0:00:00.610,0:00:04.100
Let's do another problem from[br]the normal distribution

0:00:04.100,0:00:10.120
section of ck12.org's[br]AP statistics book.

0:00:10.120,0:00:11.770
And I'm using theirs because[br]it's Open Source and it's

0:00:11.770,0:00:13.990
actually quite a good book.

0:00:13.990,0:00:16.475
The problems are, I think,[br]good practice for us.

0:00:16.475,0:00:19.070
So let's see, number 3.

0:00:19.070,0:00:20.390
You could go to their[br]site and I think you

0:00:20.390,0:00:21.690
can download the book.

0:00:21.690,0:00:26.180
Assume that the mean eight of 1[br]year old girls in the U.S. is a

0:00:26.180,0:00:28.920
normally distributed-- or is[br]normally distributed with the

0:00:28.920,0:00:32.330
mean of about 9.5 grams.

0:00:32.330,0:00:33.820
That's got to be kilograms.

0:00:33.820,0:00:35.930
I have a 10 month old son[br]and he weighs about 20

0:00:35.930,0:00:39.570
pounds which is about 9[br]kilograms not 9.5 grams.

0:00:39.570,0:00:41.040
9.5 grams is nothing.

0:00:41.040,0:00:43.900
This would be talking about[br]like mice or something.

0:00:43.900,0:00:44.940
This has got to be kilograms.

0:00:44.940,0:00:47.350
But anyway, it's about[br]9.5 kilograms with a

0:00:47.350,0:00:51.050
standard deviation of[br]approximately 1.1 grams.

0:00:51.050,0:00:56.400
So the mean is equal to 9.5[br]kilograms I'm assuming and

0:00:56.400,0:01:01.130
the standard deviation[br]is equal to 1.1 grams.

0:01:01.130,0:01:04.840
Without using a calculator-- so[br]that's an interesting clue--

0:01:04.840,0:01:08.950
estimate the percentage of 1[br]year old girls in the U.S. that

0:01:08.950,0:01:09.995
meet the following conditions.

0:01:09.995,0:01:12.910
So when they say that without a[br]calculator estimate that's a

0:01:12.910,0:01:15.250
big clue or a big giveaway that[br]we're supposed to use

0:01:15.250,0:01:16.350
the empirical rule.

0:01:20.040,0:01:27.480
Empirical rule sometimes[br]called the 68-95-99.7 rule.

0:01:27.480,0:01:29.960
And if you remember this[br]is the name of the rule

0:01:29.960,0:01:31.500
you've essentially[br]remembered the rule.

0:01:31.500,0:01:33.520
What that tells us that if we[br]have a normal distribution--

0:01:33.520,0:01:35.800
I'll do a bit of a review[br]here before we jump

0:01:35.800,0:01:36.750
into this problem.

0:01:36.750,0:01:38.750
If we have a normal[br]distribution-- let me draw

0:01:38.750,0:01:40.480
a normal distribution.

0:01:40.480,0:01:42.900
It looks like that.

0:01:42.900,0:01:44.240
That's my normal distribution.

0:01:44.240,0:01:45.940
I didn't draw it perfectly[br]but you get the idea.

0:01:45.940,0:01:47.560
It should be symmetrical.

0:01:47.560,0:01:49.980
This is our mean right there.

0:01:49.980,0:01:50.840
That's our mean.

0:01:50.840,0:01:54.810
If we go one standard deviation[br]above the mean and one standard

0:01:54.810,0:02:00.350
deviation below the mean,[br]so this is our mean plus

0:02:00.350,0:02:01.780
one standard deviation.

0:02:01.780,0:02:05.730
This is our mean minus[br]one standard deviation.

0:02:05.730,0:02:08.710
The probability of finding a[br]result if we're dealing with a

0:02:08.710,0:02:12.080
perfect normal distribution[br]that's between one standard

0:02:12.080,0:02:14.640
deviation below the mean and[br]one standard deviation above

0:02:14.640,0:02:19.320
the mean-- that would be this[br]area-- and it would be,

0:02:19.320,0:02:23.040
you could guess, 68%.

0:02:23.040,0:02:26.430
68% chance you're going to get[br]something within one standard

0:02:26.430,0:02:27.750
deviation of the mean.

0:02:27.750,0:02:30.140
Either a standard deviation[br]below or above or

0:02:30.140,0:02:31.450
anywhere in between.

0:02:31.450,0:02:34.500
Now, if we're talking about two[br]standard deviations around the

0:02:34.500,0:02:37.170
mean-- so if we go down another[br]standard deviation, we go down

0:02:37.170,0:02:39.570
another standard deviation in[br]that direction and another

0:02:39.570,0:02:41.780
standard deviation above the[br]mean-- and we were to ask

0:02:41.780,0:02:43.190
ourselves what's the[br]probability of finding

0:02:43.190,0:02:47.360
something within those two or[br]within that range, then it's,

0:02:47.360,0:02:50.740
you could guess it, 95%.

0:02:50.740,0:02:53.060
And that includes this[br]middle area right here.

0:02:53.060,0:02:56.510
So the 68% is a[br]subset of that 95%.

0:02:56.510,0:02:58.140
And I think you know[br]where this is going.

0:02:58.140,0:03:01.360
If we go three standard[br]deviations below the mean and

0:03:01.360,0:03:06.820
above the mean, the empirical[br]rule or the 68-95-99.7 rule

0:03:06.820,0:03:15.740
tells us that there is a 99.7%[br]chance of finding a result in a

0:03:15.740,0:03:19.120
normal distribution that is[br]within three standard

0:03:19.120,0:03:20.110
deviations of the mean.

0:03:20.110,0:03:23.230
So above three standard[br]deviations below the mean

0:03:23.230,0:03:26.030
and below three standard[br]deviation above the mean.

0:03:26.030,0:03:27.870
That's what the empirical[br]rule tells us.

0:03:27.870,0:03:30.960
Now let's see if we can[br]apply it to this problem.

0:03:30.960,0:03:33.140
So they gave us the mean and[br]the standard deviation.

0:03:33.140,0:03:34.550
Let me draw that out.

0:03:34.550,0:03:38.550
Let me draw my axis[br]first as best as I can.

0:03:38.550,0:03:39.600
That's my axis.

0:03:39.600,0:03:41.410
Let me draw my bell curve.

0:03:45.920,0:03:49.090
That's about as good as a[br]bell curve as you can expect

0:03:49.090,0:03:50.920
a freehand drawer to do.

0:03:50.920,0:03:54.140
And the mean here is 9.-- and[br]this should be symmetric.

0:03:54.140,0:03:55.710
This height should be the[br]same as that height there.

0:03:55.710,0:03:57.600
I think you get the idea.

0:03:57.600,0:03:59.260
I'm not a computer.

0:03:59.260,0:04:02.390
9.5 is the mean.

0:04:02.390,0:04:03.370
I won't write the units.

0:04:03.370,0:04:04.580
It's all in kilograms.

0:04:04.580,0:04:11.330
One standard deviation above[br]the mean we should add 1.1 to

0:04:11.330,0:04:14.220
that because they told us the[br]standard deviation is 1.1.

0:04:14.220,0:04:16.820
That's going to be 10.6.

0:04:16.820,0:04:19.620
If we go-- let me just draw a[br]little dotted line there-- 1

0:04:19.620,0:04:25.990
standard deviation below the[br]mean we're going it subtract

0:04:25.990,0:04:34.110
1.1 from 9.5 and so[br]that would be 8.4.

0:04:34.110,0:04:37.620
If we go two standard[br]deviations above the mean

0:04:37.620,0:04:40.400
we would add another[br]standard deviation here.

0:04:40.400,0:04:40.610
Right?

0:04:40.610,0:04:41.890
We went one standard[br]deviations, two

0:04:41.890,0:04:42.700
standard deviations.

0:04:42.700,0:04:44.435
That would get us to 11.7.

0:04:44.435,0:04:47.040
And if we were to go three[br]standard deviations

0:04:47.040,0:04:48.910
we'd add 1.1 again.

0:04:48.910,0:04:50.720
That would get us to 12.8.

0:04:50.720,0:04:53.820
Doing it on the other side,[br]one standard deviation

0:04:53.820,0:04:55.380
below the mean is 8.4.

0:04:55.380,0:04:58.480
Two standard deviations below[br]the mean-- subtract 1.1

0:04:58.480,0:05:00.910
again-- would be 7.3.

0:05:00.910,0:05:03.380
And then three standard[br]deviations below the mean--

0:05:03.380,0:05:07.280
which we'd write there--[br]would be 6.2 kilograms.

0:05:07.280,0:05:08.860
So that's our set up[br]for the problem.

0:05:08.860,0:05:12.070
So what's the probability that[br]we would find a one year old

0:05:12.070,0:05:17.730
girl in the U.S. that weighs[br]less than 8.4 kilograms.

0:05:17.730,0:05:19.330
Or maybe I should say[br]whose mass is less

0:05:19.330,0:05:21.640
than 8.4 kilograms.

0:05:21.640,0:05:25.150
So if we look here, the[br]probability of finding a baby

0:05:25.150,0:05:28.070
or a female baby who is one[br]year old with a mass or a

0:05:28.070,0:05:30.920
weight of less than 8.4[br]kilograms, that's this

0:05:30.920,0:05:31.610
area right here.

0:05:31.610,0:05:35.070
I said mass because kilograms[br]is actually a unit of mass.

0:05:35.070,0:05:36.940
Most people use it[br]as weight as well.

0:05:36.940,0:05:38.470
So that's that[br]area right there.

0:05:38.470,0:05:40.950
So how can we figure out[br]that area under this

0:05:40.950,0:05:43.900
normal distribution using[br]the empirical rule?

0:05:43.900,0:05:47.280
Well, we know what[br]this area is.

0:05:47.280,0:05:52.370
We know what this area between[br]minus one standard deviation

0:05:52.370,0:05:54.500
and plus one standard[br]deviation is.

0:05:54.500,0:05:55.920
We know that is 68%.

0:05:58.430,0:06:01.720
And if that's 68% then that[br]means in the parts that

0:06:01.720,0:06:04.360
aren't in that middle[br]region you have 32%.

0:06:04.360,0:06:07.200
Because the area under the[br]entire normal distribution is

0:06:07.200,0:06:11.380
100 or 100% or 1, depending on[br]how you want to think about it.

0:06:11.380,0:06:14.490
Because you can't have-- well,[br]all of the possibilities

0:06:14.490,0:06:17.880
combined can only add up to 1.

0:06:17.880,0:06:21.480
You can't have it more[br]than 100% there.

0:06:21.480,0:06:27.270
So if you add up this leg and[br]this leg-- so this plus that

0:06:27.270,0:06:29.490
leg is going to be[br]the remainder.

0:06:29.490,0:06:32.590
So 100 minus 68, that's 32.

0:06:32.590,0:06:33.920
32%.

0:06:33.920,0:06:37.820
32% is if you add up[br]this left leg and this

0:06:37.820,0:06:39.240
right leg over here.

0:06:39.240,0:06:41.120
And this is a perfect[br]normal distribution.

0:06:41.120,0:06:42.535
They told us it's[br]normally distributed.

0:06:42.535,0:06:44.780
So it's going to be[br]perfectly symmetrical.

0:06:44.780,0:06:48.730
So if this side and that side[br]add up to 32 but they're both

0:06:48.730,0:06:51.820
symmetrical, meaning they have[br]the exact same area, then this

0:06:51.820,0:06:56.490
side right here-- I'll do it in[br]pink-- this side right here--

0:06:56.490,0:07:00.020
it ended up looking more[br]like purple-- would be 16%.

0:07:00.020,0:07:02.700
And this side right[br]here would be 16%.

0:07:02.700,0:07:05.900
So your probability of getting[br]a result more than one standard

0:07:05.900,0:07:08.280
deviation above the mean-- so[br]that's this right hand

0:07:08.280,0:07:09.760
side, would be 16%.

0:07:09.760,0:07:13.040
Or the probability of having a[br]result less than one standard

0:07:13.040,0:07:17.050
deviation below that mean,[br]that's this right here, 16%.

0:07:17.050,0:07:19.060
So they want to know the[br]probability of having a

0:07:19.060,0:07:23.140
baby at one years old[br]less than 8.4 kilograms.

0:07:23.140,0:07:27.970
Less than 8.4 kilograms[br]is this area right here.

0:07:27.970,0:07:29.500
And that's 16%.

0:07:29.500,0:07:33.270
So that's 16% for part a.

0:07:33.270,0:07:38.280
Let's do part b: between 7.3[br]and 11.7 point seven kilograms.

0:07:38.280,0:07:41.130
So between 7.3--[br]that's right there.

0:07:41.130,0:07:47.120
That's two standard deviations[br]below the mean-- and 11.7, one,

0:07:47.120,0:07:49.100
two standard deviations[br]above the mean.

0:07:49.100,0:07:51.260
So there's essentially asking[br]us what's the probability of

0:07:51.260,0:07:54.340
getting a result within two[br]standard deviations

0:07:54.340,0:07:55.230
of the mean, right?

0:07:55.230,0:07:57.040
This is the mean right here.

0:07:57.040,0:08:00.250
This is two standard[br]deviations below.

0:08:00.250,0:08:02.630
This is two standard[br]deviations above.

0:08:02.630,0:08:04.130
Well that's pretty[br]straightforward.

0:08:04.130,0:08:07.490
The empirical rule tells us[br]between two standard deviations

0:08:07.490,0:08:13.950
you have a 95% chance of[br]getting a result that is within

0:08:13.950,0:08:15.140
two standard deviations.

0:08:15.140,0:08:17.740
So the empirical rule just[br]gives us that answer.

0:08:17.740,0:08:21.440
And then finally, part c: the[br]probability of having a one

0:08:21.440,0:08:25.510
year old U.S. a baby girl[br]more than 12.8 kilograms.

0:08:25.510,0:08:28.310
So 12.8 kilograms is three[br]standard deviations

0:08:28.310,0:08:29.770
above the mean.

0:08:29.770,0:08:34.100
So we want to know the[br]probability of having a result

0:08:34.100,0:08:36.250
more than three deviations[br]above the mean.

0:08:36.250,0:08:42.170
So that is this area way out[br]there that I drew in orange.

0:08:42.170,0:08:44.310
Maybe I should do it in[br]a different color to

0:08:44.310,0:08:45.280
really contrast it.

0:08:45.280,0:08:48.575
So it's this long tail out[br]here, this little small area.

0:08:48.575,0:08:51.020
So what is that probability?

0:08:51.020,0:08:53.420
So let's turn back to[br]our empirical rule.

0:08:53.420,0:08:56.230
Well we know the probability--[br]we know this area.

0:08:56.230,0:08:59.740
We know the area between minus[br]three standard deviations and

0:08:59.740,0:09:01.960
plus three standard deviations.

0:09:01.960,0:09:04.090
We know this-- since this is[br]last the last problem I can

0:09:04.090,0:09:08.200
color the whole thing in-- we[br]know this area right here

0:09:08.200,0:09:14.300
between minus 3 and plus[br]3, that is it 99.7%.

0:09:14.300,0:09:16.830
The bulk of the results[br]fall under there.

0:09:16.830,0:09:17.940
I mean, almost all of them.

0:09:17.940,0:09:20.320
So what do we have left[br]over for the two tails?

0:09:20.320,0:09:21.220
Remember there are two tails.

0:09:21.220,0:09:22.330
This is one of them.

0:09:22.330,0:09:24.630
Then you have the results that[br]are less than three standard

0:09:24.630,0:09:25.730
deviations below the mean.

0:09:25.730,0:09:27.480
This tail right there.

0:09:27.480,0:09:32.160
So that tells us that this,[br]less than three standard

0:09:32.160,0:09:35.280
deviations below the mean and[br]more than three standard

0:09:35.280,0:09:39.150
deviations above the mean[br]combined have to be the rest.

0:09:39.150,0:09:46.530
Well the rest, there's only[br]0.3% percent for the rest.

0:09:46.530,0:09:48.250
And these two things[br]are symmetrical.

0:09:48.250,0:09:49.620
They're going to be equal.

0:09:49.620,0:09:54.880
So this right here has to be[br]half of this or 0.15% and

0:09:54.880,0:09:59.160
this right here is[br]going to be 0.15%.

0:09:59.160,0:10:03.650
So the probability of having a[br]one year old baby girl in the

0:10:03.650,0:10:07.250
U.S. that is more than 12.8[br]kilograms if you assume a

0:10:07.250,0:10:10.490
perfectly normal distribution[br]is the area under this curve,

0:10:10.490,0:10:13.040
the area that is more than[br]three standard deviations

0:10:13.040,0:10:14.250
above the mean.

0:10:14.250,0:10:21.760
And that is 0.15%.

0:10:21.760,0:10:24.410
Anyway, I hope you[br]found that useful.