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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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