Full-time Ph.D. students receive an average of $\$ 12,837$ per year. If the average salaries are normally distributed with a standard deviation of $\$ 1500,$ find these probabilities.
a. The student makes more than $\$ 15,000$.
b. The student makes between $\$ 13,000$ and $\$ 14,000$.
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Okay, So for this problem, we're trying to find the probabilities that the student A makes more than 15,000 or be makes between 27,000. Given are different salaries. So here we're gonna set up our normal distribution curve, and we're going Tiu, find our mean in the problem is equal to 12,000 837 and our standard deviation is equal to $1500. So for problem one, we're trying to find the probability of the student makes more than $15,000. So we're trying to find this shaded area here. So what we're gonna do is be set up a see equation, which is the schools X minus the immune or the mean all over standard deviation. So, in this first problem is going to be 15,000 right here as the as our X value. So set up Z is equal to 15 opposing minus 14,837 all over 1500. So we're gonna take it to our calculator. 15 1000 minus 12,837 is equal to 2163. Uh, that answer divided by 1500 or standard deviation gives us 1.442 but were gonna round up to two decimal places. So 1.44 and we're gonna head towards the table. So we're gonna use this number here 1.44 Take it towards the table. We're gonna find the first numbers here. So 1.4 here, they were gonna go all the way out. 2.4. So it's 1.44 9251.9 to 51 So this number is basically this shaded area here, dizzy as the values ever is the probability of everything to the left of the value. So we will find the probability of the right of the value. So if you need to find the right shaded area from your ex, you're going to subtract by one or subtract from one 0.9 to 51 So take that. So one hopes. Take one minus 10.9 to 51 girls 0.749 And then finally, to wrap this problem up, we're going to multiply that by 100 in order to get a percentage in our answer is 7.4 9%. This number is this shaded area here is green shaded area Somerby. We're gonna do something a little bit different. We're trying to find the probability that it makes that the student makes between 13,000 and 14,000. So we're going to set up a normal distribution curve again. We have, um, you in the middle, and our standard deviation, which is the same, is up here. But here, we're going to be trying to find the probability that student makes between 13,000 and 14,000. So we're trying to find this shaded area here, so we're gonna call 13,000 U one I got 14,000 z2. Okay, so we're gonna do the same thing. We're gonna set up the Z, take care those the equations is the one is equal to 13,000 minus 12,000 837. All over 1500. Go back door calculator. 13,000 minus 12,837. He will the 163 take that divided by 1500 and our answer is 10,000 0.10866 Something around up. So 0.11 that's equal 2.11 ticket towards the table 0.0 point 10.1 0.11 Here 0.5438 is our dizzy value. So that is what Z one is equal to. Now we're gonna do Z two that is equal to 14,000 minus 12,837 all over 1500 calculator. I was 12,837 to give us the top the numerator 8 1163 that divide that by Take that and divided by the standard deviation 1500 the bottom to get 15000.7 eat. If we round up, we have to round up because Z table only takes two decimal places out. So it takes 0.0 point 78 So 0.7 and 0.78 point 7823 that z value it is 0.7823 So now we've found so 13,000. When we did see one, we found this area to the left of 13,014. When we did the Z value for Z two, we found the green area plus the black area. We want to subtract the black area from the green plus the black. So we're gonna take this number and this number and subtract this number from this one. So 7823 minus 0.5 for 38 0.72 3 minus 30.5438 That gives us 0.2385 Remember, Finally, we have to multiply it by 100 to get a percentage, so our answer for B should be 23.8 5%.