WEBVTT
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okay for this problem. We have some more normally
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distributed data, but this problems a little bit different
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because we don't know what the means. War is
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so with the most normal curve problems. You want
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a sketch? Uh, the normal distribution and write
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down what we know. So first I'm going to
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do is write down. We know, um,
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that standard deviation equals two ah,$3898 for junior
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executives at a company. So what's different about this
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problem in some of the earlier problems is that we
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want for this problem is we want to know for
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party. At least we want to know what,
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Thea, what the mean value is. So our
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plan is going to be to really use ah,
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the formula for the Z score to get the to
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get the mean Okay, so we're gonna have to
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find the Z score. So let's figure out how
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to get a Z score here. So first things
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first, we're gonna get ah, go to our
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calculator. And we were told in the problem that
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there's an 80% middle 80% of incomes have it a
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certain we have the high end of the salary for
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the 80% of the incomes. We know what that
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ISS. So let's do something here. We call
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the Universal Normal. We've been doing forward what I
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call forward problems we put in the the Z scores
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that we know are put in the parameters that we
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know and to get the probabilities. This case we
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know the amount of probability of the area is 80%
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. Guess so. Um, so it's 80.
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And so we're gonna get work backwards that to get
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the Z score that goes with 80%. You figure
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out that, IHS, I'm gonna leave the mean
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at zero on the stand, aviation at one.
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Because the score and we just mark this up here
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, too. Um and so the number that we
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know is the high end of the 80%. So
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it's the porn 80 number of kind of from their
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back is, um, 62,900. We note that
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case 62,900 of the high end of the middle.
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80% of the owners we want to know is the
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is the meeting here. So, um, from
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our inverse, normal calculation work backwards on the seat
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. See table Technically, so we know that we
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use a Z score, so let's go back to
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our plan here. So now we know the Z
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score that goes with the number that we have.
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The 62,900 is 0.84. So that's the Z score
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that goes with it. And so we would know
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using the Z 62,900 minus the mean divided by the
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standard deviation which we stayed over there is 30 98
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is the relationship. So we can work backwards to
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find at with that with that mean number is for
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this overall distribution. When party, we're gonna do
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a little bit more work to figure out a different
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parameter. So when he's my calculator, since it's
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up here, so see 0.84 I'm gonna solve for
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the for the mean value someone multiply both sides of
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the equation by 38 98 so we can see that
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thing. It was 3280 0.63 um, equals just
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62,900 minus the means were you okay? So now
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I'm just gonna solve Really What I would do showing
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all my work. There's at immutable sides. And
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then what happened to do is take the 60 2900
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. It's attracted 30 to 80.63. So what type
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of my calculator over here? 62,900 my ass.
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The answer. That's my stack here. So that
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tells us the mean number is 52,000. 59,000 619
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0.36 around. There's a penny there. Okay,
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so that's your answer for perch. What? I'm
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gonna really? That's what we wanted for up here
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. Okay, so we have the 59,000 619.3 sheets
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in really with that is that goes with better there
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. So I get that part done. Um,
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And now for part B. What? They asked
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us who's they asked us. Well, now use
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your information that you got a part. A to
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find an executive with an executive. The probability,
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um, that executive earns less than$50,000 so we
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wanted the probability the X is less than it 2000
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giving this distribution. Okay, well, so now
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we know this 59,000 is here um, so 50,000
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. I'd like to make a sketch of the estimate
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there, so we want to basically find the probability
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to left in this line. So I liked Always
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sketch out what we see here. Okay, I'm
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just gonna keep keep it short arms and about 50
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k All right, So when that Rooney's our calculator
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now, we're going to go forward in our calculator
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and use our second distribution function to do rz scoring
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and are, um, probability under understanding. So
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we're gonna do second distribution. Well, miss click
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their second distribution. Um, normal CDF So now
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asking our calculator. Well, let's go from negative
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infinity to the upper boundary of 50,000. And again
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, he used an apple it to do this or
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look from your table, give us the scores.
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Um, And for this problem, what we looked
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used in part A. We now know that the
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mean is 59,000 619 0.36. And standard deviation for
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this data is 38 98. So, um,
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that tells us that overall yeah, way have 0.67
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the other 60 feet. So that's the probability,
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given the information we have. But there in that
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much