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Problem 54 Hard Difficulty

There is a new working class with money to burn, according to the USA Today March 1, 2005, article "New "gold-collar' young workers gain clout." "Gold-collar" is a subset of blue-collar workers, defined by researchers as those working in fast food and retail jobs, or as security guards, office workers, or hairdressers. These 18- to 25-yearold "gold-collar" workers are spending an average of $\$ 729$ a month on themselves (versus $\$ 267$ for college students and $\$ 609$ for blue-collar workers). Assuming this spending is normally distributed with a standard deviation of $\$ 92.00$ what percentage of gold-collar workers spend:
a. between $\$ 600$ and $\$ 900$ a month on themselves?
b. between $\$ 400$ and $\$ 1000$ a month on themselves?
c. more than $\$ 1050$ a month on themselves?
d. less than $\$ 500$ a month on themselves?

Answer

a. $88.78 \%$
b. $99.82 \%$
c. $0.02 \%$
d. $0.64 \%$

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Video Transcript

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