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Numerade Educator



Problem 2 Medium Difficulty

The average annual salary for all U.S. teachers is $\$ 47,750$. Assume that the distribution is normal and the standard deviation is $\$ 5680 .$ Find the probability that a randomly selected teacher earns
a. Between $\$ 35,000$ and $\$ 45,000$ a year
b. More than $\$ 40,000$ a year
c. If you were applying for a teaching position and were offered $\$ 31,000$ a year, how would you feel (based on this information)?


a. 0.3031
b. -1.36
c. see explanation


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Gretchen G.

October 3, 2020

Thank you so much! these videos help me a ton

Video Transcript

were asked to find the probability that a randomly selected teacher turns between $80,000 so they can express. That is the probability that teacher and 35 spend 25,000 dollars a year. And we're given that the average annual salary. Uh, you, too. 47 dozen. So 50 others. That is just delicious trends that we can use from season two. Thanks. This you signal were also given signal or the standard deviation is equal to it doesn't 686 today we're gonna attempt this problem is to find the Z value spertus for these separate conditions. 5000. Perfect. So let's set this one C one this on C two Dan, we're gonna get is Z one is equal to 35 1000. Nice. You? Which is? I have 7 7050 over signal. That doesn't 600 a. I mean, this value is equal to negative 2.24 Now again to the same thing for easy to see to Jersey. Going to 45 dozen minus said 50 to first 1006 100 18. Go Mean that that is in a cool negative zero four eight. So now that we have this, we can rewrite this original expression as from Billy the negative, too. 24 Just Z value for that doesn't is less than Jeanne just less than negatives. You for eight. Now, what we're gonna do is check out the Z table, find the corresponding probabilities of these values for negative 2 to 4. Get your print you to five for natives, use for bait. Digging is your point three, but six. So now we condemn view it. This as follows as the probability that Z is less than negatives your point for and minus the probability that C is less than neck It is, too to four. We know that this is the same thing. Is this so you can just put that and share substitute values in 05 six minus You're sure 12 Which gives us ju 2.3. You're to me. What? That's fine. Apart. Be slightly different. It's asking us probability that teacher earns more than $40,000 a year so you can write. That is the probability that eggs is greater done. 40 1000 dollars. So to solve this, we're just going to use the same from love. Just Z is equal to extra for signal. We know that mule and sing Listen Same useful 50 signal respect in 680. So if you just go ahead substitute 40,000 50 uh, 6 5080 80. They're gonna get value off negative one point 36 However, I usually use this formula distance to be less than inside. And since it's not have toe do one minus, we're gonna have to do probability of said this crater 1.36 is equal to one, minus the probability probability. I see it's less than negative one from 36 That is the equivalent statement. And it's it's useful for us to write it like this because, okay, we can find this value right here using this formula and just one minus to get greater than questions. So now if you find the corresponding value for negative 36 from our see table, we're going to get zero a six. But so done. Iffy. Substitute back into here. Get one minus is your hey six, which is the corresponding probability to this. See value on that gives us a final answer. Serial 0.9 one one for Part C. The question is asking us that if you were offered there you $1000 a year for this position, how would you feel basic information? And so there's no there's no real calculation. You need to do this for this part, but you just need to look at your parts A and B on used logical thinking to terminate. Answer. So here we can tell that the probability that a teacher earns more than $40,000 a year is open 9131 which is the same thing as 91 0.3 1%. No 1.1% of teachers more than $40,000 a year. Therefore, if you were offered doing so 31 dozen dollars, you wouldn't be too happy because most teachers would be making a lot more than you