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The Association of Oklahoma Independent Oil Producers would like to prove that the average production of Oklahoma oil wells is less than 24 barrels per day. The ass...

Question

The Association of Oklahoma Independent Oil Producers would like to prove that the average production of Oklahoma oil wells is less than 24 barrels per day. The association randomly sampled seventeen actively pumping oil wells in the state. The sample produced a mean of 20.8 barrels per day and a standard deviation of 3.6 barrels. Use a 0.01 significance level to test the Associations claim:If you would like to conduct a corresponding confidence interval for #2, what percent would it be? Find th

The Association of Oklahoma Independent Oil Producers would like to prove that the average production of Oklahoma oil wells is less than 24 barrels per day. The association randomly sampled seventeen actively pumping oil wells in the state. The sample produced a mean of 20.8 barrels per day and a standard deviation of 3.6 barrels. Use a 0.01 significance level to test the Associations claim: If you would like to conduct a corresponding confidence interval for #2, what percent would it be? Find that confidence interval One approach to make green gasoline takes biomass in the form of sucrose and converts it into gasoline using catalytic reactions At one step in a pilot plant process; chemical engineer measures the output of carbon chains of length three: Nine runs with the same catalyst produced the following yields (in gallons): 0.63,2.64,1.85,1.68,1.19, 1.76,0.83,1.04,0.68. Test the claim that the mean yield is greater than 1.2 gallons: Use a=0, 05_



Answers

Fuel Catalyst To improve fuel efficiency and reduce pollution, the owner of a trucking fleet decides to install a new fuel catalyst in all his semitrucks. He feels that the catalyst will help to increase the number of miles per gallon. Before the installation, his trucks had a mean gas mileage of 5.6 miles per gallon. A random sample of 12 trucks after the installation gave the following gas mileages: $$\begin{array}{cccccc} 5.9 & 6.7 & 6.9 & 6.4 & 6.6 & 6.3 \\ \hline 6.2 & 5.9 & 6.2 & 6.4 & 6.6 & 5.6 \end{array}$$ (a) Because the sample size is small, he must verify that mileage is normally distributed and the sample does not contain any outliers. The normal probability plot and boxplot are shown. Are the conditions for testing the hypothesis satisfied? (b) Test if the fuel catalyst is effective, assuming that $\sigma=0.5$ mile per gallon at the 0.05 level of significance. FIGURE CANT COPY

Hello. Hi. Hearing this question, we have to determine the chi square value first. Okay, So the critical value at 98% of confidence, we had to find out on when you Skype sky skirt table first V c. A number off samples given here is 49. Okay, so you can find a decrease of freedom. Degrees of freedom will be getting and minus one that is equal to 48. You can see that 48 value is not there in the table on you can see 48 is greater than 40 from The terrible will be using the value 40 in order to get the values off particular values. Okay, now see values 98% so we can write physical 2.98 Then you can obtain Alfa values. I'll physical 21 minus e one minus three is one minus 10.981 minus 0.98 You can write 0.0 toe, then left. Detailed critical value. Left tail critical value that is chi square one minus alphabet will be K square one minus. Alfa by. To hear, it will be 0.99 from the table. We get it is the call to 22 point 164 Next to write tailed critical value. This guy square alphabet. Do you get square off? 0.1 from the table. We obtained the value 63.691 Okay, once we get chi square values, the next thing we can find out, we can find the confidence in trouble. Estimate four Standard deviation. We know the formula. The formula is given by what is square root off and minus one by Chi Square Alfa by two times standard deviation less than sigma. Less than squared off and minus one by chi. Squared one minus Alfa by two times standard deviation. Okay, so once we know these values, we just start substitute all these values given and minus one value. You know that is equal to 48 Chi square. Well, we found out that is equal toe 63 point on 691 times. The standard deviation value we know it is given in the question is is equal to 21. We get next to less than Sigma less than here on the right. Answer with how again? 48 by the chi square value. We know that is able to convict 2.164 times the standard deviation again. Two D one. So we just started simplified these values. Okay, when you simply find you get the value 18 point 2306 less than standard deviation value less than 13 point 9041 Okay, so from here, you can see this value does not give it any information about effectiveness. Right? You can see this value does not give any information about them effectiveness. While the value we can still stigma values between 18.23 on 13.9. Right? So that's why we cannot have any information about effectiveness off particular value. Here. I hope that this answer your question. Thank you.

So in this problem, we're looking at the total tax per gallon for gasoline. And we are going to look at a sample of 18 different gas stations. And the question here is that is there sufficient evidence to claim that the average per gallon Gas taxes less than 45 cents. So we are trying to see if the average is less than 45 cents. So the first thing we are going to have to do, since we are running this test is to write are null hypothesis. And our alternative hypothesis, and a no hypothesis is always going to be a statement of equality. And our alternative hypothesis is a statement of inequality. Since we are trying to test something that's of inequality, Our alternative is going to be that Mu is less than 45. Now, the Complimentary no hypothesis would be that μ is equal to 45. So because our alternative hypothesis has a less than symbol, this is going to end up being a left tailed test. And we are going to need a test statistic. And to calculate our test statistic, we are going to have to assume normal distribution. And we will apply the formula why bar minus mu divided by S over the square root of n. So we're gonna have to take this a set of data and find the average of that set of data as well as the standard deviation. So the most efficient way of doing that is to bring in our calculator, we're going to hit the stat feature And edit and as you can see, I have already put the 18 pieces of data into the calculator. So we're going to do stat, we're going to scoot over to the calculate menu, we're gonna do one variable statistics of everything that we have enlist one, And we are going to find that the average is approximately 39.556. And you can see our standard deviation is going to be 7.137. So we're gonna write this down, so we set our y bar Is about 39 556 And our standard deviation will be 7.13764184. So we're now ready to utilize our formula and find our standardized test statistic. So we're gonna have seven or 39 556 minus 45 all over the sample standard deviation divided by the square root of our sample size. And in doing so, we are going to get AT score of a negative 3.2 36 So now that we have set up our information, we're ready to begin answering the questions. So ultimately we've got to answer question a um so is there sufficient evidence to claim that the average per gallon Gas taxes less than 45 cents. And we want to use the table in the appendix to bound the p value associated with this test. So ultimately, well answer the test in the next section. So we want to start with bounding. Um, so we are trying to find the P value and that p value is going to be the smallest value of alpha or the level of significance for which the no hypothesis can be rejected. And in order to bound this, we are going to have to take a look at the T. Distribution in the back of your textbook. And if you look across the top of that chart, You're going to see t. Values for areas of .100, A. T. Value 4050, A. T. Value 40 to 5 A. t. 4.010 And a. 4.005. And we need to focus on the row that has the degrees of freedom of 17. Now, degrees of freedom. Since we are working with a T distribution which is a family of curves, the degrees of freedom gives us an indication of what the shape of the graph looks like And our degrees of freedom will always be found by doing the sample size -1. So our sample size was 18. So we're going to Do 18 -1 and we're going to focus on the row that has a degree of freedom of 17. And when you look across the row you're going to see these values. Yeah. Now, because this is a left tailed test, the values in this chart are basically the T scores for on the right side of the curve. But since it's a T. Distribution and it is the metric with a center of zero, these values would also apply to the left side, but they would all be negative. So we're gonna throw some negatives in here and we calculated our T. Or test statistic to be -3296, which ends up kind of like right in here. So technically we could say that T, which was negative 3236 is less than negative two point 898 but is greater than infinite or negative infinity. So if we look at the values that Go with the -298, we have an area or a probability of .005. So RP value Is going to end up being less than .005 but greater than zero. So that would be the bound of the p value associated with this particular test. Now, as we go into part B in part B, we are asked to find the exact P value. So in order to find the exact P value, we're going to have to use their applet. Or we could use the graphing calculator again. So I'm going to show you both ways in the applet you're going to see a bell shaped T distribution curve. And at the top it's going to ask you to fill in a box with the degrees of freedom. And at the bottom it's going to ask you to fill in a box for your X. Value and your probability. So when we're finding the P value, our P value is going to be the probability that our we're working with a T. Value. That is less than what we found to be our standardized test statistic. And because the T distribution is symmetric, that will also be the same as our t. Value being greater than positive 3.236. So in order to use this applet, you're gonna have to fill in the degrees of freedom Which would be 17 and we'll fill in the positive version of R. T. score 3.236. And it's going to provide you with a P. Value. It's going to fill in that remaining box with 8.002. So therefore r. p. value, our exact p value is .002. And sure enough it does fall in between the boundaries that we set in part a now another way and maybe even a more accurate way of finding that P value is to utilize. If we think about our distribution again, We came up with a (323) 6 and a negative 3.236. So in essence what we're trying to find with our p. value is the area right here. So we can do that using the graphing calculators, T. Distribution, cumulative density function. And when you do that you have to provide a lower boundary of the shaded area, an upper boundary of the shaded area along with the degrees of freedom. So I'm going to show you how to use the graphing calculator in addition to that Apple it So are lower boundary. If you think about the fact that this curve is continuing infinitely into the negative values, we're going to go super small numbers. So we're gonna use negative one times 10 to the 99th power. And our upper boundary is right here at negative 3.236 And our degrees of freedom were 17. So I'm gonna bring in our graphing calculator again and to find that T. Distribution cumulative density function, you would hit your second and then the bears button and you can see number six on this menu. Is that cumulative density function? And are lower boundary? We said negative one times 10 to the 99th power. And then our upper boundary which was our T. value which was negative three 236. And our degrees of freedom was 17. And we get a little bit more accuracy. We're gonna get .0024. So we could say that R. P. Value is also .0024. Again. They're close enough whether we use the applet or we used the graphing calculator. So let's go back and now answer the question. Since it was a right tailed test, you're sorry a left tailed test. And it looked like this right here and this value was a negative three point 236 What we want to do is we want to compare that value with What it would be if we had the .005 and at the .005 level, If we had .005 here, we would have a -2898. So here we could say that yes, we would end up rejecting the null hypothesis. So when it asked the question, is there sufficient evidence to claim that the average per gallon gas tax is less than $0.45? We could say yes. There is sufficient evidence that the average per gallon gas tax Is less than 45 cents. All right. So the final thing we want to do is to construct a confidence interval And in order to construct our confidence interval at the 95% confidence level, you've got to think about what a confidence interval is. So when we're creating a confidence interval we are focused on The 95% that's in the center of that T distribution. So if there is 95% of the curve in the center of that Um bell shaped distribution, that would mean that there is 5% left to be split evenly between the two tails because of the symmetric nature of this bell. So if I wanted to find the boundary lines for the 95% confidence interval, I can go to the table in the back of the book, I can look under um the degrees of freedom of 17 and T. Associated with a .025 Area and I'm Gonna get a 2.110. So what that is telling us is that the right Boundary of the 95% interval would be positive 2.110 and the left one would be negative 2.110. And that is what we call our um critical T value. And we're calling it really critical t of alpha divided by two. So we took the alpha the left over part of the curve from the 95%. And we split it in half. So now we want to construct our confidence interval. So I want you to think in terms of a number line. What we're going to do is we're gonna put our average in the center and we're going to create some wiggle room. So we're going to subtract a margin of error. And we're going to add a margin of error. And in order to calculate that margin of error, we're going to utilize the formula T. Of elf over to multiplied by S over the square root of N. So, in essence, to get this lower boundary, we're doing why bar minus T. Associated with alpha over two times s over the square root of N. And to get this upper boundary, we're taking that y bar and we're adding the margin of error, so we're going to Subsequute values in. So to get our lower boundary, we're gonna take our average, which was 39.556, and we're going to subtract The 2.110, multiplied by the standard deviation Divided by the Square Root of 18. And then to get our upper boundary, we're going to do the same thing. But with in addition, So we'll get plus 2.110 multiplied by the standard deviation Over the square root of 18. So therefore our Average is in the center. And we said our average was approximately 39.556. And on the low end we would end up with 36.01. And on the upper end of our confidence interval, we would end up with a 43 0.11 So that would be our 95% confidence interval for the average per gallon gas tax in the United States.

Let us read this question. A company that next cola drinks states that the mean caffeine content per 12 ounce bottle of cola is 40 mg. So my mean is 40. This will be my it's not or my Nelly apotheosis. What will be my alternative hypothesis? It will be mu is not equal to 40. All right, You want to take this claim during your test, you find the random sample off 12. So my end is 12th. Sorry, my random sample of 2012 ounce bottles. So my end is 20. My end is 20 22 Alone's borders of cola has mean caffeine content of 39.2. So my expert is 39.2. Okay, it is 39.2 mg assumed that the population is normally distributed and population standard deviation Sigma is 7.5. So sigma is seven point fight 7.5 mg and Alfa is equal to 0.1 and Alfa is equal to 0.1 Can you reject the company's claim? Okay. What is the first thing we want to identify? The null and the alternative hypothesis. We have done that the be part says find the critical values and identify the rejection regions for standardized test statistic Z. Okay, what are going to be my critical values now? This is a two tailed tests and my al 50.1 So let me be clear that this is a two tailed tests and my al 50.1 So, in each one of these tales, how much area should be there? It should be 0.50 point 005 Okay. And if I find the critical values, they are going to be plus minus 2.575 plus minus 2.575 So this is plus 2.575 and this one is minus 2.575 And what is this Z statistic that I get if I put in the formula 39.2 39.2 minus 40 minus 40 upon sanity, vision is 7.5 upon a Route 10. What is N. N. Is 20. So this is route 20. This turns out to be minus. This turns out to be minus 0.5 minus 0.5 eight. Okay? And I can see that this value falls somewhere around here. So this is not in the rejection region. Excuse me. So I will fail to reject minor hypothesis now decide whether to reject or failed. Yeah. So we will fail to reject the null hypothesis. And if you want to interpret the decision, what was on al hypothesis? It was that mean is 40. So we will say that we do not have enough statistical evidence to say, uh that the mean caffeine content 12 ounce bottle off cola is different or anything other than 40 mg, and this would be my answer.

Once again. Now, welcome to a new problem. This time you're dealing with confidence intervals. And whenever you make predictions in statistics, you, uh, dealing with confidence Interval. So what you're saying is that you could either have estimation could either have estimation or hypothesis testing hypothesis testing You could have now an alternative hypothesis. When it comes to estimation, you could use confidence Interval, where you're either say 90% confident that something is gonna happen or you're gonna make a certain prediction or your 95% confident or your 99% confidence. So these are the things that are gonna happen when it comes to, uh, confidence intervals. And your goal in making predictions is, for example, to see that the mean off a sample which you call X. But this is the mean of a sample, uh, predicts the main off the population with a certain level of confidence off course, your errors allowed to happen eso your expo will have some kind of margin off era when you're running a, uh, confidence interval. And so the purpose off building on interval is to say, if you mean is a certain value, you're gonna have some intervals that capture the mean and then you're also gonna have some intervals that don't capture the mean. But if you're 95% confidence, which is the default, uh, Interval, you'll find that off. All these intervals, 95 out of 100 of them are going to capture the mean intervals are not just for making these kinds off predictions. You could also have ah, problem like the case we have where we're dealing with gas prices, uh, gas prices say in, uh, 2003, we have gas prices and you're doing a comparative analysis between California and Florida when it comes to gas prices. And in California, your X bar for God's surprises, which is the mean of example, is two point awful. And then, for the case of Florida, the gas is a little bit cheaper. The overall average price of gas is when you take a sample is a dollar a dollar 72. So it's a dollar 72. Remember, you're using the same sample size in this case on Don't forget that when it comes to the data, uh, the sample size for California is a little bit bigger than the sample size of Florida, which is 35 on then. Off course, we do have population. Standard deviation for California is Tencent's on then population standard deviation off Florida is, uh, eight sense. And this is pretty much what we're saying is that, uh, when you're computing your margin off era, you're gonna need the standard deviations. And we're lucky enough to, uh Well, actually, you're gonna need both the sample size and the standard deviation. So the first requirement, in part A was saying way, Want to get the point, estimate, point estimate? Um, um, that differentiates that differentiates, uh, the population, I mean, gas gallon prizes between these two states. Between these are two states, you know, that's that's what you're looking for. You want to get the differences in gas prices between these two states, and then the other thing you're saying is in part B, you want to get to 95% confidence on the margin off era for making the point estimate prediction for the interval. So margin off error in part c, uh, was saying eso at at at 95% confident. What's the margin of error? That's part B. And then in part C. And just recall that you're getting the difference between the two. A gas prizes. So Expo one minus expert, too on. Then you build an interval off of that in. Patsy was saying the 95 we want to get the 95% confidence interval that estimates that estimates the difference in I was gonna say average, but I'm just gonna say population means the difference in population means between California and Florida. Well, that's what we're looking at. So we're going to jump right into it. And in part, a, um you one is the, um mean Gus prize a gallon in California. Um, YouTube is through the I mean gas prize, but gallon in Florida. Uh, expel one, as we said, is for theme the sample mean And this is this is population. Just remember that this is population the muse. Other population sample mean a gas prize. A gallon in California on export to is simple. I mean, gas prices, the gallon in Florida. So we're looking at that. And so the difference. You know, if you want to compute the difference, the point, estimate difference. Uh, the point estimate a difference becomes mm. the point. Estimate difference for me one minus mu tube is simply Expo one minus expert too, Which is the same was $2.4 minus $1.72. And that gives 0.32 dollars. That's the difference. And then, uh, in part B but be we want to get the margin of error. Three. The confidence interval or the confidence level are the confidence level, uh, is given by 95% confidence. So if it's an interval, you're saying this middle part is 95%. So this dale right here is gonna be Alfa over two, and this deal will also be off over to remember Alfa is the era, which is 20.5 So, um, since we do have standard deviations for the population and we our goal is to get the margin off error. Imagine off era mhm. We like calling it E. That's the sama's Z off over to eso. The disease caused Z off over to this is the Z score tied up to you have a positive Cisco on the left on the negatives isco on the right. So negative there was a critical values that define the computational tool for the margin off error. So Sigma squared one over N one or sigma squid to over into If you go to the table, if you look up the table Z table with an offer off Uh huh. 0.5 and then off over two off 0.5 divided by two, which is 20.4 to 5. You get to see that the Z off over two is 1.96 ZF over two is 1.96 So this becomes a computational aspect with the radical 0.10 squared over 40 plus 0.35 squared off 35. And so our margin off era E is gonna be point all four if you round it. So this is rounded. So we're rounding off the numbers on, then finally, in part C. We want to compute the actual confidence interval, uh, confidence interval for me one minus mu tube, which is the difference in the means. So to articulate the confidence interval, we take the point estimate off the difference between the mean gasoline prices or gas prizes between the difference between me mean gasoline prices between Florida and California so x by one minus expert, too. Now, this is plus minus the margin off era. So Expo One, we already got that Expo one in expert to that was 0.3 to plus minus point. +04 If you do the math on the loss side, you're going to subtract the imagine off era, and then on the upper side, you're gonna add the margin of error, okay? And that gives us a confidence level. So the 95 percent confidence interval estimate, um, full the difference. The difference in mm population mean prizes or gal, um, off gas between uh huh between California and Florida will be equivalent to 0.28 and 0.36 So we're saying what, 95% confidence that the average difference between gas prices for these two states is gonna fall in between this interval. And of course, some of the numbers around it s oh, you know, just take heart from that. So once again, we had a problem. Our goal was to determine the confidence interval off the difference in means between California and Florida. When comes to gas prices. We had, uh, ex BA for California and Florida, and the sample sizes and the standard deviations off population were given. But three requirements get the difference in terms. Off point estimates get the margin of error and then get the confidence interval The difference. 0.32 confidence level. Margin off error We took the multiplier, zesco, multiplier and the standard error. This part is the standard error portion off the two groups and this is the our multiplier, which happens to be 1.96 and that you have to look up in the table so rounded the margin of error 0.4 and then the confidence interval includes Thebe point estimate off difference in means and the, uh, margin off air on both sides. So I hope you enjoy the problem. Feel free to send any questions or comments and have a wonderful day


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A constant net force acting on an object of 5.05 kg mass for2.07 s changes the speed of the object from 4.99 m/s to 5.68 m/sand the direction of movement from the initial direction angle of63 degrees to the final direction angle of 24 degrees. Find themagnitude of the net force.Write out your result in newtons
A constant net force acting on an object of 5.05 kg mass for 2.07 s changes the speed of the object from 4.99 m/s to 5.68 m/s and the direction of movement from the initial direction angle of 63 degrees to the final direction angle of 24 degrees. Find the magnitude of the net force. Write out your r...
5 answers
Solid potassium iodide decomposes into iodine gas and solid potassium Wnte a balanced chemical equation for this reaction:
Solid potassium iodide decomposes into iodine gas and solid potassium Wnte a balanced chemical equation for this reaction:...

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