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Test # 2 Score: 1.6/12 2/12 answeredQuestion 12Suppose that the distance of fly balls hit to the outfield (In baseball) is normally distributed with _ mean 0f 259 f...

Question

Test # 2 Score: 1.6/12 2/12 answeredQuestion 12Suppose that the distance of fly balls hit to the outfield (In baseball) is normally distributed with _ mean 0f 259 feet and standard deviation 0f 44 feet Use your graphing calculator t0 answer the following questions: Write your answers in percent form: Round your answers t0 the nearest tenth of percent: a) If one fly ball is arang8rey ch chosen from this distribution; what is the probability that this ball traveled fewer than 198Pifewer than 198 f

Test # 2 Score: 1.6/12 2/12 answered Question 12 Suppose that the distance of fly balls hit to the outfield (In baseball) is normally distributed with _ mean 0f 259 feet and standard deviation 0f 44 feet Use your graphing calculator t0 answer the following questions: Write your answers in percent form: Round your answers t0 the nearest tenth of percent: a) If one fly ball is arang8rey ch chosen from this distribution; what is the probability that this ball traveled fewer than 198 Pifewer than 198 feet) b)If one fly ballis randomly chosen trom this distribution; what is the probability that this ball traveled more than 228 feet? Pimore than 228 feet) Add Work Calculator



Answers

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.
a. If $X=$ average distance in feet for 49 fly balls, then $X \sim$ _____(_____,_____)
b. What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for $\overline{X} .$ Shade the region corresponding to the probability. Find the probability.
c. Find the $80^{\text { th }}$ percentile of the distribution of the average of 49 fly balls.

All right. Um, this question has three parts ABC. So we starting from part paid part. A is interesting in the distribution for ex parte. So before we look at X bar, we want to know the distribution for X and from the question we know that axed as ally normal distribution with me. And he called to you to 15 and a center deviation being close to 50. And then we're calculating the meal for X bar. So by the central image Thera we know, um um, you off X bar is because you the mute up X So it's two clips he as well. And the standard deviation off export is he cooked Teoh Sturdy standard deviation off eggs over square, root off the sample size. And here wait, plug in the numbers. We got 15 over square rural 49 and it's 7.1 for So we refute this part. It's normal to 15 looks, I'll just rewrite it. Pecs bars, Ally. Normal with means to be 2 50 tens are deviation to be 714 Okay, so above is part day, and, uh, we're moving to heart. Be heart. Be were interesting in the probabilities. He Ah, the problem thing off X bar. Smaller, then 2 40 I'm going to use art to produce this probability. The code will be, like best. The first line. This car is what I entrant to, um to our pig Norm as a function name. And 2 48 is the, uh the number want to know such that to calculate the cumulative problem the cumulative ethnicity probability And 2 50 it's a mean for normal normal and 50/7 assists Our center deviation. The resolve we got is this number, so we just keep Okay, we can keep three decimal places, so it's going to be probability of export Smaller than where you go to to 40 is he Could Teoh there a point there? 08 Tero hate. And if we sketch at imagine, this is our power ship probability density function for normal to 50 you see, over seven and the, uh, the middle part the middle point is to 50. It's our mean we want to know. We want to know from here to where we want to know the area. The area for those parts to this This is our probability sense for. And we get that this part. The area for those park Aziz. There a point. Narrow eight there. Right. Okay, but we came hoping to you part C heart, See? Is asking the eighties, um, the eighties Bronco or persons? How sorry. 88th person towel for the for the X bar. So how we're going to do that? We first well, at Trey represents for the eighties persons home. And we need to find such We need to find the value of off the K such that the probability off X bar smaller than or equal to K as the gods. You their 0.8 again I was using I will using our to do this and before way, you know, before before the point truth out. We can just how scotch from that? So this is X bar Cand again. The middle point is to 50. We need to find the cane such that these area this shadow park is because you 80% Oh, are you can see you can say it is equal to their 0.8. All right, The cold is going, Teoh just a little bit. Have to 56. This is our answer for Okay, I won't just read it down again. Chaise it Go to to 56 point their own one. And this is our event. Final answer for Parsi. Well, I hope this we

Okay. This question gives us the fact that fly balls are normally distributed with a mean of to 50 ft and a standard deviation of 50 ft for their distance. And it's asking us some questions about a sampling distribution for 49 fly balls. So if we take a sample of 49 fly balls, we want to see what this new distribution will look like. So remember, if we're taking a sampling distribution, we're actually looking at distribution of the X bars from our sample and for means. It's still going to be normal distributed because we have 49 fly balls, which is bigger than 30 and thirties. Air cut off point for the central limit theorem on when we can start assuming normality and the mean is going to stay the same because of her sampling from the population that will eventually start to look like it. And then, as for the standard error, we take the population standard deviation of 50 and divide that by the square root of our sample size. And it just so happens that 49 is a perfect square, so we actually get normally distributed with a mean of 2 50 in a standard error of 50/7. And that's going to be the normal curve we work with when analyzing the samples. So the next part says, What's the probability that a sample of 49 balls traveled an average of less than 240 ft? So we want the probability that the mean of our sample of 49 is less than 2 40? Well, this is just a normal curve calculation now, so we can just use our calculator if we're on it, see 84 or you can find the command on your similar software. So we want normal CDF plugging in. Our numbers here are lower bound is going to be negative infinity. So just pick a very large number. Our upper bound is gonna be to 40 because that's the limit for our fly balls and are mean. We said was 2 50 in our standard deviation or standard error in this case is 50 or seven. And if we do this, we get zero point 08 eight Sorry, 0.0 a zero. That's a pretty low probability, even though it would be a pretty high probability if we had our original distribution. But when we start sampling, it just shrinks are mean on in or shrinks are standard deviation, I should say. And then part C. After we sketched the graph here I have a pretty skinny, normal curve, and the probability it's less than 2 40 is just gonna be this little region in here. Then Part C wants us to find the 80th percentile. So that would be the point on this normal curve where 80% of the data lies to the left of it. And I would estimate that that's about here and actually find this week unjust dio the inverse norm command. So if we go toe inverse norm, we're looking for an area of 0.8 with a mean of 2 50 and a standard deviation of 50/7. And if we get this, this is going to be 256 point 01 see? And again, that's pretty close to the mean because the sampling distribution really shrinks down our variability. And that's our answer

76 supposed that the distance of fly balls hit to the outfield in baseball is normally distributed with the mean of 250 feet and extended deviation of 50 feet. A if X equals distance and feet for a flat ball ex represented with the normal distribution of a mean of 250 a standard deviation of 50 be if one flat ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 220 feet? So I'm looking for on the problem of the X is less than 220. Sketch the graft scale the horizontal X axis shade the region correspondent to the probability find their probability. So this have a normal distribution that's centered at 3 50 with a standard deviation of 50. What I'm looking for here is the probability of a fly ball being less than 220. Another expire you is distance. Um, I can use my normal CDF function. My calculator this off for this. So with my normal CDF, I need to know where I started shaving. In this case, that would be negative infinity so I'm gonna insert negative 999 I stopped shaving from upper bound in this case is to 20. My mean was 250. Standard deviation was 50. And insert this calculating probability to be 0.27 for two. See, find the 88th percentile distribution of fly balls. Well, I still have the same distribution center 22 50 with standard deviation of 50. Essentially, what I'm looking for here is which value is at theeighty percentile or which value has 80% of the area to the left of it. So what I'm looking for a year is which value can be inserted in this equation where X is less than it, for the probability is 80%. So the probability of X is less than K is equal to 80. What is that kay value. So I can use my inverse norm feature here, in which 80% of the area locked to the left I mean in this case is to 50 and my standard deviation this 50. This will tell me that the value is 292.8

Neary of this probability distribution function. F of X is three over sixteen times for Minds X squared on zero to and park A We just want to find the average. So don't know that I have you That's going to be the integral from here to two of x times its average our times this distribution function which in this case okay, begin factor out through sixteens Hey! And we just have girls here, too to for X minus execute Jax Mensch is very sixteen. Here we have X squared two linus for X squared over to sit to X squared minus X cubed. So that becomes Extend the fourth over four and evaluate from zero to two. This is three sixteenths times eight minus sixteen or force of four. So this ends up being Towler sixteen, which is three. Four or point seven five. Okay, for the variance, which all just know by being do something similar says here, too, to this time we put in X squared times the distribution function and then subtract the average squared. So three four squared okay, so can take out of three sixteenths. We have the integral of from here to two for X squared minus sixty fourth. The Ex minus three fourths square Ness's three sixteenths. It's gonna be for thirds X cubed minus X to the fifth over five. Evaluated from here to two minus. This is nine hundred sixteen. So people into it get eights. That's twenty for three me for E minus thirty two five minus nine over sixteen. And when we simplify that out, we get nineteen. Hoover eighty. Rich is about point two four. All right, so next we just want the standard deviation, which is the square root of the variance. And so what do we get about? I see your point for nine kit and then party. We want the probability that, uh, this variable given by this distribution function is bigger than me. So this is just the integral from the mean up to to the right important of three sixteenths times or minus X squared, the ex and women compute that player is again. We get about zero point for six, three, nine and then for Parky, we want the probability yes, that we're within one standard deviation. So, um, you plus sigma, you mind a sigma around? It's just the integral from you might of Sigma to you plus Sigma and I just have these bodies stored. Mom's compute the interval quick of three sixteenths for minus X squared. Jax, what do we get when we compute that out? We get zero point six one three.


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