Comparing Population Proportions 2 Comparing Population Proportions 2
Comparing Population Proportions 2
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- Where we left off in the last video, we were trying to
- figure out if there's a meaningful difference between
- the proportion of men voting for a candidate and the
- proportion of women.
- We sampled 1,000 men, sampled 1,000 women, and we got a
- sample proportion for each of them.
- We got 0.642 for the men and 0.591 for the women.
- But our goal is to get a 95% confidence interval.
- So just based on our actual sample, we got-- let me write
- it over here.
- We got our sample proportion for the men minus-- let me do
- this in a neutral color.
- We got our sample proportion for the men minus our sample
- proportion for the women being 0.642 minus
- 0.591, that's 0.051.
- I just subtracted this from that.
- So what we want to do when we want a confidence interval, we
- want to be confident.
- I'll always have to say that because it's not going to be
- super precise.
- We want to be confident that there's a 95% chance that this
- thing right here-- remember, when we took the two sample
- proportions and took their difference, it's like taking a
- sample from the sampling distribution of the statistic.
- So we want a 95% chance that the true mean or the true
- value of this, that P1 minus P2 is within some range, let's
- say is within d, I'll say d for distance, of the actual
- difference that we got with our samples.
- Within d of 0.051.
- And I write this multiple times, but I always
- write it this way.
- I don't just give the formula that you
- normally see in books.
- It's very easy to memorize if you do, but this way, you
- actually see why this confidence
- interval makes sense.
- If there's a 95% chance that P1 minus P2, the actual true
- proportions, the difference of the true proportions, is
- within d of the difference between our sample
- proportions, this statement right here is the same thing
- that there's a 95% chance that 0.051 is within d of this
- actual parameter, P1 minus P2, which is the same
- thing as the mean.
- So we need to figure out some distance around this mean,
- where if we take a random sample from this, and this is
- a random sample from this distribution, it has a 95%
- chance of being within d of this mean, because if it's
- within d of the mean, then there's also a 95% chance that
- the mean is within d of our sample, and then we'll have
- our confidence interval.
- Our confidence interval would be this value plus d and this
- value minus d.
- So what are these?
- What is the distance d?
- Well, in a normalized normal distribution, I got a Z-table
- right over here, and we can assume everything is normal,
- especially the sampling distributions because our n is
- so big and also our proportion is not close to zero or one.
- It's nice and close to the middle, so we don't end up
- with all these weird cases near the edges.
- We say, OK, how do we contain the middle 95%?
- How many standard deviations in a normal distribution do we
- need to be away from the mean in order to contain 95% of the
- Now these Z-tables, and we've done it multiple times, give
- you cumulative distribution.
- We're looking for this Z-value right over here.
- If it's containing 95%, you're going to have 2.5% over here
- and you're going to have 2.5% over here.
- So from a Z-table's point of view, this Z-table gives you
- the cumulative probability up to that Z-value.
- So what we're looking for is actually 97.5%.
- We're looking for something that contains all
- of this over here.
- If we get the Z-value and then apply it on both sides, then
- we're going to have something that contains 95%.
- So let's look up the 97.5.
- 97.5 is right over there, and that is 1.96 standard
- So this is 1.96 for a normalized standard deviation,
- or a Z-score of 1.96.
- So if we looked to this normal distribution right over here,
- this distance that we care about is going to be 1.96
- times the standard deviation of this distribution, so it's
- going to be 1.96 times all of this business.
- 1.96 times the standard deviation of this
- And so we just need to calculate this and
- multiply it by 1.96.
- Now, we have a problem.
- We don't know the true parameters P1 and P2.
- We don't know the true population parameters.
- We don't know P1 and P2.
- That's part of the problem.
- We're trying to figure out if there's a meaningful
- difference between P1 and P2.
- But we've seen it multiple times.
- Since our sample size is a large, we can estimate P1 and
- P2 with our sample proportions.
- So we could change this to approximately and we can use
- our sample proportions.
- And we know what those values are.
- And actually this n over here was 1,000.
- So let's figure that out.
- Let's just get the calculator out.
- It's just going to be one big calculation here.
- So what we have is the square root, and then in parentheses,
- our sample proportion for the men is 0.642, and then we're
- going to multiply that times 1 minus 0.642, close
- That's that over there divided by 1,000.
- And then we're going to add to that plus-- do the same thing
- for the women.
- Our sample proportion is 0.591 times 1 minus 0.591.
- So that's this term right over here divided by 1,000.
- Once again, I need to get the parentheses right.
- And then we just need to close the parentheses, this original
- parentheses, because we're taking the square root of
- So we get 0.021, or maybe we'll say 0.022.
- So this value right here is approximately 0.022.
- So going back to our question, or this distance that we care
- about, this value is going to be approximately, or our best
- estimate of it, is 0.022.
- So let's just multiply that.
- 0.022 times 1.96 gives 0.043.
- I'll just round it.
- So this right here is equal to 0.043.
- And just like that, we have our confidence interval.
- We know that there's a 95% chance that the true
- difference of the proportions is within 0.043 of the actual
- difference of our sample proportions that we got.
- Or if we actually want to get an interval, we take this
- value minus 0.043.
- So let's do that.
- So we could have 0.051 minus 0.043 is
- going to give us 0.008.
- And then if we add it, so 0.051 plus
- 0.043, it gives us 0.094.
- So the 95% confidence interval between the proportions of men
- and the proportion of women who are going to vote for the
- candidate for P1 minus P2 is 0.008 to 0.094.
- I have it right here on the calculator.
- And we're done.
- So it does seem we're confident that there's a 95%
- chance that men are more likely to vote for the
- candidate than women.
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