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The mean weight of trucks traveling on IH 35 is not known and aninspector needs an estimate of the mean. He selects a random sampleof 49 trucks and finds the meanto...

Question

The mean weight of trucks traveling on IH 35 is not known and aninspector needs an estimate of the mean. He selects a random sampleof 49 trucks and finds the meanto be 15.8 tons with a standarddeviation of 3.8. What is the 95% CONFIDENCEINTERVAL? a. 14.7-16.9b. 12.0-19.6c. 14.9-16.7d. 14.4-17.2

The mean weight of trucks traveling on IH 35 is not known and an inspector needs an estimate of the mean. He selects a random sample of 49 trucks and finds the mean to be 15.8 tons with a standard deviation of 3.8. What is the 95% CONFIDENCE INTERVAL? a. 14.7-16.9 b. 12.0-19.6 c. 14.9-16.7 d. 14.4-17.2



Answers

The designer of a garbage truck that lifts roll-out containers must estimate the mean weight the truck will lift at each collection point. A random sample of 325 containers of garbage on current collection routes yielded $x-=75.3 \mathrm{lb}, \mathrm{s}=12.8 \mathrm{lb}$. Construct a $99.8 \%$ confidence interval for the mean weight the trucks must lift each time.

Told that the mean weight of luggage checked by a randomly selected passenger flying between two cities is £40 and the stair deviation is £10. We're told that the mean and standard deviation for a business class passenger, on the other hand, that was a tourist class for a business past class passenger are £30 in £6. Mhm in part A. We're told that there are 12 business class passengers and 50 tourist class passengers on a flight were asked to find the expected value of the total luggage weight in the standard deviation of the total luggage weight. Well, but X one through extra will be the waits for the business class passengers and well, that why one through 50 b, the tourist class weights in the total weight T is going to be some overall the exes and then the sum over all the wise as well, which we can actually right as the sum of two variables ex representing the total wage for the business class and why representing the total weight for the tourist class we have that the expected value of X is going to be the same as 12 times the expected value of a single tourist business Class X one and we're given that this is 30 so this is going to be 12 times 30 which is 360 pounds. Likewise, we have that the variance of X is going to be 12 square times the variance of X one, which is going to be Oh, that's actually just sorry 12 times the variance of X one since they in your combination. So this is 12 times 36 which is equal to 432. The variance is found by squaring the standard deviation for X one. We also have the expected value of why this is going to be 50 times expected value of why one so 50 times we're told that the mean for the tourist class is 40 gives us 2000 likewise thieve variants of why this is by margin 80 50 times the variance of why one which is equal to well, we had that standard deviation of libel. Miss Tens, this is going to be 10 square which is 100 a few times 100 which is 500. I'm sorry. That's not five. Hundreds of 5000. Therefore, we have the expected value of tea since cylinder combination is simply the expected value of X plus. The expected value of why which as we kept the before, is 360 in 2000. And so we get 2360 and we have the variants of the total weight is going to be. This is simply the some of the variances for X and y, because these two variables X and Y are independent. We had that this was calculated to be 132 and 5000, so we obtained 5432 and therefore the standard deviation of tea is the square root of this, which is approximately 73 0.70 to £1. Next in Part B, we're told that individual luggage weights are independent, normally distributed random variables and were asked to find the probability that the total luggage weight is at most £2500. Well, in this case, the probability that the total weight T it's less than or equal to £2500 this is going to be he standard normal, cumulative distribution function of 2500 minus the mean tweeted determined to be 2360 over the standard deviation, which is 73.70 to 1 approximately, and this is the same as five of about 1.90 And after looking this up in a book or using a computer calculated, this is approximately 0.9713

Okay for this problem we're looking at, um, some weight values, overweight men. So we're looking at a random sample of 60 and we're going to find confidence intervals. So it's, you know, it's really no over here, so we know that we have, ah, sample of 60 came. And we know that the number of men so from our sample are X bar next cream you, the sample mean equals two £30. So these men's on average were £30 based on the 60 men, and the standard deviation of the sample equals to £4.2. And what we want to know is we want to know what the point estimate ISS A. What's the point Estimate? Well, the point estimate means the population are best prediction for the population is the same thing as the sample mean, so we know our best estimate is £30. Based on the data we have, that's pretty part B asks us to find you 95% confidence interval of this. So maybe a conference in every will see, I what? We're gonna use our technology for this. We're gonna use the calculator we see up over here So when you stop and tests and we're going to get into a paged work here going to get into a ah zi interval so and we have not dated. But we have statistics. Silveria, look at our statistics here, so use what we kind of written down already. So we have been told already that the, uh, 4.2 the standard deviation that was given to us. So plus or minus 4.2 of the typical deviation from the mean and the mean value off what we were told was that £30 overweight. But these men were, and this is based on a sample size of 60 and the confidence interval of the confidence level will question be asked us to find wants us to find the 95% confidence level. So by doing this, the car this is going to tell us how wide oven interval we need to be to be confident that we're going to capture the Truman mean weight of these men. So the confidence interval for this means that the weights are between 28.9 for 95% confident that the true mean is between 28.9 and £31.1 to be 95% certain. Guess there's 95% confidence interval and then Part C asked. Justifiable. What if we want to be 99% sure 99% confident that were captured, the true meaning? Well, a good thing to know is that when you up your certainty, it's going to make it a little whiter in terms of, uh, the spread of what our prediction is. So we're gonna do the same thing. Test it is the interval and our data's deliver. Next, we just put it in 74.2 30. But we use me to change this to 8.99 when the 99% certainty certain along for 1% error in our prediction and so we can see that the for the 99% confidence interval the range of values. The true prediction is between 28.6 and £31.4 and party is the best question. It says which one is whiter? I've kind of mentioned it before, but we'll answer it out. So, uh, the 99% confidence terrible is wider because we want to have a more certainty. So look at the question here was that you have always larger and why, Uh, that was 9% confidence. Intervals is whiter because we want more certain TV. Want more certainty about possible values. So our spread of our prediction is whiter way confidence that holds for means of two different confidence levels.

The following is a solution to number 14. And this looks at used passenger cars versus pickup trucks. And we're looking at the difference in meantime used by the original owner before that, either passenger car or pickup truck is being sold. So here, see this is the normal car. So there were 40 of them, the average lifespan or the average time that the original owner had, it was 5.3 years Standard deviation to two. And then for the pickup trucks still the same sample size and 7.1 is the main, and 3.0 is the standard deviation. Were asked to find the 95% confidence interval. So I'm gonna use the T. I 84 just because it goes by a little bit more quickly. If we go to StateN tests, we're gonna go to this ninth option here. The two samples e interval. The reason why the reason is the interval is because The sample sizes are large enough, so as long as it's bigger than 40, then you can pretty much use the Z or the T. It doesn't really matter now for signal one, signal to those of your standard deviation. So just think of those as S is for now. And so you can punch in your data there and then the sea levels .95, because we're asked to find the 95% confidence interval. And then whenever we calculate this gives us our answer there, so notice zero is not contained in that in world, So negative 29 To -64. So let's go and write that down some negative 2.953 All the way up to zero 6471. So I see this as Uh since zero is not contained it looks like there is a difference. But you know it all depends on that alpha value. So the second part of this is to test at the 1% level of significance if there is a difference. And so I need to values here, I need a Z. And I need a P value and once again I'm going to use the calculator because what's nice about the T. I. T. For it saves if I go to stat and test and just go to this third option here, the two samples t test and stats is highlighted. I didn't have to changes at all. I just automatically assumed. The only thing that I need to change is the alternative potentially changes the the alternate hypothesis and it's not equal to. So I don't even have to change that. And then I calculate and that gives us a Z value. This is our test statistic and get that with a formula to but negative 3.06 and it's got a pretty small p value of zero two. All right, so the Z value is negative 306. And then the p value of 0.00 two. And what you do with that P values you explicitly compared to the alpha value in this case it's less than alpha. So when any time is less than alpha, then you reject your null hypothesis. So reject h not. So we're rejecting that there is no difference between the two and we're accepting that there is a, you know difference. And it appears that at least with the sample, that trucks tend to have um their original owner for a little bit longer than for passenger cars. And then the last part of this it says to find the significance of the test and that's just your p. Value. So the p value is also the significance significance of the test. So that's the part c there. So significance of the test is the p values 2.0 to 2.


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