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Problem 52 Hard Difficulty

6.52 According to Collegeboard.com [http://www.collegeboard.com/ $]$ the national average salary for a plumber as of 2007 is $\$ 47,350 .$ If we assume that the annual salaries for plumbers are normally distributed with a standard deviation of $\$ 5250,$ find the following:
a. What percentage earn below $\$ 30,000 ?$
b. What percentage earn above $\$ 63,000 ?$

Answer

a .0005; b .0014

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Video Transcript

okay for this problem, we have some more normal applications. So we've been given some information. The College Board has gone to some information about plumber salaries. So if any time you get a normal distribution, it's always good to draw normal curve to show your your sketch of what you're talking about. So they say the normals, the mean salary is $47,350. So we're gonna sketch the middle here. Money's Annapolis will have a nicer sketch here in a minute, but I like to get it going in show direction. So 47 that wasn't 350. That's our mean on them or asked for a couple different things to find. So we're trying to find the, uh if that is normally distributed. So for part A, they want us to find with the likelihood that Ace Benny Plumber will make less than $30,000. So say the problem, er makes less than $30,000. Not very high, if the mean is what it is, what we're told and then for party. Uh, we're gonna look at what's the likelihood they make a lot more money than the average or the mean. So the Probability X is greater than 63,000 probability that exes were and 63,000. So 47,000 is here 47,000 and 50. And the standard deviation was just write that down for effect. Here, the standard deviation equals two 50. That doesn't 150 believe yes, so one over there is only about 52,000. So let's go with the low end for self. So there's another 5000 down. That'd be like about 37,000. And then over here. So again, I'm just estimating a showing direction. So 30 thousands pretty low. So you're going to see a more scale that version. So what you see in red, there is the likelihood of that pretty small sliver and then for our more high paid plumber going up from 47,000 up another 5000 as much higher. So we look over here, we'll get a sketch of upper and Cook's tales sometimes and staffs last. So, um, but he's a stop lit thinking of a nicer diagram you're going to see in a minute here, So I like this free apple it online for a lot of different statistics. Uh, normal distributions is what we care about now. So what it does it does what? Your table three in your book would dio, um, getting Z scores and look at probabilities without the area under the normal curve, and you have to tape it in. So it's gonna give us a clear sketching what we you can analyze. So 47,003. 50 we're told, is a mean for her mean, um, salary for a plumber. And it goes up by about 50 for each to enter deviations. So they have a nice clean or sketch so we can see our green area. And if your teacher likes cleaner dime diagrams, you can use this to sketch out what you need over there. So let's use this apple it. So we want to find out below and the red. So we're gonna look it to the left of 30,000. So we're gonna do that. I can't leave. The area should be great small on in spring. Small. So looks like it's, uh, 0.0 ver five. The likelihood of making that little amount of money If your plumber your box, that sure answer so pretty small amount, which makes sense, Multiple standard deviations below the mean and let's look again. So we're not going to read rockets the same mean score and standard deviation when I were looking to the right of this value. So we're looking to the right. We want to know how I feel like we had of making over 63,000 later. We house No. Something wrong there, Right? Who? 63 0.14 Seems reasonable. So that's our probability, right? So, uh, less than 1.1% for that on day, we've answered both questions.