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As part of an interview for a research position, three applicants are asked to transfer 150 ?L of distilled water with a P-200 micropipette to a weighing paper and to determine the weight of each drop with an analytical balance. The three measurements made by each of the applicants are listed below.

Applicant A: 0.161 g, 0.147 g, 0.142 g
Applicant B: 0.158 g, 0.156 g, 0.157 g
Applicant C: 0.143 g, 0.153 g, 0.150 g

Q1. Calculate and type in the mean and standard deviation of the measurements by each applicant.
Q2. Calculate applicants' accuracy of micropipetting, and rank the applicants by accuracy.
(A (accuracy) = 100 x Vavg/V0, where A is the accuracy of the pipette, Vavg is the average calculated volume and V0 is the value you set the pipette to dispense. Accuracy should be between 99-101%)
Q3. Rank the applicants by their precision of micropipetting.
Q4 .Select one of the applicants based on 2. and 3., and explain your selection.



Okay, so I actually learned something new today. Apparently, there's a formula that kind of tells you when a fledgling bird is going to be able to fly on its own. And it's the ratio of two functions of time, one of which kind of gives an indication for how long their wings are and the other one is their body mass. Okay, so whenever these ratios kind of approach one, whenever FFT approaches one thing, the fledging fledgling is able to fly on its own. You didn't know all this question is asking us to do is to interpret the physical meaning behind those and described the units associated with them. Okay, so the 1st 2 shouldn't be too difficult, right? Because M prime of tear is the time derivative. It's the time derivative of em of tea. It's a rate of change. It's how fast this function changes with respect to time. And since we're given that the average body mass is measured in grams, this is gonna be essentially grams per per unit time and that the time is in weeks, by the way, since grams per weeks and just like I said, ah, the interpretation of physical meaning is how fast the body mass is changing with respect to time. It's the rate of change of body mats. Okay, for w prime of tea, that's again the time derivative of W T. Which is the length of the wings. And since they said that wing length was gonna be measured in millimeters, this rate of change is millimeters per week. And again, it's just how fast the length of the ones you're changing F prime of tea has to have special analysis because, um, f prime of TIA is the time derivative of f of tear. But f of tia is the ratio. Oh, the length of the wings and the average body mass. These air two functions that change. So we actually do have to use the quotient rule, which is lo de I minus high. Do you low swearing the bottom and we're gonna determine the units of this function As far as a physical meeting goes out, I said half of Tia as FFT approaches one, then the fledgling is gonna be more able to fly. This is a rate of change of that. So if this is a really, really positive number, then it's going to rapidly go towards, and it's gonna take a lot less time for to be able to fly. Maybe it's some sort of growth spurt. So all that remains to do is to find the units of this I'm of Tia. Ah was measured in grams w prime of tea we said was millimeters per week. Okay, W of tea waas um millimeters and M prime of tea is grams per week, divided by AM of T squared M of T is measured in grants. This is just grams squared. Okay, so we have a grams times of millimeters minus of grams, times of millimeters, all divided by a week. Okay, so this is gonna be some sort of of grams times millimeters. We don't know. Of course you know how exactly these variables are changing, so they're obviously not gonna be the same necessarily. So when you subtract two things that are like each other, you're gonna get something else that's like each but that's like those two things. So this is gonna be some grams millimeters divided by weeks, all divided by Graham Sward, which is gonna be grams times, millimeters times, weeks times one over. Graham squared. One of these grams cancels with one of those, and then we're left with millimeters Her grands week. Here we go.

So in this question of firework was launched at some height? Right? So the graph of the projectile is given. And the equation of the projectile would be why it goes to minus five X square plus 20 X plus one here. Why is the height of the firework from the ground and access to time? Right. So we have four parts of question here. The first part is saying that uh what is the highest point that the firework would be the would reach? Right? So to find the highest point we have to find divide by the X. And we have to equate this to zero because at the highest point that slope of the cover will be whole gentle to the X axis. Right? So David by the access across digital, which means that minus 10 X plus 20 is equal to zero. So from here we get access equals to two seconds. So after two seconds the the firework will reach the maximum point right now to find the height of the maximum point, we can find the value of Y had X equals to two seconds. Light soil substitute X equals to do in the aggression. We will get minus five to his square plus 20 multiplied by two plus one. So when we simplify this will get why he goes to 21 fits. So this is the highest point that the firework will reach. Now coming to the second part, the second part is how long the firework was in the air. Right? So uh at the end the fireworks will be reaching the goal of reaching the ground. Right? So at that point the height of the firework would be judo, which means why we liquid 20 Right? So we have to equate why equals agenda. Which means that minus five X squared plus 20 X plus one is equal to zero. So some the situation will get the values of X. Right? So it is a quadratic equation. We can find the roots of the equation by minus 20 plus minus the square root of 20 square mm minus fall into minus five. Divided by to into minus five. Yeah. Right. So from here we'll get the values of X one would be 4.3 to 5 seconds, 34.3 or 49 to be more like you did. Mhm. And another value would be X equals two minus zero point Beautiful night. Since this cannot be possible, time cannot be negative. So the correct value is 4.249 So after 4.49 seconds the firework will reach around. Right? Mhm. Now coming to the sea but ah we have to find the time we've just taken to reach the highest point. Right? So two at the highest point, the slope of the cover studio that we have already found in the first part of the question. So by creating slow because we have God the value of access two seconds as in the first part. Right? So after two seconds the firework will reach the highest point. Now coming to the depart, uh we have to find the height of the firework at the initial mind where it is launched. Right? So in this uh this equation we have to find the value of Y. A. Toxic was which means that a time it was zero seconds. That is the initial point. We have to find the height of the firework. So we can we can substitute judo forex in the equation. So we'll get y equals two minus five geo plus 20 multiplied by zero plus one. So this will give the value of my years one ft late. So at uh height of one ft, the vibe work was launched, Right? So this uh this is a solution for the given question.

So a reminder about how you get the electric field given a potential or potential function as the case may be, The electric field is defined to be the gradient of the potential with a minus sign in front of it, reminder of what a gradient is a gradient is in Cartesian coordinates will write that out. Uh The partial of some function with respect to X. For an X. Component. The same for the why and the same for the Z. Now we can simplify this. If we know that we just have one component, we can say something like E C. For example, the Z component of the electric field is minus the derivative of the potential with respect to the spatial coordinates. Z. So as an example of this, we have a potential function and it has shown in the graph, there is a portion that is quadratic and then parts that are linear. And just knowing that the electric field is the slope, the minus sign in front of it minus the slope of potential. It's probably good to go ahead and find the slope of the linear portions. And we see that in this case it's dropping on one side, in the positive sea region. It is going down with a slope of minus 10 volts per meter. And on the other side it is increasing with a positive slope of 10 volts per meter. And so if we demarcate our regions as one on the left, two in the middle and three on the right, We can quickly figure out in region one that easy is minus 10 volts per meter and we'll go ahead and figure out region too. Um but there we do need a derivative easy is minus um The derivative of minus five Z squared would be minus tendency. And that would mean that the uh constant there has to have units of volts per meter squared. Um So yeah, that gets a little tricky, but we'll assume that it has overall units of volts per meter. Yeah, but yeah, that constant has to have Some interesting units. And then in three our electric field in the Z direction is equal to a positive 10 volts per meter. Okay, now we are given a clue that this is a slab. And if you want to think about the way slabs or plates work, electric field is uniform outside a slab or a plate. The only difference is a plate is usually a conductor with a surface charge. Um whereas a slab has a finite thickness and some sort of perhaps volume charge density. So we can see we have the makings of a slab where our electric field is uniform and this is a slab that's we're going to pretend this infinite. So we'll pretend it has no edges, no real edges. It goes on forever. Okay, that's hard to draw. Hard to draw infinity. Anyway, let's make a Z axis. This is a slab and RC access points up and usually they like to center a slab With its equals to zero and going upwards from there. Yeah. Um but we see that what the electric field would look like is above the slab. It would be uniform and positive. This is Region one. And I should have used green arrows on my pardon? My bad. Um Yeah, we'll go ahead and use green arrows for the region one. Actually this is region three. I beg your pardon? It's above the slab with Z greater than we'll say greater than one. There's a where is it equal to one? Probably right at the border of the slab. But here we see that the slab extends from one up above two minus one below. Okay, and the electric field and goes uniform with the same strength down below. Um Now what does the electric field do in between Is it starts at zero at the origin and it grows proportionately as you get away from the origin. And to understand that we probably want to look at Galaxies law. So the second piece of the puzzle, his gases law, a reminder of how gases all works is it says that the electric flux through a closed surface is proportional to the enclosed charge. Now you almost never do the integral on the left and seldom even anything having to do with the integral on the right. Um For a slab we expect the electric field through a surface perpendicular to the electric field to remain constant. And let me kind of blow up the slab and show such a surface. So let's let's take a microscope magnifying glass and look inside the slab. Um What we basically have Is this the positions equal zero Where the electric field is zero. And I am going to draw a little calcium surface. So I'll draw it in blue. And this is an imaginary surface which uh is typically used to try to imagine an enclosed charge. Um The electric field. Uh huh. He was red again inside that slab. So the electric field should point perpendicular to that surface and there should be nothing going into the bottom bottom has zero electric field at that point. Um And so what we can say is the electric field Z component times the area of that surface has to equal the enclosed charge inside of that little surface. Um And if we are assured that there is a uniform density, uh sorry, I forgot to include the absolute zero. We'll do that. But if we are assuming that there is a constant uniform density, we can say that the enclosed charge is the density times the volume of that little surface. And yes, we do have to divide by absolutely not. But the volume of that little surface, it's a cylinder with area A. For the face, which is pi r squared and then Z for the length. Okay, so um the electric field then times the area is equal to row a. See And and don't forget the absolute or not. Um and fortunately that arbitrary area cancels, but the Z is not arbitrary. What this tells us is that the electric field inside the slab is a function of Z. And the constant in front of Z is rho times epsilon? Not. Yes. Now, up here we found that the constant was 10. After doing the derivative, we found that that constant was equal to 10. So by looking and compare to our region to, we found E C. Was 10 times see with 10 having units. Yes, it did have units. We won't worry too much, but that tells us that we must conclude that row over epsilon Is equal to 10. Yeah. And we could put in numbers absolute knots. Just a reminder is 8.85 times 10 to the -12. And it has some units Coolum squared over newtons metre squared. So yes, it does have some units. And we can see that Columns will come in there, that the 10 is not unit list either. So as long as we're using things in S. I. Units, the rose should have units of columns per cubic meter. Yeah,

This question is asking us to calculate the mean and standard deviation for each of these students and to relate this to precision and accuracy. So our formula for me is each of our sample values divided by the number of samples we conducted. Standard deviation. The formula for this is the square root of this, sum of each of our values minus the mean, that value squared for each of these, and then you divide it by the number of samples minus one. This question, it's also important to note that the density of water is one g per mil leader, So if you draw saying 150 0.150 mL of water, this is equivalent to drawing 0.150 g of water. So moving on to the calculations for each student, the participant A had values of 0.161 0.147 and 0.142 participant B. Had values of 0.158 0.156 and 0.157 participant C. Had values of 0.143 0.153 and 0.150 We can add each of these up to get the total and divide it by three because we have three samples for each participant to get our mean value. So here we have 0.161 plus 0.147 plus your 0.142 which equals 0.450 We can divide 0.450 by our three samples. And you get a mean of 0.150 g per participant B. We do the same. We add up our values get our total value which here equals 0.471 and 0.471 divided by our sample amount. And we get a mean of 0.157 g here for C. Again, same process, we get a total of 0.446 and divided by three. You get a mean of 0.149 About this is rounded from there we can calculate our standard deviations so far participant A. We have the square root of our first value is your 0.161 minus the mean of 0.150 squared. You add it to r 0.147 minus 0.150 squared and 0.14 to minus 0.15 also squared. And we divide this by end minus one. So we divided by two. When you calculate this out, you get a standard deviation value of zero 0.98 On average we do the same for participant B. Here we have 0.158 and I'll just abbreviate this 0.156 0.157 And we subtract 0.157 which was the mean from each of these values divided by two. And our standard deviation value for participant B is 0.1 We do the same for participants. See I'll abbreviated again for three to a 0.153 and 0.150 subtract the mean of 0.149 divided by two. Remember each of these values needs to be squared after subtracting the mean and you should get a standard deviation of 0.54 Standard deviation is just a basic calculation of the amount the participant is off from the mean, from each of their values. So, using this information. All right, this is an abbreviated format. We have our mean and standard deviation for each of our participants, participant A If you remember the mean was 0.150 Under standard deviation of 0.0 98 participant B 0.14 sorry 0.157 and zero and zero point 001 Lastly for participants, see army, see army was zero point 149 after rounding and 0.54 Now, in order to calculate our accuracy, what they're telling us to do is our mean value divided by our volume that we were supposed to draw and we multiply this by 100%. So here for participant A we have 0.150 divided by 0.150 times 100% equals 100% participant B. We have 0.157 divided by 0.150 And they have an accuracy value of 104.7. No the max accuracy that you want is 100%. Even if you are over 100% that is not better for participants. See we have 0.149 Fight it by 0.150 and we get an accuracy value of 99.3%. Finally they want us to evaluate the precision of each of our participants. And precision is how close their values are each time. So oftentimes when relating precision and accuracy, accuracy is how close you get to the center of the target, whereas precision is how close you get each attempt. So this would be high precision here and this would be high accuracy here. So for student A. Our accuracy was 100% in our standard deviation was 0.98 For participant be accuracy of one oh 4.7 precision of 0.1 participant C accuracy of 99.3% and precision of 0.54 If we consider we want our accuracy to be between 99 and 101% we can eliminate participant be for accuracy, so since we can eliminate participant B, we can pick our best participant based on their standard deviation and precision students see has the better standard deviation. Each of their values varies less with each attempt to draw water into the pipe. It so our best choice for hiring based on precision and accuracy students see.


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