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5. Recently; researchers analyzed the 700 most popular songs since 2012 (as ranked Onl Billboard s Hot 100) , and found that a small group of just 10 songwriters wa...

Question

5. Recently; researchers analyzed the 700 most popular songs since 2012 (as ranked Onl Billboard s Hot 100) , and found that a small group of just 10 songwriters was responsible for 23% of these songs Suppose you sample n = 35 songs randomly; with replacement, fron the 700 most popular songs_ What is the exact probability that YOur 35 songs will include at least 11 songs from the small group of songwriters? What is the approximate probability; using the CLT (with continuity correction)?

5. Recently; researchers analyzed the 700 most popular songs since 2012 (as ranked Onl Billboard s Hot 100) , and found that a small group of just 10 songwriters was responsible for 23% of these songs Suppose you sample n = 35 songs randomly; with replacement, fron the 700 most popular songs_ What is the exact probability that YOur 35 songs will include at least 11 songs from the small group of songwriters? What is the approximate probability; using the CLT (with continuity correction)?



Answers

Jukebox You look over the songs on a jukebox and determine that you like 15 of the 56 songs. (a) What is the probability that you like the next three songs that are played? (Assume a song cannot be repeated.) (b) What is the probability that you do not like the next three songs that are played? (Assume a song cannot be repeated.)

So in getting the probability that E can occur, we have two things needed. It's the number of ways went Beacon her. Okay, so first you are to get the probability that the random song plays a rap song. Hey, now the number of ways that we can get a rap song or that songs play so the lumber wraps holes in our set. So we have seven rap songs in our set so that this seven over the total possibilities that this just equal to the total number songs we have in the said that this 10 with seven was the which is equal to 20. Therefore, the probability that the random song please it's a rap song is equal. 7/20. No, we are to get the probability you the random some place on not country saw. Hey, so not country is the complement off country. So recall the probability off a compliment off season. So the problem e that the song with the random song places, not a country. It's just equal to the compliment getting a country song so that this country ways to see so it is denoted by this one a equals to one minus the probability the random someplace a country saw So that is equal to one minus the probability random someplace country song way have three country songs set, so we have three over the total possible peace, which is also off the training. Therefore, we have one minus the overtime or just equal to 17 over train.

Concert has three pieces of music to be played before the intervention. We're told the time taken to play. Each piece has a normal distribution and we're told to assume that three times are independent of each other. Were given the mean times are 15 30 20 minutes, respectively. And the standard deviations are 12 and 1.5 minutes and were asked to find the probability that this part of the concert takes at most one hour. Well, if we treat X one as the time taken to play the first piece, X two is time taking the second piece. Next three is the time taking to play the third piece. Then we have that. It's part of the concert. The time it takes is going to be given by x one plus x two plus x three. So we're asking for probability that this part of the concert takes at most one hour. We want to find yeah probability distribution to do this, want to find expected value So they expected value of excellent 6263 is some of the expected values So expected value of X one, plus expected value of x two plus expected value of x three were given that these air 15 30 20 minutes and so this sums up to 65 minutes. We also want to find the variance of X one plus x two plus 63 This is going to be the variance of X one, plus the variance next to plus the variance of x three and then because the times are independent, follows that the co variances for all these times or zero. So this is what the variance is equal to, and we're given that thes variances well. The standard deviations are 12 and 1.5. This is equal to one squared plus two squared plus 1.5 squared, and this comes out to be 7.25 minutes squared. And therefore the standard deviation of X one plus x two, plus x three is the square root of 7.25 which is approximately 2.69 to 6 minutes now. With the mean and standard deviation, you have the probability well, once again because each piece at a time which has a normal distribution, follows that x one plus x two plus x three has a normal distribution as well, with the above mean and standard deviation. And so the probability that X one plus X two plus x three probability that this is less than or equal to an hour, which is 60 minutes. This is the same as the probability that the standard variable Z is less than or equal to 60 minus the mean tree determined was 65 over the standard deviation, which is approximately 2.69 to 6. And this could be simplified to approximately probability. Let's see, is less than or equal to negative 1.86 And using the table computer, you find that this is approximately 10314 were asked if there are reasons to question the independence assumption from this exercise. So this is less a question about numbers, and it is about our interpretation. And I would say that yes, I would say there is reason to question Independence assumption or the longer do you first peace takes to play, um, or tired. The musicians are when they play. The 2nd and 3rd pieces, therefore may take longer to finish these other pieces

Okay, so let's define it X as the number of defective of defectives, and that's in the batch of 2 50 So X is approximately equal to the binomial 2 50 comma 0.5 and we can approximate X by a normal distribution. So since NP is equal to 12.5, which is greater than or equal to 10 this case it's greater than, um, the mean and standard deviation of X is the following. So the mean is 12.5 and the standard deviation is three point 446 If we consider the fact that 10% of 2 50 is 25 we could now fear out this probability probability of X being greater than or equal to 25 is equal to one, minus the probability of ex being less than 25 which is the same thing as one minus the probability of X being less than or equal to 24.5. And this is approximately equivalent to the following. And now we could simplify this even further to get the following and our final answer would be one minus nine 997 which is equal 2.3 and, um, the exact, uh, by no mo binomial probability from the software given in the textbook is 0.86 So either one of these works again. These are very small numbers, so this might seem like it's a big difference, but they're they're very, very close to one another. So this is a and now for B, um, we're using the same normal approximation with the continuity correction. So we have the probability of X equal to 10 the same thing as a probability being between 9.5 reclusive and 10.5 inclusive. Okay. And this is approximately equal to the following, just plugging it into our standard formula. And then we subtract the following. This is how we find the distribution it's actually move this to another line. Can be could add 9.5 minus 12.5 over 3.44 six. And this is all equivalent to 0.2 810 minus 2.192 to, and that is equivalent to 0.88 Just wanna put it out there of a software that the textbook uses gives you an answer of 0.963 This is how we do this problem, and an answer in this general ballpark would be correct.

So we're given that a random sample. 15 baseball players are chosen to be tested for drugs. You're also giving the 750. It's just there to see if you can actually read the problem and not just pull out all the numbers and stick them in. But the 15 is the more relevant number because that's our sample. Now every question is the same. What's the probability that one or more tests positive for drugs? The only difference is the probability of people using drugs. Now we have a random sample. 15. So if we want to do this the normal way, you would have to take the probability of 12345 all the way up the 15 and add all them up. But all probabilities add upto one. So instead of doing that, we can just do one minus all over the probabilities that we aren't looking for. So one or more. Instead of doing that, we can just do one minus less than wine. So for all three of these problems, we're going to do one minus probability of zero because that is equal to one or more. That is the same pain. So the only thing we have to change each time is our probability. So for here, probability So one minus probability of zero is equal to one minus so and choose acts off and is our sample. It's 15 x zero. That's what we're looking for. The probability is 00.5 to the X Power times one minus 10.5 to the 15 minus zero Power. So this part over here, that is 0.46 free. So one minus. That is point by 37 You'll do the same thing for B except you has changed the 0.52 point wine. Remember, we can't write these numbers as a percentage. We need to convert them to a decibel. So that's why I put 0.5 instead of 5%. Same thing here. Just change this to a point wine. So one minus probability of zero is equal to one minus 0.206 which gives us 0.79 or and finally just changed the point. Wine to now a point to so one minus probability of zero is equal to one minus 10.35 which is 0.965 and we are done


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