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Two-sample t test for difference of means | AP Statistics | Khan Academy - YouTube
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kaito grows tomatoes in two separate
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fields when the tomatoes are ready to be
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picked he is curious as to whether the
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sizes of his tomato plants differ
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between the two fields he takes a random
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sample of plants from each field and
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measure the and measures the heights of
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the plants
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here is a summary of the results
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so what i want you to do is pause this
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video and conduct a two sample t test
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here and let's assume that all of the
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conditions for
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inference are met the random condition
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the normal condition and the
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independence condition and let's assume
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that we are working with a significance
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level of 0.05
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so can pause the video and conduct the
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two sample t test here to see whether
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there's evidence that the sizes of
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tomato plants differ between the fields
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all right now let's work through this
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together so like always let's first
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construct our null hypothesis and that's
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going to be the situation where there is
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no difference between the mean sizes so
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that would be that the mean size in
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field a is equal to the mean size in
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field b
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now what about our alternative
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hypothesis
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well he wants to see whether the sizes
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of his tomato plants differ between the
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two fields he's not saying whether a is
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bigger than b or whether b is bigger
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than a and so his alternative hypothesis
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would be around his suspicion that the
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mean of a is not equal to the mean of b
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that they differ
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and to do this two sample t test now we
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assume the null hypothesis
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we assume
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our null hypothesis and remember we're
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assuming that all of our conditions for
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inference are met and then we want to
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calculate a t statistic based on this
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sample data that we have and our t
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statistic
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is going to be equal to the differences
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between the sample means
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all of that over our estimate of the
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standard deviation of the sampling
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distribution of the difference of the
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sample means
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this will be the sample standard
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deviation from sample a squared
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over the sample size from a
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plus the sample standard deviation from
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the b sample squared over the sample
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size from b
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and let's see we have all the numbers
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here to calculate it this numerator is
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going to be equal to
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1.3 minus 1.6 1.3 minus 1.6
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all of that over
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the square root of let's see the
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standard deviation the sample standard
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deviation from the sample from field a
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is 0.5 if you square that you're going
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to get 0.25
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and then that's going to be over the
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sample size from field a
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over 22
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plus 0.3 squared so that is 0.3 squared
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is 0.
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all of that over the sample size from
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field b all that over 24.
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the numerator is just going to be
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negative 0.3
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negative 0.3
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divided by
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the square root of .25
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divided by 22
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plus
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.09
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divided by 24
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and that gets us
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negative 2.44
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approximately negative 2.44
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and so if you think about a t
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distribution
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and we'll use our calculator to figure
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out this probability so this is a
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t-distribution right over here
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this would be the assumed mean of our
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t-distribution and so we got a result
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that is negative we get a t statistic of
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negative 2.44 so we're right over here
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so this is
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negative 2.44
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and so we want to say what is the
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probability from this t distribution of
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getting something at least this extreme
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so it would be
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this area and it would also be and it
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would also be this area if we got 2.44
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above the mean it would also be this
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area
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and so what i could do is i'm going to
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use my calculator to figure out this
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probability right over here and then i'm
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just going to multiply that by 2 to get
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this one as well
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so the probability
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of getting
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a
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t value i guess i could say where it's
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absolute value is greater than or equal
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to 2.44
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is going to be approximately equal to
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i'm going to go to second
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distribution i'm going to go to the
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cumulative distribution function for our
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t distribution
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click that
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and since i want to think about
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this
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tail probability here and then i'm just
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going to multiply by 2. the lower bound
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is a very very very negative number you
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could view that as functionally negative
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infinity
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the upper bound is negative 2.44
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negative 2.44
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and now what's our degrees of freedom
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well if we take the conservative
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approach it'll be the smaller of the two
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samples minus one
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well the smaller of the two samples is
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22 and so 22 minus 1 is 21.
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so put 21 in there
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2
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21
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and now
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i can paste
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and i get
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that number right over there and if i
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multiply that by 2 because this just
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gives me the probability of getting
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something lower than that but i also
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want to think about the probability of
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getting something 2.44 or more above the
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mean of our t distribution so times 2
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is going to be equal to approximately
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0.024
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so approximately
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0.024
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and what i want to do then is compare
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this to my significance level and you
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can see very clearly this right over
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here this is equal to our p-value
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our p-value in this situation
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our p-value in this situation is clearly
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less than our significance level
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and because of that we said hey assuming
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the null hypothesis is true we got
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something that's a pretty low
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probability below our threshold so we
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are going to reject
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reject our null hypothesis which tells
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us that there is so this suggests
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this suggests
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the alternative hypothesis that there is
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indeed a difference between the sizes of
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the tomato plants in the two fields
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