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The probability is (Round to four decimal places as needed )(c) What are the parameters and of the beta distribution illustrated here?(Type an integer Or decimal: D...

Question

The probability is (Round to four decimal places as needed )(c) What are the parameters and of the beta distribution illustrated here?(Type an integer Or decimal: Do not round )(Type an integer Or decimal: Do not round )(d) The mean of the beta distributionWhat is the mean proportion of impurities the batch?(Round to four decimal places as needed )(e) The variance of a bela distributed random variable is 62 What is the variance of Y in this problem? (a + 0)? (a + B+1)(Round SIX decimal placesnee

The probability is (Round to four decimal places as needed ) (c) What are the parameters and of the beta distribution illustrated here? (Type an integer Or decimal: Do not round ) (Type an integer Or decimal: Do not round ) (d) The mean of the beta distribution What is the mean proportion of impurities the batch? (Round to four decimal places as needed ) (e) The variance of a bela distributed random variable is 62 What is the variance of Y in this problem? (a + 0)? (a + B+1) (Round SIX decimal places needed: )



Answers

A random variable is normally distributed with a mean of $\mu=50$ and a standard deviation of $\sigma=5 .$
a. Sketch a normal curve for the probability density function. Label the horizontal axis
with values of $35,40,45,50,55,60,$ and $65 .$ Figure 6.4 shows that the normal curve
almost touches the horizontal axis at three standard deviations below and at three
standard deviations above the mean (in this case at 35 and $65 ) .$
b. What is the probability that the random variable will assume a value between 45
and 55$?$
c. What is the probability that the random variable will assume a value between 40
and 60$?$

For the given binomial distribution identify and equals eight equal 80.5. We want to draw the probability instagram constructed binomial probability distribution and compute the mean and standard deviation. So I've already drawn the hist a gram here on the left. Note how it's symmetric. This makes sense because P is 0.5. Not what we want to do is we want to compute the binomial probability distribution so we can proceed to do this first. The binomial probability distribution remember is probably ox equals K or the probability of finding a specific number of objects out of the n trials. So probably of X equals K is a factorial over K. Factorial over eight minus three factorial 20.5 to the K 10.5 38 minutes. Kay we can also write 380.5 DK because 108 minutes case simply it's 1080.5 to 8. Now we want to look for mu and sigma note that I've already marked a meal for on the history of the left. So the menu is some X p X. This is the formula for a discrete probability distribution mean, or n P equals for the standard deviation. Sigma X is for a discrete probability distribution square root, some expert, PX minus the square or root mp tens, 11 S p equals route to.

We want to draw the probability instagram constructed binomial probability distribution and compute the mean and standard deviation for a binomial distribution with unequal six and P equals 60.3 on the left. I've already drawn the probability instagram in order to understand its probability instagram, it would be appropriate to identify the binomial probability distribution. So let's do that. Next. So we've binomial probability distribution, p of x equals k equals six factorial over K factorial from six honesty factorial times 0.3 D k. 10 0.7 of the six minus k. Next, let's compute the menu which have already marked here on the instagram as well as standard deviation. So we have um you equal some XP x or mp, which gives me equal 1.8. The standard deviation is some expert, p x minus mu over x squared all square rooted or root mp times one minus p, giving 1.1360 At this point, we can make a note of the fact that our distribution is skewed left and we already identified this new on the instagram

We want to draw the probability instagram constructed binomial probability distribution and compute mu And sigma for by naming experiment with n equals nine trials. Probably success P equals 90.8. So, as you can see, I've already drawn the instagram to left, it's huge. The right. This makes a lot of sense because P equals 0.8 means a high probability success, meaning were more likely to see a higher number of successful trials out of our number of trials. So now it's computer, binomial probability distribution, the number or rather probability of defining exactly K equals success is so probably X equals K is nine factorial over K factorial. Nijinsky factorial or 90.8 K. Kind of 0.229 minus K. Now, if you compute mu and sigma knows that I've already marking you on the ground the left. So mu is some X p X. This is the formula for a discrete probability distribution, or n p equals 7.2. Then we have sigma X. Standard deviation is equals square root, some expert, P x minus x squared, or root mp times one minus P equals 1.2, So sigma X equals root, Mp, one minus P equals 1.2 is the standard deviation, so.

For this problem. We're dealing with binomial probabilities, and you have to construct a binomial distribution. You have to calculate the mean and standard deviation of that distribution in multiple ways. And then we have to draw a probability, hissed a gram, talking about its shape and where the mean is located. And for this particular problem, we're going to use and being 10 and the P value of 0.5. So let's first talk about what a binomial distribution is. It's going to be a probability distribution whereby you calculated probabilities on a fixed number of trials. There were only two outcomes of the event either success or failure, and that each of those trials were independent. So for part A, we need to actually create the distribution. And the distribution is going to be a two column chart where the first column is the X, or the possible outcomes. And the second column is the probabilities associated with each. Now, if you're running an experiment 10 different times, the outcomes could be that you had zero success is, or one or two or three all the way up to and including 10 possible successes, and now we want to calculate the probability of each of those outcomes. And the way we're going to do that is we're going to apply the formula and the formula for the probability of a binomial experiment is N. C X multiplied by P to the X power multiplied by the quantity of one minus p to the n minus X power. So for our situation, all the end values are going to be filled in with the value 10. We don't know the X is they're going to change as we go down this chart. We know that P is 0.5 in both instances here, and we know that in is 10 we don't know X. So for the first problem, we are going to substitute a zero in. So we're going to do 10 C zero multiplied by five to the zero multiplied by one minus 10.5 toothy 10 minus zero power. So I'm gonna bring in my calculator. They were going to type that in, So I have a home screen right now and I'm going to do 10 c zero multiplied by five raised to the zero power multiplied by one minus five. Raised to the 10 minus zero power. And I'm going to get a value of 000 97656 to 5. Now, that could be a very tedious process. So there is a faster approach if you have a graphing calculator. So I'm going to show you the fast approach in the graphing calculator, and I'm going to give us a little bit of space here. So in the graphing calculator, we would hit this stat, feature and edit. And as you can see, I have put all of our possible outcomes are values of X from 0 to 10 into list one. So I'm going to scoot over and I'm gonna sit up on top of list, too. And I'm going to type my generic formula in this formula down here. But every time I see an ex, I'm going to use list one, because that is where I have stored all my ex values. So I'll have 10 see NCR of list one multiplied by 0.5. Raised to the list one power multiplied by one minus five, raised to the 10 minus list one power. And as you can see, the number up on the top associate it with a X value of zero corresponds to what we did when we use different the formula alone. So we can now copy down the remaining values. And keep in mind that there are more decimal places than show. As you can see, down here looks very different than the highlighted up the top. So the highlighted up the top drunk eights and who arounds are answers. So I recommend you use the actual value. Um, And as to how maney decimal places you carry your answer out to is dependent on your professor or your teacher. So for two, if I scoot down, I get 0.4394 53125 And we're just gonna keep going on that 1171875 0.205078125 0.24609375 And as you can see, some of our probabilities are starting to repeat themselves. And the Onley reason that that is starting to repeat itself is because we were using a P value of 0.5 Any other P value that would not have happened. So I don't want you to fall into a false sense of security, expecting it to happen all the time. So that is our probability distribution for n being 10 and P being a 100.5 now. We could have also used a table in the back of most textbooks. There's a binomial distribution table, and if you locate in that table the n value of 10 and the P value of 0.5, you should also get these same values. Let's move on to part B. So in part B, we are going to determine the mean of our data, and we're going to use a method that you studied back a couple sections ago are actually in the previous section and section 61 and your formula for the mean new sub X is equal to the sum of X times P of X. So what that means is we have to go back to our chart and we have to create an additional column. So I'm going to extend my chart, and I'm going to create an additional column, and I'm going to call that column X times p of X. So that means you're going to take this X value times its corresponding p of x value, and we're gonna get zero. We're gonna take this X value and multiply by its corresponding value, and we're gonna keep that going. And again, there's a fast way of approaching it by bringing in your graphing calculator. So we're going to sit on top of list three and we're going to tell it to take everything from list one and multiply by everything enlist to, and you're going to see our first two numbers matched. So let's copy down the remaining values. So we get 0.878 906 to 5 0.3515 6 to 5 0.8203125 one point 23 046 875 And that duplicates. And then we get 0.8203125 and again, you can see we have duplicating values. And again, the only reason for those duplicates is because of the P value being point five. So now our formula for the mean says we have to add up these values, so the fastest way to add them up is to let the calculator add them up. So again, I'm going to bring in the calculator and I'm going to tell the calculator we need to add up list three. So I'm going to quit out of here, and I'm gonna say, Second stat, I'm going to move over to the math subsection and say some up list three and we're finding out that our average of this probability distribution is going to be five. So down here, we could say equals five. Now we need to find the standard deviation. And again, we're going to use the formulas from section 61 And there's two different formulas you can choose from. You get to take your picks. So the first formula is to take the square root of the some of the quantity X minus mu sub x that quantity ISS squared and multiplied by p of X, or you can do the square root of the sum of X squared times p of X. And then from that, you would subtract Musa X squared so you would get to take your picks which method you want to use. I'm going to select this method right here, So that means I need to go back to my chart, and I've got to create an additional column. And I want you to keep in mind that all of our X values were stored Enlist one, our, um, use of X. We just calculated to be five and r p of X was stored enlist to, So we're going to go back up. We're going to add on a column. We're going to call it the quantity of X minus mu sub X squared times p of x, and we're gonna let calculator do some work for us. So I'm gonna bring it back in. I'm gonna hit my stat and get back to my data. I'm gonna sit up on top of list for this time, and I'm going to tell it to take every value Enlist one. Subtract five from that square that result before multiplying by everything enlist to, and I'm going to get another set of numbers. So again, depending on what your professor and or teacher recommends, we're either going to copy them down in their entirety or we can round them for the purpose of just translate referring them to your paper. But again, when you're doing the math, we want to incorporate those numbers. So I'm gonna move the calculator out of the way so that we can record our values. So for zero, we got 00.2441406 to 5. And then we had 0.156 to 5. And then we had 0.395 5078125 0.46875 0.205078125 Then we get uneasy when we get zero. And then again, we start seeing those duplicate values. Now, if you rounded these and then you added them up, your answer is going to be slightly off what it should be. So that's why again, I say, Don't round. You might round as your recording on your paper, but definitely utilize the decimal answers in their entirety when we sum them up. So to find our standard deviation, we have to some these numbers up. That means we have to sum up list four. So we're going to say, Second stat, some up, everything enlist for And when we do that, we get a nice clean value of 2.5 so we can come down here and we could say our standard deviation is equal to the square root of 25 And if we calculate that out, we get 1.58113 883 So that was our answer to Part B. Was using the formulas from section 6.1 to calculate your mean and to calculate your standard deviation. Let's go on to part C in part C. There's too neat, easy formulas to use, but they only use if we're working with a probability distribution that is binomial in nature. And the two shortcut formulas will be, um, use of X equals n times p and the standard deviation where Sigma sub X equals the square root of en times p times the quantity of one minus p. So earlier we knew that end was 10. So if I substitute in, I could say Meuse sub X equals 10 times are P value, which was five. So lo and behold, 10 times 100.5 is five. So we get use of X equals five and it gets us what we got a long way in section B. And if we do that for the standard deviation, our N value was 10 RPI value was 100.5. So we're going to put five in two different locations. And sure enough, we end up with the square root of 2.5, which results in 1.58113883 Same is what we got right here. So again, these formulas Onley work. If you're dealing with a binomial distribution, any other probability distributions, we can rely on the old set of formulas. Or there's some additional formulas that you could learn and wrapping up this problem We need to draw a probability distribution, hissed a gram. So letter D. So in letter d, we're constructing a history, Graham. So history Graham is made up of a vertical axis and a horizontal axis. The vertical axis is going to be our probability axis. And if we look back at our chart, our highest probability that we had was the 0.246 almost 25%. So we're going to label our vertical axis as 0.5 0.10, 0.15, 0.20 and 0.25 and then we recall that up. Here are probabilities started to match up thes matched up here. So I'm going to start by drawing the tower for five at 25 or close to 25%. So we're going to put that first tower smack dab in the middle. So I'm going to say the first tower goes up to about here, goes across and down. So this was our tower for five, and our probability was 50.246 ish. And then we're going to work our way backwards and do the four, the three, the two. The one. Now, I probably made that a little bit too fat, so I'm going to just really quickly skinny that up a little bit. There we go. All right. So four had a probability of almost 20.5%. So for four, I'm going to go up to about 20.5. So someone right in here and that was 0.205 and then the same thing for six. Six goes up to about that same height. And then as we go back to three, it was 11.117 So 0.11 is just slightly above the 10 that was three and the same thing on the other side, 0.117 and then, as I go to to to waas 0.4 So I'm coming up to about here. So it was 0.44 and then the same thing with eight and let's keep going. For one, it was just around 1%. So we're only going up very little clips, and that's going to be point 01 and the same thing for nine. And then finally the zero in the 10. We're at minute values, so it's gonna be really hard to draw them. But we're talking very little here, and if you really wanted to, you could say 0.1 So that would be the hissed a gram to represent this binomial distribution. Now let's comment on its shape. Its shape is symmetric. It's mound shaped even more specific. It's bell shaped, it looks like a bell. So if I were to take let's go with the pink, you can almost see the bell shaped going on right there so it would be bell shaped. And where would the mean be the mean is going to be right here? right in the middle of that tower. And lo and behold, it was at five, and it would be at the peak of the belt.


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