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The U.S. Department of Transportation provides the number of miles that residents of the 75 largest metropolitan areas travel per day in a car_ Suppose that for a r...

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The U.S. Department of Transportation provides the number of miles that residents of the 75 largest metropolitan areas travel per day in a car_ Suppose that for a random sample of 60 Buffalo residents the mean is 22.5 miles a day and the standard deviation is 8.8 miles a day, and for an independent random sample of 30 Boston residents the mean is 18.8 miles day and the standard deviation is 7.2 miles a day_ Round your answers to one decimal place.a. What is the point estimate of the difference b

The U.S. Department of Transportation provides the number of miles that residents of the 75 largest metropolitan areas travel per day in a car_ Suppose that for a random sample of 60 Buffalo residents the mean is 22.5 miles a day and the standard deviation is 8.8 miles a day, and for an independent random sample of 30 Boston residents the mean is 18.8 miles day and the standard deviation is 7.2 miles a day_ Round your answers to one decimal place. a. What is the point estimate of the difference between the mean number of miles that Buffalo residents travel per day and the mean number of miles that Boston residents travel per day? What is the 95% confidence interval for the difference between the two population means?



Answers

The U.S. Department of Transportation provides the number of miles that residents of the 75 largest metropolitan areas travel per day in a car. Suppose that for a simple random sample of 50 Buffalo residents the mean is 22.5 miles a day and the standard deviation is 8.4 miles a day, and for an independent simple random sample of 40 Boston residents the mean is 18.6 miles a day and the standard deviation is 7.4 miles a day.
a. What is the point estimate of the difference between the mean number of miles that Buffalo residents travel per day and the mean number of miles that Boston residents travel per day?
b. What is the 95$\%$ confidence interval for the difference between the two population means?

Hello. Welcome to this. Pleasant. In this lesson we are looking for an interval within which We would construct the the 95 interval of the mean. And that's the formula. So we have the meanest 25, So last of minus 1.96. Okay. Times 74.5. All of us square it of 64. All right. So by solving all of the you have 252.45 glass or minus 18 point 25 25 All right. So this gives an interval of 234 .198 And 2 7 0.703. Okay, So clearly if we look in this integral we see that 215.6. So does the interval falls outside of the interview? Okay. So this is no the reason is that it is outside oh uh interval because the lowest bound is higher than the value. So it means that falls outside the body should be between this and that. All right, so thanks for your time. This is the end of the lesson.

This question problem. 30 is all about the number of cars per household on. It also has to replace the category eight or more with eight, which is what I've done here. And I've ran out the values of X and that probabilities which was given to in the question. So we now need to find the mean and standard deviation from this data. So it for in the mean when you do x p x so we need to times ah, value of X over the top wrote by the one ballot. So going through them all, we're gonna get no point six to no 0.66 I'm not going to fall no 0.15 these air fairly easy to do in your head. But you can always check things on a calculator, which I would always recommend. If you have a calculator, definitely use it. So the mean is this some? That symbol means some off x p a fax and we now have well world values for that. So you need to add them together. So anything No 0.34 There's no 0.6 to 1.66 That's not point to for There's no 0.15 That's no 0.12 plus no point, no 7% no point No. Eight. Which gives us 2.2 eight now, first on a deviation, willing to first work out the variants. So to do that, we're gonna need exquisite P X. So once great it still want so that one stays the same. But now I need to do 0 00.31 times two squared, which is full so that now comes out 1.24 I know. Did you know point to to you times three square. So it's no 30.22 times in nine, which is 1.99 The next one is not putting not six times four squared and foreswear the 16 it comes at north 0.96 Five. Squared is 25 times it by no point No. Three. Do you know 0.7 foot six squared is 36 3rd 6 times. No point or two. No 0.72 seven squares is 49. Everyone do 49 times. No point No one. There's no point for known on it. Squared is 64 so that would be no 640.64 Now, just like the previous question, we're gonna need to add this whole road together Trade, you know, 0.34 times ad. Sorry. Definitely. No times. 1.24 plus 1.98 plus 9.96 plus no 0.75 I probably would do this on a calculator, so you know, you're going to get it right and just go across the whole Ray. And you can always make a little prediction of what you think. They would come out as just a since Check your answer, and that comes out as 7.12 So that some of ex word p x 7.12 So are very ins. And this is a formula you should know Variances E X squirt, which is that number minus e of X. It's good. Which is that number? The square. So we need to do 7.1 to minus 2.28 Sward in 2.28 squared. It's 5.198 So if you do 7.1 to minus 5.1984 you're variants off one point 9 to 16 and Now we just need to square the answer because are stunned. Deviation his square area of our parents. So we need to do square root 1.9216 which equals 1.3862 No, as all standard deviation um, the next part poppy also. So how does the mean calculated empire Ice, the mom of just calculated, correspond to a national average of 2.82 point 28? So 2.23 is the national average? It is also exactly what we works out. Andi, Exactly equal to calculated part. C asked us to explain the effect that replacing catch, create or more with eight has on the main has done a deviation. So in most cases, when you change category that's eight class or three class or four plus wherever is. If you exchange that with the actual number the mean under standard deviation, it tend to be smaller because you've got rid of any massive outliers that could be included in that eight plus category. But in this case, we're told that there is a national average of 2.28 which means that replacing eight or more all with A hasn't actually changed the mean so so because there's been no change in meeting, we can assume no or very little change in standard deviation. There is almost certainly also no change in seven deviation, but we don't know that that absolute certainty. So I would say assume no or very little change in the Stone Division and that is everything.

In Problem 36. We have the mean and the standard division for the commute. Time to work where the mean equals 27.3 minutes and the standard division equals 8.1 minutes for birthday. You want to get the minimum percentage of commuters that has a commute time within two standard divisions, then we need to get the percentage or proportion for the mean to be within two standard divisions, plus or minus two seconds. We can use the shape shift inequality, which says that the percentage percentage equals one minus one, divided by K square, the right multiplied by 100 to get in in percentage. Where K is the number, we multiply it by the Thunder Division, which here is to then the percentage equals one minus one, divided by two squared multiplied by 100 which equals 75%. This is the minimum percentage knew that receptions inequality gives a minimum percentage and for K greater than or equals greater than one for birth. Three. We want to do the same to get the minimum percentage for for the commuters within 1.5 standard divisions, then mu plus or minus 1.5 Sigma, which means K equals 1.5. Then the minimum percentage equals one minus one, divided by 1.5 square or multiplied by 100 which equals 15 5 0.6 percentage. Which means we have 55.6 percentage of the commuters that travel or their time to work is between you plus or minus one 0.5 seconds for birth. C. Let's continue both to be first. We need to get the times within 1.5 standard divisions to get the times we just substitute in these values or in these equations to get the times the first time. The minimum time equals new, which equals 27.3, minus 1.5 multiplied by Sigma, which is given 8.1. These gifts 15.15 minutes. This is the minimum time, and the maximum time equals 27.3 plus 1.5 sigma, which equals 39.45 minutes. This means minimum 55.6 percentage of the commuters have travel time for their work between 15.15 minutes and 39.45 minutes. Let's go to varsity. We need to get the minimum percentage of commuters. Well, commute who have a commute time between three minutes and 51.6 minutes. What if the percentage or the minimum percentage, the minimum percentage equals? What for? The time is between three minutes and 51.6 minutes. We can see that the interval is very big. That's fine. What does it mean? To have a minimum value of three to have a minimum value of three. Let's get the K value. The key is the number adjacent to Sigma here. Then we have three equals. Um, you minus one point. Sorry, minus K sigma. And we know new equals 27.3. Then we have three equals 27.3, minus K multiplied by sigma 8.1. Then we can get Kay equals 27.3, minus three, divided by 8.1. Then K equals three. This means three is about three standard divisions from the mean And to make sure that the same value here is three of the mean we get 51.6 equals. Um, you plus K sigma. Then it equals 27.3 plus K multiplied by 8.1. Then, to get key, we use the same formula. But with the last 27 point. Sorry we have here 51.6, minus 27.3, divided by 8.1, which is three. This means now we have the time here is within three Sigma New plus or minus city segment. And to get the minimum percentage to get the minimum percentage here, minimum percentage we use the same shape shifts inequality one minus one divided boy, three square multiplied by 100 which gives 88.9% which means 88.9% of the commuter have a travel time to work from three minutes to 51.6 minutes, and these are the final answers of our problems.

The following is a solution for number 20 and we're seeing if there are more people on the 8 15 train or the 8 30 train. So they did a sample of 30 and 45 days for the 8 15 train in the 8 30 train and they found these sample means to be 323 and 356 respectively. And then the sample standard deviations of 41 45 Passengers were asked to find the 90% confidence interval For the difference between the two means now we're gonna use a two samples the interval, because these sample sizes are so large, as long as the sample size is greater than 30, the Z and the t interval roughly approached the same numbers. So that's plenty big. We're going to go and use the Z. Interval. So let's go to the T I T four and if we go to stat and tests its give me this ninth option down here, the two samples the interval, so nine. And then just make sure stats is highlighted there. And so for signal one and sigma two, that's where you're gonna put your S one and S two. So 41 45 there's the sample mean, with the sample size and then the other sample mean with the sample size. Okay, so that's all you know, given and then the sea levels 90% of .9. So whenever we calculate its negative 49.5 it's negative 16.5. So let's go and write those down. So I won't round some negative 40 A953 All the way up to negative 16.47. Okay. So the confidence interval says that there is a difference in zero is not contained. But let's see about the test. So our alpha value here is five and we're testing if there is a difference or this one the null hypothesis saying there's not a difference and the alternative is that new one minus me too is less than zero meaning that more people are on the 8 30 train as opposed to the 8 15 train. So we need two things we need are test statistic and we need the p value. Okay, so to find the test statistic you can use the formula. I'm gonna use the T. I. T. For because a lot of the data is already in there. So if we go to test, it's going to be this third option here. The two samples e test and stats is highlighted. So here you can see that everything is already done for you. You may need to change the alternate hypothesis and it will be greater than since new one minus me too is greater than zero. All right, at least. That's the actually no that's not right. So it should be less than you. one is less than you two. I'm sorry. So mu one minus me too is less than zero. So change that to less than and whenever we calculate that gives us a a test statistic of negative 3.283. And then a p value is something very very small. So that p values not five, it's actually Five times 10 to the negative force. So there are four zeros in front of it. Um So just remember PD is never bigger than one because it's a probability. Let's go and write those down and then we'll kind of look at them and see what we can say. So the test value would be negative three 283. And then the p value of 0.0005. Okay. So what we do with that P values we explicitly compared to the alpha value and it is less than alpha at any time. It's less than alpha. We reject h not. So we're rejecting the null hypothesis, saying that that's the same amount of people on average. And we can accept the fact or accept the statement saying that There are more people in the 830 train. Okay? And then part to see, it says, what's the significance of the test? That's just the p value. So, any time and ask for the significance of the test significance of the test. Just know it's just talking about the p value there. Okay. So that's a to sample the interval. Nz test for the difference of two


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