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The time it takes NFL running backs to run the 40-yard dashfollows a normal distribution with a mean time of 4.59 seconds anda standard deviation of 0.12 second. Le...

Question

The time it takes NFL running backs to run the 40-yard dashfollows a normal distribution with a mean time of 4.59 seconds anda standard deviation of 0.12 second. Let x = the time it takes anNFL running back to run the 40-yard dash. (Use the Empirical Rule.Write your answers as a percent.)A) P(x < 4.35) (in a %)B) P(4.35 < x < 4.83) (in a %)

The time it takes NFL running backs to run the 40-yard dash follows a normal distribution with a mean time of 4.59 seconds and a standard deviation of 0.12 second. Let x = the time it takes an NFL running back to run the 40-yard dash. (Use the Empirical Rule. Write your answers as a percent.) A) P(x < 4.35) (in a %) B) P(4.35 < x < 4.83) (in a %)



Answers

Use the Empirical Rule. The mean speed of a sample of vehicles along a stretch of highway is 67 miles per hour, with a standard deviation of 4 miles per hour. Estimate the percent of vehicles whose speeds are between 63 miles per hour and 71 miles per hour. (Assume the data set has a bell-shaped distribution.)

We are going to be evaluating the empirical rule on looking at different approximate percentages of a normal distribution based on some intervals that were given in standard deviations on evaluating the correlations between them. So to start off, I always like to create a visual. Think that's easiest. We're talking about these, so you're mean, Were you? Mu is going right in the middle because again it's an average. Then we're gonna draw out our standard deviations. So 12 and three negative one negative, too and negative three. So let's say we want to see in this normal distribution which again just a reminder it is symmetrical. So we want to see what less than the mean would look like. And what percentage of our data would be less than the mean. So if you wanted to draw a line right down the middle there so you can kind of see and Segway off your bell curve? What percentage would be less than the mean? So that's all this over here, this shaded region, so 100% would be everything, and just that shaded region is 50%. And then, if you want to see what's more than the mean. That would be 50% because again it's symmetrical. So everything is the exact same. So 50% would be under the mean 50% for data would people are. But what do we see when it's greater than one standard deviation above the mean So the normal distribution based on the empirical rule we already know within one standard deviation all the data and their 68% of your data is gonna fall within one standard deviation. So based off of that, we want to see the 100% of the bell curve minus 68% which is what balls in that standard deviation that we shaded in. Okay. And then we're actually going to divide that by two because of the two sides or two different tales, and that should give you 16%. So again, if you want to look at it in a visual standpoint, render almost over your really quick. So we already have this shaded region right here, which 68% falls with them. What we're trying to find is over here and over here. So it's 100% minus the 68 to get these circles from here. But then we need to divide by two. Because we don't really want to find that we want to find greater than so this one right over here, that would be positive. OK, so that's kind of gonna put a visually what we just did with math. Okay, so just get rid of that. Makes it a little cleaner. So let's say we wanted to then find what it would look like for anything less than one standard deviation above the mean. So again, find visuals. Very helpful. Oops. We're trying to find we already know 68% is within one, and we would want to find were prompted to find anything less than one standard deviation above the mean. Okay, so one just secrecy it. So anything less than one standard deviation. So from your mule up upto one standard deviation positive and then all of this over here. Last man. So we have 68% already, but we need to find the additional over here. So you want to look at it in math? We already have the 68% but we need to add what's over here or question mark the rest of that. So it would be the 100% left of the bell curve minus 2 68 So again, little visual over here, we already have, you know, 68 shaded in here out of the whole thing. We're subtracting that out. So, what that would give us Are these right here? Okay. And then you divide it by two. Because we don't care about that one over there. Were trying to find this one right here with less than and that would give you after you add on to the 84 excuse me. 68. It should give you 84. Okay. So just do it really quickly. It would be the 100 minus 68 would be 32 by that by, to which we found at the top will be 16. So it's basically 68. Plus the 16 over here, which would give you 84% is all that over there. Let's shaded. Okay, so we've already been prompted to find those, and then what we're gonna dio the last one was Keep that there in case you need it for reference. The last one we're gonna look at is we want to toe see between one standard deviation below the mean and two standard deviations above the mean, so again visualisation standpoint from you for me just smack in the middle. And then we want to find out between one standard deviation below the mean and two above. So again, let's just write it out to be consistent. 12 and three, negative one negative to negative three. So between one standard deviation below. So that's gonna be over here and then two standard deviations above. So it's all that and there. So, using the empirical rule, we already knew that between long two on three they want to negative three between the first inter deviation 68%. But if we go out to two here, 95% falls within that. So we can use that and look at the 68% so the ones in the middle can shaded 68 the middle right there, shaded one center deviation over, plus 95% minus that 68% and divide it by the two different tales. So, again, that's one standard deviation. We know what two standard deviations we're here and we're just subtracting out the redundancies of the 95 the 68 dividing by two and then re adding on to the 68. And that should give you 81 0.5% would fall within one standard deviation below, so that negative one over here and positive two standard deviations, so 81.5% would fall within that in that original sheeted region.

This problem, we are given that the cumulative distribution function F of X is equal to zero Effects is less than zero and one minus E to the negative four X. The fact is greater than or equal to zero. We want to find the probability of waiting Fewer than 12 minutes 1st. Using the cumulative distribution point. Now, the probability that X is less than or April to 12 Is just the definition of capital f. of 12. So to evaluate capital f of 12, we just put in 12 forex down here on the bottom. It would be 1 -1 To the -48. Let's see. Using the cumulative distribution function. That's the probability. Mhm. Now, and be we want to use the probability density function. Well, the probability density function is equal to the derivative of the cumulative distribution function, derivative of one minus E. With an annual four, X is four. Either the -4. And so this is 40. The native forests In fact is greater than or equal to zero. So the probability that actually less than 12, It's April to the integral from 0 to 12 of 42 the negative four X. Dx. And this is negative E. To the negative four X. As it goes from 0 to 12. Just negative either 1948 minus. Thank you to veto the zero, Just negative to negative 48 postman, which is one minus E. To the negative 48 just like we found above.

So this question were given exa normally distributed random variable with mean zero and standard diminishment .75 and we're asked to find the probability is indicated, so he's driven to transform X into our standard normal random variable x minus mean over standard deviation. We're going to apply this transformation to X. So part day we basically have probably TZ is between -5.36 95.0933. Now we know, call it easy, less than 5.0933 is one. Probably easy, less than -5.360. So the probability here is one. Our theme for asked for the probability that looks is more than 4.11, which is the same as probably to Z. more than 5.48, That is 1- authorities, Z And minus the probability of c less than 5.4 ft. That here is zero Less than frequent, courageous one. So the answer here is zero.


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