.
- Everything that I started all the way from up here in brown
- to yellow and all this magenta, I'm just restating
- the same thing inside of this.
- It became a little bit more loosey-goosey once I went from
- the exact standard deviation of the sampling distribution
- to an estimator for it.
- And that's why this is just becoming-- I kind of put the
- squiggly equal signs there to say we're reasonably
- confident-- and I even got rid of some of the precision.
- But we just found our interval.
- An interval that we can be reasonably confident that
- there's a 95% probability that p is within that, is going to
- be 0.43 plus or minus 0.1.
- Or an interval of-- we have a confidence interval.
- We have a 95% confidence interval of, and we could say,
- 0.43 minus 0.1 is 0.33.
- If we write that as a percent we could say 33% to-- and if
- we add the 0.1, 0.43 plus 0.1 we get 53%-- to 53%.
- So we are 95% confident.
- So we're not saying kind of precisely that the probability
- of the actual proportion is 95%, but we're 95% confident
- that the true proportion is between 33% and 55%.
- That p is in this range over here.
- Or another way, and you'll see this in a lot of surveys that
- have been done, people will say we did a survey and we got
- 43% will vote for number one, and number one in this case is
- candidate B.
- And then the other side, since everyone else voted for
- candidate A, 57% will vote for A.
- And then they're going to put on margin of error.
- And you'll see this in any survey that you see on TV.
- They'll put a margin of error.
- And the margin of error is just another way of describing
- this confidence interval.
- And they'll say that the margin of error in this case
- is 10%, which means that there's a 95% confidence
- interval, if you go plus or minus 10% from that value
- right over there.
- And I really want to emphasize, you can't say with
- certainty that there is a 95% chance that the true result
- will be within 10% of this, because we had to estimate the
- standard deviation of the sampling mean.
- But this is the best measure we can with the information
- you have. If you're going to do a survey of 100 people,
- this is the best kind of confidence that we can get.
- And this number is actually fairly big.
- So if you were to look at this you would say, roughly there's
- a 95% chance that the true value of this number is
- between 33% and 53%.
- So there's actually still a chance that candidate B can
- win, even though only 43% of your 100 are
- going to vote for him.
- If you wanted to make it a little bit more precise you
- would want to take more samples.
- You can imagine.
- Instead of taking 100 samples, instead of n being 100, if you
- made n equal 1,000, then you would take this number over
- here, you would take this number here and divide by the
- square root of 1,000 instead of the square root of 100.
- So you'd be dividing by 33 or whatever.
- And so then the size of the standard deviation of your
- sampling distribution will go down.
- And so the distance of two standard deviations will be a
- smaller number, and so then you will have a
- smaller margin of error.
- And maybe you want to get the margin of error small enough
- so that you can figure out decisively who's going to win
- the election.
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