[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.50,0:00:02.62,Default,,0000,0000,0000,,It never hurts to get\Na bit more practice.
Dialogue: 0,0:00:02.62,0:00:05.60,Default,,0000,0000,0000,,So this is problem number five\Nfrom the normal distribution
Dialogue: 0,0:00:05.60,0:00:11.56,Default,,0000,0000,0000,,chapter from ck12.org's\NAP statistics FlexBook.
Dialogue: 0,0:00:11.56,0:00:16.03,Default,,0000,0000,0000,,So they're saying, the 2007 AP\Nstatistics examination scores
Dialogue: 0,0:00:16.03,0:00:20.75,Default,,0000,0000,0000,,were not normally distributed\Nwith a mean of 2.8
Dialogue: 0,0:00:20.75,0:00:23.96,Default,,0000,0000,0000,,and a standard\Ndeviation of 1.34.
Dialogue: 0,0:00:23.96,0:00:25.63,Default,,0000,0000,0000,,They cite some College\NBoard stuff here.
Dialogue: 0,0:00:25.63,0:00:27.17,Default,,0000,0000,0000,,I didn't copy and paste that.
Dialogue: 0,0:00:27.17,0:00:29.17,Default,,0000,0000,0000,,What is the approximate z-score?
Dialogue: 0,0:00:29.17,0:00:31.53,Default,,0000,0000,0000,,Remember, z-score\Nis just how many
Dialogue: 0,0:00:31.53,0:00:33.98,Default,,0000,0000,0000,,standard deviations you\Nare away from the mean.
Dialogue: 0,0:00:33.98,0:00:35.95,Default,,0000,0000,0000,,What is the approximate\Nz-score that
Dialogue: 0,0:00:35.95,0:00:39.34,Default,,0000,0000,0000,,corresponds to an\Nexam score of 5?
Dialogue: 0,0:00:39.34,0:00:40.92,Default,,0000,0000,0000,,So we really just\Nhave to figure out--
Dialogue: 0,0:00:40.92,0:00:42.63,Default,,0000,0000,0000,,this is a pretty\Nstraightforward problem.
Dialogue: 0,0:00:42.63,0:00:45.72,Default,,0000,0000,0000,,We just need to figure out how\Nmany standard deviations is
Dialogue: 0,0:00:45.72,0:00:48.34,Default,,0000,0000,0000,,5 from the mean?
Dialogue: 0,0:00:48.34,0:00:53.37,Default,,0000,0000,0000,,Well, you just take\N5 minus 2.8, right?
Dialogue: 0,0:00:53.37,0:00:54.40,Default,,0000,0000,0000,,The mean is 2.8.
Dialogue: 0,0:00:54.40,0:00:56.12,Default,,0000,0000,0000,,Let me be very\Nclear, mean is 2.8.
Dialogue: 0,0:00:56.12,0:00:56.87,Default,,0000,0000,0000,,They give us that.
Dialogue: 0,0:00:56.87,0:00:58.80,Default,,0000,0000,0000,,Didn't even have\Nto calculate it.
Dialogue: 0,0:00:58.80,0:01:00.23,Default,,0000,0000,0000,,So the mean is 2.8.
Dialogue: 0,0:01:00.23,0:01:03.76,Default,,0000,0000,0000,,So 5 minus 2.8 is equal to 2.2.
Dialogue: 0,0:01:03.76,0:01:06.37,Default,,0000,0000,0000,,So we're 2.2 above the mean.
Dialogue: 0,0:01:06.37,0:01:08.54,Default,,0000,0000,0000,,And if we want that in terms\Nof standard deviations,
Dialogue: 0,0:01:08.54,0:01:10.77,Default,,0000,0000,0000,,we just divide by our\Nstandard deviation.
Dialogue: 0,0:01:10.77,0:01:14.86,Default,,0000,0000,0000,,You divide by 1.34.
Dialogue: 0,0:01:14.86,0:01:17.29,Default,,0000,0000,0000,,Divide by 1.34.
Dialogue: 0,0:01:17.29,0:01:20.71,Default,,0000,0000,0000,,I'll take out the\Ncalculator for this.
Dialogue: 0,0:01:20.71,0:01:31.28,Default,,0000,0000,0000,,So we have 2.2 divided\Nby 1.34 is equal to 1.64.
Dialogue: 0,0:01:31.28,0:01:34.97,Default,,0000,0000,0000,,So this is equal to 1.64.
Dialogue: 0,0:01:34.97,0:01:37.59,Default,,0000,0000,0000,,And that's choice C. So this was\Nactually very straightforward.
Dialogue: 0,0:01:37.59,0:01:40.62,Default,,0000,0000,0000,,We just have to see how far\Naway we are from the mean
Dialogue: 0,0:01:40.62,0:01:42.93,Default,,0000,0000,0000,,if we get a score of\N5-- which hopefully you
Dialogue: 0,0:01:42.93,0:01:44.72,Default,,0000,0000,0000,,will get if you're\Ntaking the AP statistics
Dialogue: 0,0:01:44.72,0:01:46.24,Default,,0000,0000,0000,,exam after watching\Nthese videos.
Dialogue: 0,0:01:46.24,0:01:48.45,Default,,0000,0000,0000,,And then you divide by the\Nstandard deviation to say,
Dialogue: 0,0:01:48.45,0:01:50.85,Default,,0000,0000,0000,,how many standard deviations\Naway from the mean
Dialogue: 0,0:01:50.85,0:01:52.23,Default,,0000,0000,0000,,is the score of 5?
Dialogue: 0,0:01:52.23,0:01:53.54,Default,,0000,0000,0000,,It's 1.64.
Dialogue: 0,0:01:53.54,0:01:55.67,Default,,0000,0000,0000,,I think the only tricky\Nthing here might have been,
Dialogue: 0,0:01:55.67,0:01:58.40,Default,,0000,0000,0000,,you might have been tempted\Nto pick choice E, which says,
Dialogue: 0,0:01:58.40,0:02:01.30,Default,,0000,0000,0000,,the z-score cannot be calculated\Nbecause the distribution is not
Dialogue: 0,0:02:01.30,0:02:01.80,Default,,0000,0000,0000,,normal.
Dialogue: 0,0:02:01.80,0:02:04.70,Default,,0000,0000,0000,,And I think the reason why you\Nmight have had that temptation
Dialogue: 0,0:02:04.70,0:02:07.43,Default,,0000,0000,0000,,is because we've\Nbeen using z-scores
Dialogue: 0,0:02:07.43,0:02:10.30,Default,,0000,0000,0000,,within the context of\Na normal distribution.
Dialogue: 0,0:02:10.30,0:02:12.86,Default,,0000,0000,0000,,But a z-score literally\Njust means how many
Dialogue: 0,0:02:12.86,0:02:15.95,Default,,0000,0000,0000,,standard deviations you\Nare away from the mean.
Dialogue: 0,0:02:15.95,0:02:18.29,Default,,0000,0000,0000,,It could apply to\Nany distribution
Dialogue: 0,0:02:18.29,0:02:21.82,Default,,0000,0000,0000,,that you could calculate a mean\Nand a standard deviation for.
Dialogue: 0,0:02:21.82,0:02:23.91,Default,,0000,0000,0000,,So E is not the correct answer.
Dialogue: 0,0:02:23.91,0:02:27.04,Default,,0000,0000,0000,,A z-score can apply to a\Nnon-normal distribution.
Dialogue: 0,0:02:27.04,0:02:29.17,Default,,0000,0000,0000,,So the answer is C. And I\Nguess that's a good point
Dialogue: 0,0:02:29.17,0:02:31.09,Default,,0000,0000,0000,,of clarification to\Nget out of the way.
Dialogue: 0,0:02:31.09,0:02:33.26,Default,,0000,0000,0000,,And I thought I would do\Ntwo problems in this video,
Dialogue: 0,0:02:33.26,0:02:35.46,Default,,0000,0000,0000,,just because that\None was pretty short.
Dialogue: 0,0:02:35.46,0:02:36.90,Default,,0000,0000,0000,,So problem number six.
Dialogue: 0,0:02:36.90,0:02:39.35,Default,,0000,0000,0000,,The height of fifth grade\Nboys in the United States
Dialogue: 0,0:02:39.35,0:02:41.48,Default,,0000,0000,0000,,is approximately\Nnormally distributed--
Dialogue: 0,0:02:41.48,0:02:45.69,Default,,0000,0000,0000,,that's good to know-- with\Na mean height of 143.5
Dialogue: 0,0:02:45.69,0:02:46.41,Default,,0000,0000,0000,,centimeters.
Dialogue: 0,0:02:46.41,0:02:50.96,Default,,0000,0000,0000,,So it's a mean of\N143.5 centimeters
Dialogue: 0,0:02:50.96,0:02:56.64,Default,,0000,0000,0000,,and a standard deviation\Nof about 7.1 centimeters.
Dialogue: 0,0:03:01.70,0:03:04.62,Default,,0000,0000,0000,,What is the probability that\Na randomly chosen fifth grade
Dialogue: 0,0:03:04.62,0:03:09.13,Default,,0000,0000,0000,,boy would be taller\Nthan 157.7 centimeters?
Dialogue: 0,0:03:09.13,0:03:10.80,Default,,0000,0000,0000,,So let's just draw\Nout this distribution
Dialogue: 0,0:03:10.80,0:03:13.76,Default,,0000,0000,0000,,like we've done in a\Nbunch of problems so far.
Dialogue: 0,0:03:13.76,0:03:15.60,Default,,0000,0000,0000,,They're just asking\Nus one question,
Dialogue: 0,0:03:15.60,0:03:19.32,Default,,0000,0000,0000,,so we can mark this\Ndistribution up a good bit.
Dialogue: 0,0:03:19.32,0:03:21.41,Default,,0000,0000,0000,,Let's say that's\Nour distribution.
Dialogue: 0,0:03:21.41,0:03:28.27,Default,,0000,0000,0000,,And the mean here, the\Nmean they told us is 143.5.
Dialogue: 0,0:03:28.27,0:03:30.41,Default,,0000,0000,0000,,They're asking us\Ntaller than 157.7.
Dialogue: 0,0:03:30.41,0:03:32.08,Default,,0000,0000,0000,,So we're going in the\Nupwards direction.
Dialogue: 0,0:03:32.08,0:03:35.36,Default,,0000,0000,0000,,So one standard\Ndeviation above the mean
Dialogue: 0,0:03:35.36,0:03:37.74,Default,,0000,0000,0000,,will take us right there.
Dialogue: 0,0:03:37.74,0:03:40.51,Default,,0000,0000,0000,,And we just have to add 7.1\Nto this number right here.
Dialogue: 0,0:03:40.51,0:03:42.70,Default,,0000,0000,0000,,We're going up by 7.1.
Dialogue: 0,0:03:42.70,0:03:45.98,Default,,0000,0000,0000,,So 143.5 plus 7.1 is what?
Dialogue: 0,0:03:45.98,0:03:49.44,Default,,0000,0000,0000,,150.6.
Dialogue: 0,0:03:49.44,0:03:51.05,Default,,0000,0000,0000,,That's one standard deviation.
Dialogue: 0,0:03:51.05,0:03:52.88,Default,,0000,0000,0000,,If we were to go another\Nstandard deviation,
Dialogue: 0,0:03:52.88,0:03:54.95,Default,,0000,0000,0000,,we'd go 7.1 more.
Dialogue: 0,0:03:54.95,0:03:57.50,Default,,0000,0000,0000,,What's 7.1 plus 150.6?
Dialogue: 0,0:03:57.50,0:04:02.95,Default,,0000,0000,0000,,It's 157.7, which\Njust happens to be
Dialogue: 0,0:04:02.95,0:04:04.22,Default,,0000,0000,0000,,the exact number they ask for.
Dialogue: 0,0:04:04.22,0:04:06.24,Default,,0000,0000,0000,,They're asking for\Nthe probability
Dialogue: 0,0:04:06.24,0:04:08.30,Default,,0000,0000,0000,,of getting a height\Nhigher than that.
Dialogue: 0,0:04:08.30,0:04:10.47,Default,,0000,0000,0000,,So they want to know, what's\Nthe probability that we
Dialogue: 0,0:04:10.47,0:04:12.83,Default,,0000,0000,0000,,fall under this area right here?
Dialogue: 0,0:04:12.83,0:04:15.98,Default,,0000,0000,0000,,Or essentially more than\Ntwo standard deviations
Dialogue: 0,0:04:15.98,0:04:16.63,Default,,0000,0000,0000,,from the mean.
Dialogue: 0,0:04:16.63,0:04:18.67,Default,,0000,0000,0000,,Or above two\Nstandard deviations.
Dialogue: 0,0:04:18.67,0:04:21.42,Default,,0000,0000,0000,,We can't count this\Nleft tail right there.
Dialogue: 0,0:04:21.42,0:04:24.48,Default,,0000,0000,0000,,So we can use the\Nempirical rule.
Dialogue: 0,0:04:24.48,0:04:26.63,Default,,0000,0000,0000,,If we do our standard\Ndeviations to the left,
Dialogue: 0,0:04:26.63,0:04:29.83,Default,,0000,0000,0000,,that's one standard deviation,\Ntwo standard deviations.
Dialogue: 0,0:04:29.83,0:04:32.01,Default,,0000,0000,0000,,We know what this whole area is.
Dialogue: 0,0:04:32.01,0:04:35.66,Default,,0000,0000,0000,,Let me pick a different\Ncolor so that I don't.
Dialogue: 0,0:04:35.66,0:04:39.17,Default,,0000,0000,0000,,So we know what this\Narea is, the area
Dialogue: 0,0:04:39.17,0:04:40.78,Default,,0000,0000,0000,,within two standard deviations.
Dialogue: 0,0:04:40.78,0:04:42.02,Default,,0000,0000,0000,,The empirical rule tells us.
Dialogue: 0,0:04:42.02,0:04:46.82,Default,,0000,0000,0000,,Or even better, the\N68, 95, 99.7 rule
Dialogue: 0,0:04:46.82,0:04:48.83,Default,,0000,0000,0000,,tells us that this\Narea-- because it's
Dialogue: 0,0:04:48.83,0:04:55.30,Default,,0000,0000,0000,,within two standard\Ndeviations-- is 95%, or 0.95.
Dialogue: 0,0:04:55.30,0:04:59.74,Default,,0000,0000,0000,,Or it's 95% of the area under\Nthe normal distribution.
Dialogue: 0,0:04:59.74,0:05:02.40,Default,,0000,0000,0000,,Which tells us that what's\Nleft over-- this tail
Dialogue: 0,0:05:02.40,0:05:04.88,Default,,0000,0000,0000,,that we care about and\Nthis left tail right here--
Dialogue: 0,0:05:04.88,0:05:08.34,Default,,0000,0000,0000,,has to make up the\Nrest of it, or 5%.
Dialogue: 0,0:05:08.34,0:05:12.22,Default,,0000,0000,0000,,So those two combined\Nhave to be 5%.
Dialogue: 0,0:05:12.22,0:05:13.57,Default,,0000,0000,0000,,And these are symmetrical.
Dialogue: 0,0:05:13.57,0:05:14.59,Default,,0000,0000,0000,,We've done this before.
Dialogue: 0,0:05:14.59,0:05:16.33,Default,,0000,0000,0000,,This is actually a little\Nredundant from other problems
Dialogue: 0,0:05:16.33,0:05:17.25,Default,,0000,0000,0000,,we've done.
Dialogue: 0,0:05:17.25,0:05:20.01,Default,,0000,0000,0000,,But if these are added, combined\N5%, and they're the same,
Dialogue: 0,0:05:20.01,0:05:22.58,Default,,0000,0000,0000,,then each of these are 2.5%.
Dialogue: 0,0:05:22.58,0:05:24.79,Default,,0000,0000,0000,,Each of these are 2.5%.
Dialogue: 0,0:05:24.79,0:05:26.25,Default,,0000,0000,0000,,So the answer to\Nthe question, what
Dialogue: 0,0:05:26.25,0:05:29.16,Default,,0000,0000,0000,,is the probability that a\Nrandomly chosen fifth grade boy
Dialogue: 0,0:05:29.16,0:05:32.82,Default,,0000,0000,0000,,would be taller then\N157.7 centimeters.
Dialogue: 0,0:05:32.82,0:05:34.32,Default,,0000,0000,0000,,Well, that's literally\Njust the area
Dialogue: 0,0:05:34.32,0:05:35.93,Default,,0000,0000,0000,,under this right green part.
Dialogue: 0,0:05:35.93,0:05:37.51,Default,,0000,0000,0000,,Maybe I'll do it in\Na different color.
Dialogue: 0,0:05:37.51,0:05:39.66,Default,,0000,0000,0000,,This magenta part that\NI'm coloring right now.
Dialogue: 0,0:05:39.66,0:05:40.92,Default,,0000,0000,0000,,That's just that area.
Dialogue: 0,0:05:40.92,0:05:43.60,Default,,0000,0000,0000,,We just figured out it's 2.5%.
Dialogue: 0,0:05:43.60,0:05:47.78,Default,,0000,0000,0000,,So there's a 2.5% chance we'd\Nrandomly find a fifth grade
Dialogue: 0,0:05:47.78,0:05:51.26,Default,,0000,0000,0000,,boy who's taller than\N157.7 centimeters,
Dialogue: 0,0:05:51.26,0:05:53.65,Default,,0000,0000,0000,,assuming this is the mean,\Nthe standard deviation,
Dialogue: 0,0:05:53.65,0:05:56.68,Default,,0000,0000,0000,,and we are dealing with\Na normal distribution.