0:00:00.500,0:00:02.620
It never hurts to get[br]a bit more practice.

0:00:02.620,0:00:05.600
So this is problem number five[br]from the normal distribution

0:00:05.600,0:00:11.560
chapter from ck12.org's[br]AP statistics FlexBook.

0:00:11.560,0:00:16.030
So they're saying, the 2007 AP[br]statistics examination scores

0:00:16.030,0:00:20.750
were not normally distributed[br]with a mean of 2.8

0:00:20.750,0:00:23.964
and a standard[br]deviation of 1.34.

0:00:23.964,0:00:25.630
They cite some College[br]Board stuff here.

0:00:25.630,0:00:27.170
I didn't copy and paste that.

0:00:27.170,0:00:29.170
What is the approximate z-score?

0:00:29.170,0:00:31.530
Remember, z-score[br]is just how many

0:00:31.530,0:00:33.980
standard deviations you[br]are away from the mean.

0:00:33.980,0:00:35.950
What is the approximate[br]z-score that

0:00:35.950,0:00:39.337
corresponds to an[br]exam score of 5?

0:00:39.337,0:00:40.920
So we really just[br]have to figure out--

0:00:40.920,0:00:42.628
this is a pretty[br]straightforward problem.

0:00:42.628,0:00:45.720
We just need to figure out how[br]many standard deviations is

0:00:45.720,0:00:48.340
5 from the mean?

0:00:48.340,0:00:53.370
Well, you just take[br]5 minus 2.8, right?

0:00:53.370,0:00:54.400
The mean is 2.8.

0:00:54.400,0:00:56.121
Let me be very[br]clear, mean is 2.8.

0:00:56.121,0:00:56.870
They give us that.

0:00:56.870,0:00:58.800
Didn't even have[br]to calculate it.

0:00:58.800,0:01:00.230
So the mean is 2.8.

0:01:00.230,0:01:03.760
So 5 minus 2.8 is equal to 2.2.

0:01:03.760,0:01:06.374
So we're 2.2 above the mean.

0:01:06.374,0:01:08.540
And if we want that in terms[br]of standard deviations,

0:01:08.540,0:01:10.770
we just divide by our[br]standard deviation.

0:01:10.770,0:01:14.860
You divide by 1.34.

0:01:14.860,0:01:17.290
Divide by 1.34.

0:01:17.290,0:01:20.710
I'll take out the[br]calculator for this.

0:01:20.710,0:01:31.280
So we have 2.2 divided[br]by 1.34 is equal to 1.64.

0:01:31.280,0:01:34.966
So this is equal to 1.64.

0:01:34.966,0:01:37.590
And that's choice C. So this was[br]actually very straightforward.

0:01:37.590,0:01:40.620
We just have to see how far[br]away we are from the mean

0:01:40.620,0:01:42.929
if we get a score of[br]5-- which hopefully you

0:01:42.929,0:01:44.720
will get if you're[br]taking the AP statistics

0:01:44.720,0:01:46.242
exam after watching[br]these videos.

0:01:46.242,0:01:48.450
And then you divide by the[br]standard deviation to say,

0:01:48.450,0:01:50.850
how many standard deviations[br]away from the mean

0:01:50.850,0:01:52.230
is the score of 5?

0:01:52.230,0:01:53.545
It's 1.64.

0:01:53.545,0:01:55.670
I think the only tricky[br]thing here might have been,

0:01:55.670,0:01:58.400
you might have been tempted[br]to pick choice E, which says,

0:01:58.400,0:02:01.300
the z-score cannot be calculated[br]because the distribution is not

0:02:01.300,0:02:01.800
normal.

0:02:01.800,0:02:04.700
And I think the reason why you[br]might have had that temptation

0:02:04.700,0:02:07.430
is because we've[br]been using z-scores

0:02:07.430,0:02:10.300
within the context of[br]a normal distribution.

0:02:10.300,0:02:12.860
But a z-score literally[br]just means how many

0:02:12.860,0:02:15.950
standard deviations you[br]are away from the mean.

0:02:15.950,0:02:18.290
It could apply to[br]any distribution

0:02:18.290,0:02:21.820
that you could calculate a mean[br]and a standard deviation for.

0:02:21.820,0:02:23.910
So E is not the correct answer.

0:02:23.910,0:02:27.045
A z-score can apply to a[br]non-normal distribution.

0:02:27.045,0:02:29.170
So the answer is C. And I[br]guess that's a good point

0:02:29.170,0:02:31.094
of clarification to[br]get out of the way.

0:02:31.094,0:02:33.260
And I thought I would do[br]two problems in this video,

0:02:33.260,0:02:35.460
just because that[br]one was pretty short.

0:02:35.460,0:02:36.900
So problem number six.

0:02:36.900,0:02:39.350
The height of fifth grade[br]boys in the United States

0:02:39.350,0:02:41.480
is approximately[br]normally distributed--

0:02:41.480,0:02:45.690
that's good to know-- with[br]a mean height of 143.5

0:02:45.690,0:02:46.410
centimeters.

0:02:46.410,0:02:50.960
So it's a mean of[br]143.5 centimeters

0:02:50.960,0:02:56.635
and a standard deviation[br]of about 7.1 centimeters.

0:03:01.700,0:03:04.620
What is the probability that[br]a randomly chosen fifth grade

0:03:04.620,0:03:09.134
boy would be taller[br]than 157.7 centimeters?

0:03:09.134,0:03:10.800
So let's just draw[br]out this distribution

0:03:10.800,0:03:13.755
like we've done in a[br]bunch of problems so far.

0:03:13.755,0:03:15.600
They're just asking[br]us one question,

0:03:15.600,0:03:19.320
so we can mark this[br]distribution up a good bit.

0:03:19.320,0:03:21.410
Let's say that's[br]our distribution.

0:03:21.410,0:03:28.270
And the mean here, the[br]mean they told us is 143.5.

0:03:28.270,0:03:30.414
They're asking us[br]taller than 157.7.

0:03:30.414,0:03:32.080
So we're going in the[br]upwards direction.

0:03:32.080,0:03:35.360
So one standard[br]deviation above the mean

0:03:35.360,0:03:37.740
will take us right there.

0:03:37.740,0:03:40.510
And we just have to add 7.1[br]to this number right here.

0:03:40.510,0:03:42.700
We're going up by 7.1.

0:03:42.700,0:03:45.980
So 143.5 plus 7.1 is what?

0:03:45.980,0:03:49.440
150.6.

0:03:49.440,0:03:51.047
That's one standard deviation.

0:03:51.047,0:03:52.880
If we were to go another[br]standard deviation,

0:03:52.880,0:03:54.950
we'd go 7.1 more.

0:03:54.950,0:03:57.500
What's 7.1 plus 150.6?

0:03:57.500,0:04:02.950
It's 157.7, which[br]just happens to be

0:04:02.950,0:04:04.220
the exact number they ask for.

0:04:04.220,0:04:06.240
They're asking for[br]the probability

0:04:06.240,0:04:08.304
of getting a height[br]higher than that.

0:04:08.304,0:04:10.470
So they want to know, what's[br]the probability that we

0:04:10.470,0:04:12.830
fall under this area right here?

0:04:12.830,0:04:15.980
Or essentially more than[br]two standard deviations

0:04:15.980,0:04:16.630
from the mean.

0:04:16.630,0:04:18.670
Or above two[br]standard deviations.

0:04:18.670,0:04:21.420
We can't count this[br]left tail right there.

0:04:21.420,0:04:24.480
So we can use the[br]empirical rule.

0:04:24.480,0:04:26.630
If we do our standard[br]deviations to the left,

0:04:26.630,0:04:29.830
that's one standard deviation,[br]two standard deviations.

0:04:29.830,0:04:32.010
We know what this whole area is.

0:04:32.010,0:04:35.660
Let me pick a different[br]color so that I don't.

0:04:35.660,0:04:39.170
So we know what this[br]area is, the area

0:04:39.170,0:04:40.780
within two standard deviations.

0:04:40.780,0:04:42.020
The empirical rule tells us.

0:04:42.020,0:04:46.820
Or even better, the[br]68, 95, 99.7 rule

0:04:46.820,0:04:48.830
tells us that this[br]area-- because it's

0:04:48.830,0:04:55.300
within two standard[br]deviations-- is 95%, or 0.95.

0:04:55.300,0:04:59.740
Or it's 95% of the area under[br]the normal distribution.

0:04:59.740,0:05:02.400
Which tells us that what's[br]left over-- this tail

0:05:02.400,0:05:04.880
that we care about and[br]this left tail right here--

0:05:04.880,0:05:08.340
has to make up the[br]rest of it, or 5%.

0:05:08.340,0:05:12.216
So those two combined[br]have to be 5%.

0:05:12.216,0:05:13.570
And these are symmetrical.

0:05:13.570,0:05:14.590
We've done this before.

0:05:14.590,0:05:16.330
This is actually a little[br]redundant from other problems

0:05:16.330,0:05:17.250
we've done.

0:05:17.250,0:05:20.010
But if these are added, combined[br]5%, and they're the same,

0:05:20.010,0:05:22.580
then each of these are 2.5%.

0:05:22.580,0:05:24.792
Each of these are 2.5%.

0:05:24.792,0:05:26.250
So the answer to[br]the question, what

0:05:26.250,0:05:29.160
is the probability that a[br]randomly chosen fifth grade boy

0:05:29.160,0:05:32.820
would be taller then[br]157.7 centimeters.

0:05:32.820,0:05:34.320
Well, that's literally[br]just the area

0:05:34.320,0:05:35.927
under this right green part.

0:05:35.927,0:05:37.510
Maybe I'll do it in[br]a different color.

0:05:37.510,0:05:39.660
This magenta part that[br]I'm coloring right now.

0:05:39.660,0:05:40.920
That's just that area.

0:05:40.920,0:05:43.600
We just figured out it's 2.5%.

0:05:43.600,0:05:47.780
So there's a 2.5% chance we'd[br]randomly find a fifth grade

0:05:47.780,0:05:51.260
boy who's taller than[br]157.7 centimeters,

0:05:51.260,0:05:53.650
assuming this is the mean,[br]the standard deviation,

0:05:53.650,0:05:56.680
and we are dealing with[br]a normal distribution.