WEBVTT
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right. This problem is about water taxi safety and
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a scenario of an incident that happened in Baltimore's inner
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harbor. And there's Cem, given information that we
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have to start with before we can answer the four
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parts to this question. And the given information is
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that waits of men are normally distributed that the average
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weight of a man is£189 with a standard deviation
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of£39 and we were also told about the occupancy
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of the boat. The stated occupancy of the boat
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was 25 people, and the load limit was£3500
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. And keep in mind, those 25 people could
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have been men, women and Children. So when
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they do the stated occupancy, they have to talk
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in terms of both a weight limit as well as
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a person limit. So let's go to problem A
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and in problem A. It asks, given that
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the water taxi that sank was rated for a load
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limit of£3500. What is the maximum mean weight
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of passengers if the boat was filled with the 25
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passengers? So if we take those 25 passengers and
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we divided into the 3500 possible Wait, We're going
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to end up with an average of 140 pounds again
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. Keep in mind, that's average. So that
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meant that some of the 25 people could have been
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weighing more than 100£40 and some of them could
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have been weighing less than£140 in part B.
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The problem is asking you if the water taxi is
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filled with 25 randomly selected men, so we're going
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to focus on the heaviest class of people We've got
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the men, the women and the Children and usually
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the men. Average weight is higher than any women
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, average weight or child's average weight. So we're
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drawing a sample from our population and our sample size
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is going to be 25. And we're trying to
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determine what is the probability that they're mean. Wait
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exceeds, which means greater than the value from Part
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A, which we found to be£140. And
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in order to solve this problem, we are going
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to have to discuss the average of the sample means
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because again we are finding a sample of 25.
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We're talking about their mean wait and we need to
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discuss the standard deviation of those sample meets. And
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we're going to let the central limit theorem guide us
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in calculating these. And the central limit Theorem says
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that the average of the sample means will equal the
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average of the population, which in this case was
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£189. And the standard deviation of sample means will
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be equivalent to the standard deviation of the population divided
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by the square root of the sample size. So
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in this case, is going to be£39 divided
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by the square root of 25 and because they told
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us that it was normally distributed. We can start
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this process by drawing are bell shaped curve, and
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when we draw our bell shaped curve, we're going
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to put the average in the center, and we're
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trying to calculate the probability or the likelihood that the
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average is greater than 140. We will need a
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Z score formula to assist us in solving this problem
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, and since we're dealing with sample means the Z
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score formula that we're going to use is going to
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be X bar minus the average of the sample means
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it's that over a little bit all over. The
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standard deviation of the sample means. So we need
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to calculate R Z score for 140. So we're
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going to do Z equals 140 minus 189. And
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in place of the standard deviation of sample means we're
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going to use the expression 39 divided by the square
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root of 25. In doing so, you get
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a value of negative 6.28 So when we're solving this
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problem, if I put that negative 6.28 up here
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on my bell, when I discuss the chances of
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the average of thes 25 men's being greater than£140
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it's no different than saying. What's the probability that
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the Z score is greater than negative 6.28? And
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because our standard normal table, which is found in
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Table 82 in the back of your book, always
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discusses theme area or the probabilities into the left tail
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of the bell, and as you see our picture
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is going or extending into the right tail. We're
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going to have to rewrite this probability as one minus
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the probability that Z is less than negative 6.28 So
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when we go to our standard normal table, um
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, the negative 6.28 is not found in the table
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. But you do have a statement in that table
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that says that a Z score that is less than
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or equal to ah, 3.5 or negative 3.5 is
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going to have a probability of 0.1 So when we
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subtract the one minus the 10.1 we're getting a probability
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of 0.9999 So let's recap Part B in part B
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. It's saying if the water taxi was filled with
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25 randomly selected men, what is the probability that
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they're mean? Weight exceeds that£140 from part A
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, and that probability is going to be 0.9999 So
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let's move on to part C and in part c
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, you are asked. After the water taxi sank
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, the weight assumptions were revised. So the new
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capacity, instead of being 25 people, we now
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are only going to do 20 people. So now
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we're going to have a new sample size and this
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time our sample size is going to be 20.
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And if the water taxi was filled with 20 randomly
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selected men and again, we chose the men because
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the man's weight is usually heavier than the woman or
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the Children. If the water taxi is filled with
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those 25 randomly selected men, what is the probability
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that they're mean wait exceed, which is greater than
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£175? So again, we're going to need to
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calculate with this new sample, the average of the
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sample means and the standard deviation of the sample means
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. So again we can apply our central limit theorem
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. The average of the sample means is equal to
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the average of the population, which in this case
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was£189. As the average weight of men and
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the standard deviation based on the central limit theorem would
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be equivalent to standard deviation of the population divided by
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the square root of the sample size. And our
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standard deviation of the men's weight was 39 and our
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sample size here is going to be the square root
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of 20 again. We're going to draw our US
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bell shaped curve. We're going to put the average
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in the center, and we're trying to determine the
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probability that we are greater than 1 75. So
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again, the Z score we're going to use is
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going to be Z equals X bar, minus the
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average of the sample means divided by the standard deviation
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of sample means. And in this case, it's
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going to read to see equals 1 75 minus 1
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89 divided by Here's that expression for the standard deviation
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of the sample means we're gonna have 39 over the
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square root of 20. And that's going to yield
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a Z score of negative 1.61 so we can put
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a negative 1.61 up here on our bell. And
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then when we're talking with our problem being, what's
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the probability that the average is greater than 1 75
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? It's no different than saying What is the probability
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that your Z score is greater than negative 1.61 and
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again, our our picture is extending into the right
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table tail. But the table in the back of
139
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your book discusses the areas or the probabilities as we
140
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extend into the left tail. So we're going to
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have to rewrite this as one minus the probability that
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Z is less than negative 1.61 And when you look
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that value up in the table, you're going to
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get one minus 10.537 for an overall probability of 0.9463
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So recapping part C. After the water taxi accident
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, they downgraded the capacity number from 25 passengers on
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Lee, allowing 20 passengers. And when we put
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20 men on that water taxi, the probability that
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their average weight exceeded£175 would be the 0.9463 And
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then finally, Part D. In this problem in
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part D, the question is saying, Is this
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new capacity of 20 passengers safe? Um, when
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you look at the new capacity, the new capacity
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of 20 passengers still does not appear safe because the
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probability of being over the load limit is still high
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. At a value of 9.9463