# The Chi Square Test for Homogeneity
# https://www.youtube.com/watch/ImUKhLqBmns

00:00:00.000 hey everybody today we're getting into
00:00:01.680 the chi-squared test for homogeneity
00:00:03.560 we're in this situation where we're
00:00:05.700 measuring one categorical variable
00:00:07.560 across two groups and asking whether the
00:00:10.349 groups are different or not whether they
00:00:11.550 come actually from the same population
00:00:13.790 here's an example
00:00:15.599 we've got people under 50 and people
00:00:17.940 over 50 and we're asking them do you
00:00:20.130 agree with the statement I prefer to
00:00:22.260 work from home as much as possible you
00:00:24.359 can either agree be neutral or disagree
00:00:26.730 here are our totals do young and old
00:00:29.550 people prefer working from home in equal
00:00:31.410 proportions so we've got a null
00:00:36.510 hypothesis that the probability of
00:00:38.219 agreeing with that statement is the same
00:00:40.079 in either group the probability of being
00:00:41.940 neutral is the same in either group and
00:00:44.010 so on we want to use a chi-square test
00:00:47.640 on this table and in order to do that
00:00:49.829 we're gonna need expected cell counts of
00:00:51.780 course as usual so how are we going to
00:00:55.020 do it well under the null hypothesis we
00:00:57.629 expect that the corresponding
00:00:59.940 respondents are going to have equal
00:01:01.199 proportions in each group the same
00:01:03.000 proportion in group a and Group B is
00:01:04.979 going to agree the same proportion is
00:01:06.960 going to be neutral the same proportion
00:01:08.520 is going to disagree and if in fact
00:01:10.860 those proportions are the same they'll
00:01:12.479 also be the same as the pooled
00:01:14.760 proportions
00:01:16.530 so the overall proportion that agreeing
00:01:19.830 neutral disagree so let's get those
00:01:24.119 pooled proportions we take the total
00:01:26.909 number that agree and take it over the
00:01:29.189 total number period the total number
00:01:31.079 that our Neutral take it over the total
00:01:32.430 number period and so on we're going to
00:01:35.700 multiply those by the different row
00:01:37.049 totals and those are going to give us
00:01:38.549 expected cell counts here's what we get
00:01:41.369 as usual we don't worry if our expected
00:01:44.159 counts aren't integers and now we're
00:01:47.759 going to compute a chi-squared statistic
00:01:48.840 in the usual way we're gonna do the
00:01:51.299 values we got - the ones we expected
00:01:53.430 square it and divide by the predicted
00:01:57.630 number the expected number add all those
00:01:59.549 up to get the chi-squared statistic
00:02:01.820 there's the calculation we get 6.8 for
00:02:05.750 the last thing we got to do before we
00:02:07.979 compute a p-value is to talk about
00:02:09.869 degrees of freedom so in this case we
00:02:12.420 have two independent
00:02:13.240 samples one for each row so we're
00:02:15.430 starting with four degrees of freedom we
00:02:18.070 had six total cells we take away those
00:02:20.320 two for the two independent samples we
00:02:24.250 also have to take away two degrees of
00:02:25.720 freedom because we estimated two
00:02:27.730 parameters one for each column except
00:02:30.040 the last which is determined by the
00:02:31.570 other two so overall we've got two
00:02:33.370 degrees of freedom we compute the
00:02:36.190 p-value as usual in a chi-squared
00:02:38.290 distribution probability that
00:02:40.120 chi-squared is greater than or equal to
00:02:41.710 that 6.8 for that we got in chi square
00:02:45.010 root of 2 and we get 0.03 3 so this
00:02:47.830 provides good evidence against the null
00:02:49.330 hypothesis we have good reason to
00:02:51.400 believe that in fact these two groups
00:02:53.320 young people and old people are
00:02:55.090 different in their opinions about
00:02:56.440 working from home