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The length of the thorax in a population of male fruit flies isapproximately normal with mean 0.7 millimeter (mm) and a standarddeviation of 0.03 mm.(a)Approximatel...

Question

The length of the thorax in a population of male fruit flies isapproximately normal with mean 0.7 millimeter (mm) and a standarddeviation of 0.03 mm.(a)Approximately, what proportion of male fruit flies havethorax length between 0.73 and 0.79mm?(b)What is the 75th percentile of thorax lengths among malefruit flies? (Use R or table) (c)One male fruit fly was randomly selected from this populationand the length of its thorax was measured as 0.66mm. Suppose thepopulation size is provided as 10000.

The length of the thorax in a population of male fruit flies is approximately normal with mean 0.7 millimeter (mm) and a standard deviation of 0.03 mm. (a)Approximately, what proportion of male fruit flies have thorax length between 0.73 and 0.79mm? (b)What is the 75th percentile of thorax lengths among male fruit flies? (Use R or table) (c)One male fruit fly was randomly selected from this population and the length of its thorax was measured as 0.66mm. Suppose the population size is provided as 10000. How many fruit flies have thorax length less than this chosen fruit fly?



Answers

In a population of Drosophila melanogaster reared in the laboratory, the mean wing length is $0.55 \mathrm{mm}$ and the range is 0.35 to $0.65 .$ A geneticist selects a female with wings that are $0.42 \mathrm{mm}$ in length and mates her with a male that has wings that are $0.56 \mathrm{mm}$ in length. a. What is the expected wing length of their offspring if wing length has a narrow-sense heritability of $1.0 ?$ b. What is the expected wing length of their offspring if wing length has a narrow-sense heritability of $0.0 ?$

I. The life span of this particular species of fruit flies has a bell shape so we can draw the bell. It's got an average of 33 days, and that goes in the center of your bell, and there's a standard deviation of four days. So therefore, if we add 4 to 33 we'd be at 37. And if we add four to that, we'd be at 41. If we subtract four from 33 we'd be at 29. And if we subtract four from that would be at 25. Now. If I wanted to transition this into Z scores, I could use a formula. But I also can use the concept that whatever the averages transitions to a zero Z score, and if you add the standard deviation, then that would be a Z score of one. So we're one standard deviation above the average and then 41 would be two standard deviations above the average 29 would be negative. One standard deviation or one standard deviation below the average and 25 would be at negative two. So for part A, it's asking you to calculate the Z scores for 34. So it's saying X is 34 and then it's asking you to find thes e score associated with X being 30 and then finally, party is asking you to find the Z score for X being 42. So let's look at the picture First 34 would be somewhere right here. So we should expect an answer somewhere bigger than zero, but smaller than one. And the formula that you're going to use is that Z equals X minus mu over sigma. So for us to find the Z score associated with 34 we're going to take 34 minus the average. Which was that 33 we put in the center and divided by the standard deviation, which was four. And in doing so, we get an answer of 0.25 And that makes sense, because again, we were expecting it to be slightly bigger than zero. And sure enough, that's slightly bigger than zero. Now let's think about where 30 is. 30 is right around here. So the answer we get to this should be smaller than zero, but bigger than negative one. So we're going to use our formula so we're gonna take 30 minus 33 and divided by four, and we get negative 0.75 And again, we already expected something close to negative one, but not quite there. And then if you think about where 42 is, 42 is right up in this area, so we're going to expect a number bigger than two. And when we use our formula 40 to minus 33 all over four, you're going to get a Z score of 2.25 Now, this question also asked determine whether any of thes lifespans are unusual and we could say yes. It is unusual for a lifespan to last 42 days, and the reason we could say that is because it's Z score is more than two standard deviations away from the mean. So it wouldn't have mattered if we were two standard deviations above or below the mean anything outside of that region is going to be classified as unusual. Let's look at part B. Part B wants us to do the same thing, find the Z scores, but then we need to use the empirical rule to find the percentile. So let's refresh our memory on what the empirical rule says Mhm. So the empirical rule states that as long as you're data is normally distributed. And of course, we're going to put, um, you in the center. Then you can count on 68% of the data falling within one standard deviation of the mean. So from here to here would be 68% of the data. The next part of the empirical rule is that if you are two standard deviations out So, um, you plus two Sigma and Mu minus two sigma. Then you can count on 95% of the data to fall between two standard deviations below and two standard deviations above the mean. And if we were to go three standard deviations out. So right here is gonna be mu plus three standard deviations and back here is going to be mu minus three standard deviations than from here. Thio here is going to account for 99.7% of the data now, because of the symmetric nature of this curve, since this is 68% we could break each of these regions into 34% or 340.34 of the curve. And if you take that 95% and take away the 68% you will be left with 27% of the curve still left. So therefore from here and here is going to be 27% of the curve. So if we divide that by two, then it puts 13.5% in here and 13.5% in there. And then if we take the 99 7 and we subtract out the 95 we will, we will be left with 4.7% of the curve. So 4.7% of the curve is in here and in here combined. So if I divide that by two, I end up with 0.235 in each of these sections. And keep in mind that the entire curve represents 100% mhm. So now that we have discussed what the empirical rule says, let's answer the rest of part B. So you try to keep this picture visible. And for part B, you had to start with X equals 29. So if I transition that Intuit Z score keeping in mind that the average was 33 and these Sorry, try again. Let's do the correct symbol. The average was 33 and the standard deviation was four. And the formula for Z score is X minus mu over Sigma. Then we'd have 29 minus 33 divided by four, which would be equivalent to a Z score of negative one. So if I were to look at that bell and here would be the Z score of negative one, I'm trying to find what's the area of the curve in here? Well, if I think about the fact that the entire curve is 100 and I take away this part of the curve, which represents half of the curve, so I take away 50% and then I take away this part of the curve, which was 34%. I am going to be left with 16% of the curve shaded in. So therefore, the percentile associated with an ex score of 29 or a lifespan of 29 days would be the 16th percentile. Let's look at Part B or the second part of part P. In this case, X is 41 days. So let's calculate it Z score. So we would do 41 minus 33 divided by four and we will get a Z score of two. So if you think in terms of the picture, the average is in the center. A Z score of two would be out here and we are trying to find this shaded region. So if we were to think in terms of the values from the empirical rule, we'd have 50% in here, we'd have 34% in here and 13.5% in that region. So therefore the percentile can be found by adding 50% plus 34% plus 13.5% and we will get a percentile of 97.5 percent. And then finally, the third part of Part B, we want to talk about X being 25 so as a Z score that would be 25 minus 33/4, which gives us a negative too. If we think in terms of the picture, here's theatric. Here's one standard deviation below, so here would be Z equals negative too. And we are trying to find this shaded area right here. So the entire bell is 100. We're gonna take away this section, which is 50% of the curve. We're going to take away this section, which is 34% of the curve, and we're going to take away this section, which is 13.5% of the curve, and in doing so, you will be left with 2.5% so therefore the percentile associated with a life span of 25 days would be 2.5%.

Okay, were continuing in section 6.5. So let's look at the problem we have in front of us. We've got one with lager of them, so Ah, problem number 63. So number 63 number 63 in section 6.5. Okay, let's give us an expression here that deals with the population of fruit flies. But they give us an equation. That is the log of why is equal to, um what is at one point when we get 1.54 1.54 minus 0.8 X and then minus 0.658 times the log of X, and they want us to transform in this into an expression for why so basically we're asked to solve for why so? Easiest way to solve a long equation like this is to get both of the log expressions on one side. So I could write this as the log of why. Plus, and let's just write this as the log of X rays to the point 658 power. Okay, that's using the power rule. And this is equal to 1.54 minus 0.8 x. Now I can combine the issues in the product property of lager of them. So the log of why x to the 0.658 is equal to 1.54 minus 0.8 x. So anytime I get an equation where I had the log equal to some number. Now I can convert this to exponential, So convert this an exponential equation. You have to remember, this is log base 10. Okay, so that means that 10 raised to the 1.54 minus 0.8 axe is equal to why x to the 0.6 five eight. So this is the question that I am solving for why so 10 to the 1.54 s. So what is that? So 10 raised to the 1.54 is 34.67 Okay, so are 36.34 point seven. So this gives me 34 0.7. Okay. And then I've got this is, um, times 10 to the minus 100.8 x is equal to why extra the 0.658 So simple. My what is 10 to the negative point. 08 So 10 raise to the point. 08 is gonna be, what, 1.186 So continuing over here I'm gonna get 34.7 times one point. And that waas one point 1861.0, 18 six Raised to the negative X power Good. And then that equals why X to the 0.658 So to solve this for why you get wise equal to 34.7 times 1.186 raised to the minus x times X to the minus 0.65 eight. Okay, so this is solving that equation for why, in terms of X now we're asked to do a couple things there were asked to take that equation that we just solved. Why is equal to 34 0.7 1.18 six to the negative x times X to the negative 0.65 eight and were asked to evaluate that What happens when X is 20 and what happens when X is equal to 40? Okay, so evaluating that expression Ah, for, um, two different values. So what would happen if I evaluate this at X equal to 20. Okay, so that is 34 point seven times 1.186 Raised to the minus 20 times 20 raised to the minus 0.65 eight equal 3.343323 point three for three. So when x equal 20 we know that why is equal to 3.34332 when X is equal to 40. Then what? Well, why be equal to I could just go back to my calculator. And everywhere I see a 20 let's just change that to a 40 1.46561 point 4656 Okay, so that is evaluating why at 20. And why at 40 now, the next thing were asked to do is to find why prime so fine, Why prime and evaluate that when X is 20 and excess 40. So what was why again? Why is equal to 34.7 times one 0.186 to the negative X and then X to the negative 0.65 eight. So what is why prime? Why prime is I've got a constant out front. So let's just simplify and keep that constant out front. And then I have to use the product rule. What's the derivative of one point? 0186 to the negative X. So the derivative of that first expression, Um so that is going to be negative. 1.1 86 to the negative. X times the natural log of one point oak 18 six. Okay, that's the drill. Bit of the Sievers's expression times a second expression, plus the first expression times the derivative of the second expression. So that first expression is, what one point 0186 to the negative. X The derivative of that second expression is negative. 0.658 times X to the negative 0.658 minus one. All of that times 34.7. And so let's just see this. This exponents right here it's gonna be minus point, I think 342 So I can factor out a little bit. Why? Prime is 34 0.7 and then I can factor out Um, I think I'm gonna leave. Uh um, if I factor out, eh? 1.1 86 to the minus X. Then that leaves me with a negative natural log. One point no. 186 extra The minus 0.6 58 plus negative 0.658 extra, The minus 0.342 And now I just need to evaluate this when X is equal to 40. Excuse me, 20 when X is equal to 40 I know that. Why prime find the value of y prime in this expression at 20 in at 40. So I will go to my calculator and try to do this very carefully. So let's just see 34 0.7 times 1.186 raise to the negative 20. And then all of that is times negative. Okay, Natural log, 1.186 times. Um X, which is 2020. Raise to the negative 20200.658 minus 0.658 times 20. Raise to the negative. 0.34 two. And then I will end that parentheses. Um, and there we go. So I get negative. 0.573 So negative. Excuse me. I was negative. 5.73 So negative. 5.73 It's what happened when I evaluate that derivative at 20. Let's go back and evaluate that derivative at 40. So I'm just gonna recall and hope I did everything right. So, um, instead of a 20 I'm gonna have a 40 there. So that is the first expression. And if I go backwards, uh, let me just go back and do it quick here. Um, everywhere I see a 20 I'm going to replace that with a 40. So that goes a 40 here and over here as well. 40 negative, 3.12 So at 40 I get negative. 3.12 Has the value the derivative, Okay.

So we're looking at having five male and five females. Mhm. For a total of 10 of these Dressel phileas. And we want to know based on time in days. If we start with 10 and they're going to double the population every 2.4 days. We want to find what happens after a week. So seven days. And so it will double seven divided by 2.4 times. So we start with 10 and Mhm. Two to the power of seven divided by 2.4. What ends up there 75 0.5. So roughly 76 of these bugs as opposed to if we go for two weeks, which will be 14 days. Now we're going to have a lot more doubling this quotient is the number of times we're going to have double. So 10 times two to the power of 14 divided by 2.4. And we're going to end up with 570 one more week. So look almost 500 more. Yeah.

Okay, So for this problem will still use the data for doing off mail 68.6 and the standard of imported two point. So, according to you, the problem or part age Hey, so it has adultery. Height. That's equal to 51.6. And we're looking for the percentage that adult men can fit through that high 15.6 without even bending. Okay, so we're actually looking. So since this is 51.6, so we're looking for the area less than 51.6. It's a probability that access less than 51.6. Okay, but we have to standardize this to get the value of Z. Okay, So disease he called to explain this new all over sigma is equal. Thio 51.6, my nurse 68.6 over 2.8. Okay, so calculate the Z score for this one. Hmm. Yeah. Oh, okay. So you must get a value off Z. Mhm. That's equal to negative. Uh, yeah. Oh, active 6.7 Uh, okay. Right. Mhm. So now we're looking for the probability that disease less than negative 6.7 and since the standard normal Kerr happens to be from, uh, negatively three. Then the probabilities is actually equal to zero. Hey there for no men can pass through that door without them. So they have to bend first in order to bash through it. Uh huh. So So, yeah, the door design is actually adequate. So this is adequate. Yeah. The 51.6 is interest. It's actually add it because it will consume a lot off space in the job if if the door high iss, you know, higher. Okay. So, yeah, it's just good to have. It's just good to have a door height that a minimal Yeah, in order to avoid, uh, large space for just or, um And then now we're looking for the value of height that could fit 40% without even then Mhm, you know. Oh, let's just call this down. All right? So we're looking for mhm the value of X. Uh huh. We're in 40% fete without many, so this is 40 percent. So we're looking for the value off X. Less than a certain value. Give us an area equal to zero point 40. Okay. Oh, are in terms of the we have the lessons. Z score, Give us an area off 0.4 or 0.40 me. So you look at the Z table. Look at the corresponding Z score given area. That's all too. 0.4. Uh huh. And that Z score is actually negative. 0.25 Yeah. Okay, So in order to get the X value here, Okay, use the formula. Z equals x minus were over sigma. So we have Z. That's still at a 0.25. Okay, is equal to x minus the mean 60.6 or over two point. So multiply 2.8 on both sides of the equation. We have negative 2.5 negative 0.25 times. 2.8 This actually know that there was zero point, so it's equal to X minus 68 0.6. So we add 68.6 on both sides off the creation. Okay. Lets scare the value that's equal to 67 point line. INGE's Hey,


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