WEBVTT
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in this problem. We have information about the gold
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collar workers, and they spend an average of$729
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on themselves with a standard deviation of$92 and this
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problem has quite a few parts to it. So
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Part A is asking for a percentage of gold collar
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workers who spend between 609$100 on themselves. So
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in order to find the percentage refers, gonna find
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the probability. So the probability that X is between
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609 100 is the way we'll start. Um,
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they did say throughout that we're going to assume normal
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distribution, so therefore we can create a normal shaped
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curve and we'll put 729 in the center. So
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we need to find the probability between 600 and 900
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. So we will need to use these scores and
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to refresh your memory to find a Z score.
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We do the ex score minus the population average divided
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by the standard deviation. So the Z score associated
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with 600 would be found by doing 600 minus 729
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divided by 92 and that ends up giving you a
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Z score of negative 1.40 and the Z score for
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900 would be 900 minus 7 29 divided by 92
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that Z scores approximately 1.86 Now I like to put
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those e scores on the bell shaped curve, so
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we have negative one point for zero that's associated with
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601.86 which is associated to 900. So when we
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talk about the probability between 609 100 in terms of
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the gold collars spent it or the gold collar workers
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spending money themselves, we can also say then that
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Z will be between negative 1.40 and 1.86 And in
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order to solve this problem, we would have to
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rewrite our probability statement as the probability that Z is
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less than 1.86 minus the probability that Z is less
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than negative 1.40 And at that point, you're going
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to have to utilize your standard normal table in the
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back of the book. In the area to the
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left of 1.86 would be 0.9686 and the area to
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the left of negative 1.40 is 0.808 And when we
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subtract those two, we get a value of 20.8878
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So that means 88.78% of gold collar workers are going
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to spend between 609$100 on themselves. So let's
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look at Part B and in part B. We
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are asked to find the percentage of gold collar workers
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spending between 400 and 1000 month on themselves. So
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we're still using the same bell curve, the same
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average and standard deviation. So the average was 729
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and this time we're talking between 400 and 1000.
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So we need the Z score associated with 400.
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We should be 400 minus 729 divided by our standard
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deviation of 92 and that's the score would be negative
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3.58 so that would be appear negative 3.58 and the
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Z score associated with 1000 would be 1000 minus 729
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divided by 92 and that is going to get you
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2.95 So again, when you were talking about the
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number of gold collar workers spending between 400 1000 you
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can say, Well, that's the same thing as
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the Z score being between negative 3.58 and positive 2.95
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And to solve that, we're going to first evaluate
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the probability that Z is less than 2.95 And from
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that we're going to subtract the probability that Z is
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less than negative 3.58 You're going to utilize your standard
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normal table, and we're going to get an area
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to the left of 2.95 to be 0.9984 and the
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area to the left of negative 3.58 to be 0.1
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resulting in 0.9983 or 99.83% of gold collar workers are
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going to spend between$401,000 on themselves, per but
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in part C. We're trying to find the percentage
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off gold collar workers that spend more than 1050 on
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themselves, so that would be that X is greater
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than 1050. So here is our bell curve,
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with the average of 729 in the center, and
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we're trying to go above 1050. So refined the
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Z score for 1050. So we'll do 1050 minus
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729 divided by 92 which is a Z score value
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of 3.49 So we have 3.49 and when we're talking
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about being greater than 1050 it's the same as the
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Z score being greater than 3.49 And to solve that
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will have to do one minus the probability of Z
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being less than 3.49 And by doing that, we
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end up with one minus 10.9998 or 0.2 So that
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would translate into 0.2% of gold collar workers spending more
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than$1050 on themselves in a month. In Part
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D. You're asked to determine how many or what
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percentage of gold collar workers spend less than 500 a
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month on themselves, so that's going to translate into
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X is less than 500 again. I'm going to
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construct that bell just to get a visual of what's
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going on. We're gonna have 700 29 in the
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center, and our 500 is to the left of
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that. So we need a Z score for 500
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. So do 500 minus 729 divided by the standard
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deviation of 92. You're going to get a Z
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score of negative 2.49 So I put it up here
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on my bell. So when I am referring to
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spending less than 500 then it's also the same as
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saying. What's the probability that Z is less than
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2.49? You can look for that in your chart
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in your normal distribution chart in the back of your
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book, and you're going to find a value of
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0.64 which translates into 0.64% of gold collar workers spending
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less than$500 on themselves. Emma