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  • Hi, I'm Adriene Hill, and welcome back to Crash Course Statistics.

  • In the last episode we dove into the logic surrounding test statistics and talked about

  • a general formula that allows us to create them for lots different situations.

  • There are so many questions we might want to answer, and it would be rough if we had

  • to memorize a new formula for EVERY Single One.

  • And sometimes Statistics is taught in a way that makes it seem like there's a different

  • formula you need to know if you want to test whether your bus is late more often than the

  • average bus in your town.

  • Or if burns treated with aloe heal faster than those that are left alone.

  • But! Hah-zah.

  • We can adapt the general formula...in all sorts of situations.

  • INTRO

  • Let's say that you just moved to a new place, and you're looking for the BEST coffee in town.

  • Since you've been watching Crash Course Statistics, you decide to do a little impromptu experiment.

  • Word on the street is there are two really popular coffee places near you, Caf-fiend

  • and The Blend Den.

  • So one Sunday after brunch, you grab a random sample of 16 of your new friends, and randomly

  • give half of them an unmarked cup with coffee from Caf-fiend, and the other half an unmarked

  • cup with coffee from The Blend Den.

  • You made sure to get the same roast--dark--to keep things as even as possible.

  • After delicate sniffs and sips of coffee in a process known ascupping”, the tallies are in.

  • On a scale of 1 to 10, Caf-fiend got a mean score of 7.6 and The Blend Den got a mean

  • score of 7.9

  • So we observe a difference between the coffee scores.

  • Coffee from Caf-fiend scored 0.3 points lower than Coffee from The Blend Den.

  • So coffee from The Blend Den is better?

  • Right?

  • Done and done.

  • Nope not yet.

  • Maybe it's just random chance.

  • So first we need to define our null.

  • There's no difference between the two coffee shops.

  • And then our alternative hypothesis, that there is a difference.

  • One is better than the other.

  • In this case, we're interested in whether the mean scores for coffee are different between

  • Caf-fiend and The Blend Den.

  • With a little algebra, we can see that this is the same thing as asking whether the difference

  • between the two means is not zero.

  • Now that we have our hypotheses, we can do a t-test.

  • Specifically, we'll do a two sample t-test, also called an independent or unpaired t-test.

  • The formula for a two sample t-test follows our general test statistic formula:

  • The difference we observed is 0.3.

  • If the null hypothesis were true and there's no difference between the coffee shops, we'd

  • expect a difference of 0.

  • So the numerator of our t-test is 0.3.

  • For this kind of t-test, our measure of average variation is the standard error.

  • For two groups, the standard error is calculated a bit differently since we have to account

  • for the sample variance of two groups.

  • Here, we're squaring the standard deviation to get the variance and n1 and n2 are the

  • sizes of the two groups--both are 8 here.

  • Now that we have our t-value, we can figure out if there's a statistically significant

  • difference between the two coffee shops and there are two ways to do this.

  • We can calculate the critical t-value and if our t-statistic is GREATER than the critical

  • value we reject the null hypothesis.

  • Or we can calculate the p-value from our t-statistic and we can reject the null hypothesis if the

  • p-value is SMALLER than our chosen alpha level.

  • To do either of these things, we'll need to choose our alpha level.

  • Again, our alpha is arbitrary.

  • But usually people will use 0.05 since that means that in the long run, only 5% of tests

  • done on groups with no real difference will incorrectly reject the null.

  • So, we'll conform :) and use an alpha of 0.05 here.

  • To calculate our critical t-value we need to find the t-values which correspond to the

  • top 5% most extreme values in our t-distribution.

  • Usually a computer or a calculator will do this for you, so we won't go into the formula,

  • but here are the cutoffs:

  • The cutoffs for our specific problem are about -2.145 and 2.145.

  • We have two cutoffs because we're doing a two tailed test.

  • We want to reject the null if coffee from Caf-fiend is better or if coffee from The

  • Blend Den is better.

  • We can already tell that we should fail to reject the null.

  • That there's no clear difference between the quality of the coffee.

  • Our t-statistic of about 0.44 is isn't close to -2.145 OR 2.145.

  • The critical value and p-value approach will give you identical results, so we don't

  • really need to do both.

  • But for the sake of showing we get the same outcomeour calculated p-value is 0.6684.

  • We reject the null if the p-value is smaller than alpha, so again we fail to reject since

  • 0.6684 is WAY bigger than 0.05.

  • One thing that's nice about the p-value approach, and the reason we'll mainly rely

  • on it throughout the rest of these examples, is that p-values are easier for us non-computers

  • to interpret.

  • A p-value of 0.6684 means that if there were NO difference in scores between coffee from

  • Caf-fiend and coffee from The Blend Den, we'd still expect to see a difference in our sample

  • means that's 0.3 or greater pretty often...

  • 66.84% of the time.

  • Since our observed difference of 0.3 or greater is pretty common under the null hypothesis,

  • we haven't found evidence that it's a bad fit.

  • That's why we failed to reject it.

  • So right now we don't have any evidence that one coffee shop is better than the other.

  • But remember, absence of evidence is not evidence of absence.

  • And while our coffee excursion and experiment were well designed, we can probably improve it.

  • If you look at the scores that your friends gave the coffees, you'll see that there's

  • one person who tried coffee from Caf-fiend and really hated it.

  • After looking through your scorecards, you realize it's Alex , who has mentioned in

  • the past that she just doesn't love coffee.

  • Which gets you thinking.

  • Even though you randomly assigned your friends to get either coffee from Caf-fiend or coffee

  • from The Blend Den, that design didn't account for the fact that some people just like coffee

  • more than others.

  • Alex might give the best coffee in the world a measly 6 point rating just because...coffee's

  • not really her thing.

  • Whereas your always caffeinated friend Cameron would probably give that day old coffee in

  • the breakroom a score of 7 just because he loves coffee.

  • So in addition to any true difference in scores between coffee from Caf-fiend and coffee from

  • The Blend Den, our sample means are also affected by how much the people in each group like coffee.

  • You randomly assigned your friends to groups, so you don't expect that there's some

  • systematic difference between the average coffee enjoyment of the groups.

  • But random assignment adds variation, which can make it harder to see a true difference

  • between the coffee scores.

  • One solution to this issue is a paired t-test.

  • You could try to pair up your friends based on how much they like coffee and then randomly

  • assign one to coffee from Caf-fiend and the other to coffee from The Blend Den, and repeat

  • this over and over until everyone had been assigned.

  • The best match, of course, for a person is themselves.

  • I'm just like me.

  • So you decide to call another random sample of 16 of your friends.

  • This time you give all of them both Caf-fiend coffee AND The Blend Den coffee and they record

  • their scores.

  • Now that everyone has scored both coffees, you can be sure that the two groups have the

  • exact same level ofcoffee affinitysince it's the exact same people.

  • The mean scores are still affected by variation due to individual coffee preferences, but

  • since the exact same people are in both groups, we can extract that variation andthrow

  • it awayso to speak.

  • One way to do this, is to make a difference score for each person.

  • This will tell you how much more they like coffee from Caf-fiend than coffee from The Blend Den.

  • Now that we have only one list of values--the difference scores--our matched pairs t-test

  • will look surprisingly similar to the one sample t-test that we've seen before.

  • We observed a mean difference (Caf-fiend - The Blend Den) of -0.18125, which means that on

  • average, people rated coffee The Blend Den 0.18125 points higher than coffee from Caf-fiend.

  • The null hypothesis here is that there's no difference between ratings for coffee from

  • Caf-fiend and coffee The Blend Den, so we'd expect our mean difference to be 0.

  • And our measure of average variation is just the standard error of the difference scores:

  • Putting it together, we get a t-statistic of about -3.212.

  • Before we get to the corresponding p-value that our computer spit out, let's consider

  • another way to think about what t-statistics are actually telling us.

  • T-statistics tell you how many standard errors away from the mean our observed difference is.

  • Though the t-distribution isn't EXACTLY normal, it's reasonably close, so we can

  • use our intuition about normal distributions to understand our t-values.

  • Normal distributions have about 68% of their data within one standard deviation from the mean.

  • And about 95% within 2 standard deviations.

  • That means that t-scores around 3, like ours, are about 3 standard errors away from the

  • mean...only around 0.3% of scores are that far away!

  • So it makes sense that our p-value is very small: 0.00582.

  • Which allows us to reject the null hypothesis that there is no difference between the scores

  • for Coffee from Caf-fiend and coffee from The Blend Den.

  • Which means that from now on, I'll be buying my coffee from The Blend Den.

  • Except for when I'm meeting up with Alex, then I'll buy` tea.

  • Statistical tests help us wade through the murky waters of variability, and our goal

  • should be to get rid of as MUCH of that variability as possible so that we can see patterns.

  • We can see whether exercise improves sleep...which your friends might be lacking after all that coffee.

  • Or whether your hearing could be hurt by listening to loud music by Cream or Ice Cube or Vanilla Ice

  • or some other musician that sounds like it belongs in coffee.

  • Like Spoon! Spoon. Yeah? Brandon Spoon.

  • But more importantly, we're learning that all those formulas you may have seen floating

  • around, really aren't that different.

  • We're just comparing what we see, to what we think we should see.

  • We're always comparing the way things are to how we expect them to be.

  • And statistics is no exception.

  • We now have the tools to design experiments and answer a lot of interesting questions

  • and do our own experiments even if we over caffeinate some of our friends in the process.

  • Thanks for watching. I'll see you next time.

Hi, I'm Adriene Hill, and welcome back to Crash Course Statistics.

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B1 CrashCourse fiend caf den blend difference

T-Tests: A Matched Pair Made in Heaven: Crash Course Statistics #27

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    林宜悉 posted on 2020/03/30
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