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%.