5

Problem 2 of tutorial 1:The tensile strength of Portland cement being studied Four different mixing techniques can be used economically. The following data have bee...

Question

Problem 2 of tutorial 1:The tensile strength of Portland cement being studied Four different mixing techniques can be used economically. The following data have been collected:Mixing techniqueTensile strength3129 3200 2800 26003000 3300 2900 27002865 2975 2985 26002890 3150 3050 2765For the cotton weight percent example in Problem 2 of Tutorial suppose that the difference in tensile strength great enough such that the standard deviation has increased by 20 percent What sample size should be uscd

Problem 2 of tutorial 1: The tensile strength of Portland cement being studied Four different mixing techniques can be used economically. The following data have been collected: Mixing technique Tensile strength 3129 3200 2800 2600 3000 3300 2900 2700 2865 2975 2985 2600 2890 3150 3050 2765 For the cotton weight percent example in Problem 2 of Tutorial suppose that the difference in tensile strength great enough such that the standard deviation has increased by 20 percent What sample size should be uscd if we wish t0 detcct this difference with - probability of at least 0.90?



Answers

The compressive strength of samples of cement can be modeled by a normal distribution with a mean of 6000 kilograms per square centimeter and a standard deviation of 100 kilograms per square centimeter. (a) What is the probability that a sample's strength is less than $6250 \mathrm{Kg} / \mathrm{cm}^{2} ?$ (b) What is the probability that a sample's strength is between 5800 and $5900 \mathrm{Kg} / \mathrm{cm}^{2} ?$ (c) What strength is exceeded by $95 \%$ of the samples?

All right. In this problem we are told that the random variable is normally distributed and has means 6000 and standard deviation sigma equals 100. Basically information. Hello, we want to answer a three seat. This question is testing our understanding of random variables. Specifically normal variables or normally distributed variables. Facade, let's first review relevant information to the normal distribution which will allow us to answer agency. So we cannot these scores on the probabilities using a Z table. Yeah, because the tables are used for standard distributions. We have to make the conversion from our random variable X here to a standard normal distribution. We do that using conversion Z equals x minus mu over sigma. Specifically, you can utilize the symmetry of the normal curve as well as total area in the normal curve being one to answer questions eight through see below as well. So first to find the probability that x is less than 60 to 50 we're looking for the probably that Z is less than a certain value. Probably that 60 we can convert 62 52. The Z score of 62 59 6100 equals 2.5. That's the probably the less than 2.5 is not 0.62 but rather one minus 10.62 by symmetry. So if we can fix this answer now we write this as one minus 10.62 equals 0.9938. Next proceeding to part B one of the probably between 585901st as I've done here, we convert to the relevancy scores as I did in part, a probability falls between these scores. 1.1359 from is the table finally, and see what strength is exceeded by 95% of samples. We want to find the sample where 95% is to the right. That is probably the greatest seen articles. 950.95 from a normal distribution table. This is not equals negative 1.64. To convert to X. We reverse this equation to obtain X equals 100 times the Not for 6000 equals 58 36.

In this exercise, we are given the sample data For the compressive strength of concrete for 12 specimens. And for part A were asked to check the assumption that the compressive strength is normally distributed. So one way to do this is to do a normal probability plot. And I have done my normal probability plot in our so I will show you the steps that I took in our to do this. So, first to enter the data that was given in the question for the sample and I put it into a variable called X. So this is the day that right from the question here and then the function QQ norm makes a probability plot of the argument. As you can see this normal probability plot represents a pretty straight line from the bottom left to the top right, which is indicative of a population that is normally distributed. So the assumption of a normal population is reasonable. And then for part B We are asked to construct a 95% 2 sided constant confidence interval on the population mean? Now, for a two sided confidence interval, when we do not know the population standard deviation, it can be given by this formula. And so in order to find the confidence interval, we must find the values for all of these parameters in our we can find the sample mean by simply typing in the function mean and the argument X We get 2,259.917. And we can also find the sample standard deviation in our the function is SD It's 35.5693. And the simple size is 12. Now for 95% confidence interval, Yeah Alpha is equal 2.05. So the percentile that we want to find is T of alpha over two equals .025. And with N -1° of Freedom. So looking at a tea table, It's an area of .025 and one area in one tail And the degrees of freedom is 11. So we get 2.201. So now we have all the values for this formulas and we simply just plug them into the formula to get our confidence interval. And this gives us an interval ranging from 2,237.3 To 2282.5. So that is the 95% confidence interval for the population mean or for the mean strength of the concrete. And then for part C were asked to find a 95% confidence lower bound for the mean strength of the concrete. Now for a confidence lower bound, it's given by this formula and we know all of these parameters from Part B Except for this one. So just t for 0.5, 12° of Freedom. And going back to the tea table, We get 1.796. And that should actually be 11° of freedom. And this gives us a lower bound of 2,241 0.48. So the 95% confidence lower bound is 20-41.48, and the lower bound of the two sided confidence interval is 20- 37.3. And the reason why the lower bound on the one sided interval is greater is because we have an area of alpha in the lower tail below this bound, whereas for the two side at interval we have an area of alpha divided by two in the lower tail. So in this case the tail is smaller so the bound is further to the left or is smaller.

This exercise, we are given a sample size of 12, Sample average of 34, And we're also told that the population and the standard deviation is 60 for part A whereas, to test the hypothesis that the mean is 3500, I'm gonna assume that's the null hypothesis, I mean is 3500. And the alternative hypothesis Is that it is not 3500. And were asked to do this hypothesis test at a significance level of Alpha equals 0.01. Now it is a two sided hypothesis test, since the alternative hypothesis is not equal to and therefore are critical value is going to be zed sub alpha over to. So are critical values are plus and minus 2.576. Now, since we know the population standard deviation our test statistic will be, is that not? It's the sample average, modestino hypothesized mean, over the population standard deviation over the square root of the sample size And this comes out to -2 887. So this is more extreme than negative 2.576. And therefore we would reject the null hypothesis. No hypothesis. Now, for part B were asked for the smallest level of significance at which we would be willing to reject the null hypothesis, that is by definition the p value, P value is the the devil is the smallest level of significance that we'd be willing to reject the null hypothesis if the level of significance is smaller than the p value than we do not reject the null hypothesis. However, if the p value is smaller than the level of some significance, then we do reject the null hypothesis. Now the p value is the probability of getting A test statistic at least as extreme as the one We got. So it's the probability that zed is less than or equal to -2.887 plus. The probability that said is creator than or equal to 2.887 And this comes out 2.0039 next for see, we are asked for the beta era error for the testing part aid, if the true mean is 34 70 So if the Truman is 34, what is data? Remember that beta is the probability that we fail to reject the null hypothesis. When the no hypothesis is actually false for the two sided hypothesis test where the population standard deviation is known, beta is given by this formula. So we have already found this critical value, we have the population standard deviation from the question, we have the sample size from the question, delta is the difference between the true mean and the no hypothesized mean. If we plug all these numbers into our formula for beta, It is approximately zero 80. The Truman is close enough to the null hypothesized. Main That the probability of committing a type two error is. And then for part E we were asked to explain how we could answer the hypothesis testing part A With a two sided confidence interval in the mean. So the strategy we can use is It's a two sided hypothesis test At a significant level of .01 Significance level of points or one Corresponds to 99% confidence. So what we could do is making a two sided 99% confidence interval on the mean. And if 3500 is not inside that interval, then we would reject the null hypothesis. If 3500 is inside the interval, then we failed to reject the null hypothesis. So we make a 99 confidence interval on the mean. Now 99% confidence interval on the mean, when we know the population standard deviation is given by this formula and we have all of these parameters from earlier in the question. So 34 50 runs two point 576, I'm 60 over a square 12 is lower bound and the upper bound is given by the following. And when we do this we get a confidence interval for the mean ranging from 34 05.4 2, 34, 94 0.6. Now 30 500 Does not lie within the 95% confidence interval. Therefore, we reject the null hypothesis, 99% confidence interval

Mhm. In this video, let's look at constructing a 90% confidence interval for the population mean tensile strength. If we randomly selected 72 items and we found the sample mean tensile strength To be 242.2 newtons With a sample standard deviation of 70.6 newtons. So when we're looking at trying to make an estimate of the population mean with our constructing our confidence intervals, the first thing we do is find our point estimate for the population mean. And the point estimate for the population mean is the sample mean? Which here is given to us as 242.2. Now, once we have that, we look and see that we have a sample size that is 72, which is a large sample size. So we don't need to have any information about the distribution of the population when our sample size is large because by the central limit theorem that the sampling distribution of our sample means will be approximately normal with large sample sizes. So we can use our formula for finding our lower bound of our confidence interval by X bar minus T sub alpha over to times S over the square root of n. Um and then also finding our upper bound of our confidence interval by X bar plus teeth of alpha over two times S over the squared event. And x bar we have is the to 42.2 s. We have is the 70.6 And we have is the 72. So what we need to calculate before we can do the full amount with the confidence interval, is this test evolve over to now some courses are set up so the students or the person working with the information would just be allowed to use a scientific calculator and tables and charts to be able to find their critical values. Other courses are set up where the students are allowed to use a graphing calculator and you can utilize the information and the um programs within the graphing calculator to do these statistical inferential statistical methods here, I'm going to show you both ways. So if we're looking at finding this teeth of alpha over to value the first thing I want to do is find alpha, alpha is one minus the confidence level in decimal form. So for 90% confidence interval my confidence level is 90% and in decimal form that's a 900.90. And when i subtract that 1 -10 is 0.10. Now alpha over to you just actually take that alpha and specifically divided by two. So 0.10 divided by two is 0.05. Now, next with this, we also need our degrees of freedom degrees of freedom for this application is and -1. So for this particular question, my n recall is 70 two. So it's 72 -1 or my degrees of freedom is 71. Now when you look at AT chart it will skip over that um 71. So you need to find your critical value by looking across the column heads and with the student's T distribution page, the legend of it for many of them. Is that the area to the right of your critical value is what's the column heading? And then you look under the .05 for that and across from 71 if it's on the table. If not, you get to your closer value and find your teeth the buffalo over to If you have your graphing calculator that you can use, you would go second and the bars key to get two distributions, you would curse her down to inverse T. And the area in your calculator is area left of the critical value. What alpha over two is our area right of it. So area left of it is one minus that. 10.5, which would be 0.95. So it's already in there. From a previous problem 0.95 is the area that we will have. Now notice you don't put in the confidence level, You have to find your alpha divided by two and go 1- that number for the entry for your area. Now, degrees of freedom I have is 71. And then when I curse her and paste it And then push enter one more time to have it calculate it. It gives me that my degrees that my critical value to stable for over two is equal to 1.67. Now we are going to go ahead and put our values in. So the sample mean Is the 2 42.2. So we have 242.2 minus the 1.67 critical value that we found. And that's lower down here. I'm sorry, it's right here. And then times s Which in this case is 70.6. Yeah, divided by The Square Root of an N. is 72. And then close up parentheses. That's a lower bound. And then the upper bound is 242.2 then plus The 1.67 times The 70.6 divided by the square root of 72. So this amount that we're subtracting off of the point estimate and adding to the point um point estimate, that's your margin of error. So after uh ever just asked you for the margin of error you would calculate that part of it. But now going through the calculations here, you're going to get a lower bound of to 28.33 And an upper bound of to 56.07. Newton's. Now if you are allowed to use your graphing calculator, you can also find this interval without having to go through the formulas. So if you go stat button that's right underneath your delete cursor right to tests. Now we're doing confidence intervals. So we curse her down to the intervals and we have the tea is our critical value. So we do t interval push enter. We have stats not data because we've gotten the pre calculated mean and standard deviation data would be if we had the individual numbers. So we push enter on the stats Then for our sample mean that's the 242 Point to our sample. Standard deviation is 70.6, Our N is 72. Our confidence level is .90 remember you get that from what it told you to do the construct the confidence interval for cursor, down push enter to calculate and when it reports it it gives it to you as an open interval. So the lower bound is our to 28.33 and then comma the upper bound is to 56.07 like we had. So if you were to write this in a sentence you would say a 90 confidence interval four. The population mean tensile strength Yeah, Perfect. Mhm. Is between yes, 228.33 and 256.07. Newtons. Another way they might ask you to interpret it is that you're 90% confident that the population mean is somewhere between 288-28.33 and 256.07


Similar Solved Questions

4 answers
ResourcesGIA Upssignment Score: 733/1700Question 5 0f 17Suppose you havcsamples that an: equal wcight; 26.8 g V and 26.8 g Al,Os.Calculate thc number of moles of cach subslancemolmolAL,0,TOOLS
Resources GIA Up ssignment Score: 733/1700 Question 5 0f 17 Suppose you havc samples that an: equal wcight; 26.8 g V and 26.8 g Al,Os. Calculate thc number of moles of cach subslance mol mol AL,0, TOOLS...
2 answers
16. (6 points) Define T .2,R'by Tlv) = Av where A = isomorphism? Prove YOur answer .Is T an
16. (6 points) Define T .2,R'by Tlv) = Av where A = isomorphism? Prove YOur answer . Is T an...
5 answers
(1ipau? JO} waiqojd zseI 341 uo Jsnu Jamsue pue XJOM Jnok nq '3134 JaMSUE 01 paau jou op noA)D -X D~X Dz = lu![ z z*:aOd '"HIUII e JO UOQuYap eJlap uolsda a41 Bus0 :WJOJ 8u07Jlie
(1ipau? JO} waiqojd zseI 341 uo Jsnu Jamsue pue XJOM Jnok nq '3134 JaMSUE 01 paau jou op noA) D -X D~X Dz = lu![ z z* :aOd '"HIUII e JO UOQuYap eJlap uolsda a41 Bus0 :WJOJ 8u07 Jlie...
1 answers
S): Find the Fourier sine integral of the following function; xz 4X <1, f(x) = 1 < x < 2 x >2.
s): Find the Fourier sine integral of the following function; xz 4X <1, f(x) = 1 < x < 2 x >2....
5 answers
Thc average crcdit card debt for collcge seniors is S3173. If the debt is normally distributed with standard deviation of S1120. Find thesc probabilities 11) What is thc probability that rndomly selected undergraduate has credit balance less than S2700?-0.42-2.113372D) .017412) You randomly select undergraduatc_ What is the probability that their mean balance less than S2700?~0.42-2.11C) .3372D) .0174
Thc average crcdit card debt for collcge seniors is S3173. If the debt is normally distributed with standard deviation of S1120. Find thesc probabilities 11) What is thc probability that rndomly selected undergraduate has credit balance less than S2700? -0.42 -2.11 3372 D) .0174 12) You randomly sel...
2 answers
0, Let f () ={ 2,~T < I < 0 0 < I < TThen the Fourier series of f (x) isA) 1+2 2 1-(C1) cosnx n=1B) 1 + 4 2, 1 sin nxC) 1+?='64"sin nD) 1 +4 1cosnx n=1
0, Let f () ={ 2, ~T < I < 0 0 < I < T Then the Fourier series of f (x) is A) 1+2 2 1-(C1) cosnx n=1 B) 1 + 4 2, 1 sin nx C) 1+?='64"sin n D) 1 +4 1cosnx n=1...
5 answers
Which of the following reactions can evolve phosphine?(a) White $mathrm{P}+mathrm{Ca}(mathrm{OH})_{2} longrightarrow$(b) $mathrm{AlP}+mathrm{H}_{2} mathrm{O} longrightarrow$(c) $mathrm{H}_{3} mathrm{PO}_{4} stackrel{ext { Heat }}{longrightarrow}$(d) $mathrm{PH}_{4} mathrm{I}+mathrm{NaOH} longrightarrow^{-}$
Which of the following reactions can evolve phosphine? (a) White $mathrm{P}+mathrm{Ca}(mathrm{OH})_{2} longrightarrow$ (b) $mathrm{AlP}+mathrm{H}_{2} mathrm{O} longrightarrow$ (c) $mathrm{H}_{3} mathrm{PO}_{4} stackrel{ ext { Heat }}{longrightarrow}$ (d) $mathrm{PH}_{4} mathrm{I}+mathrm{NaOH} longri...
5 answers
Prove that the centroid of a parallelogram is the point of intersection of the diagonals of the parallelogram. IHint.Choose coordinates so that the vertices of the parallelograme are located at $(0,0),(0, a),(b, c), ext { and }(b, a+c) .]$
Prove that the centroid of a parallelogram is the point of intersection of the diagonals of the parallelogram. IHint.Choose coordinates so that the vertices of the parallelograme are located at $(0,0),(0, a),(b, c), \text { and }(b, a+c) .]$...
5 answers
Solve the equation by completing the square.$x^{2}+2 x-5=0$
Solve the equation by completing the square. $x^{2}+2 x-5=0$...
5 answers
Discuss the difference between Michael addition and Diels-Alderreaction,.
Discuss the difference between Michael addition and Diels-Alder reaction,....
5 answers
Which of the following 'pairs . compounds are isomers? CH3-CH2- 0-CH2-CH3 and CH2-0 - CH2 - CH3CH3CH3-CHz-C- CH3andCH3 - CH2 - CH2- CHOHCH3-C CH2-€ CH-OH and CH3 - CH- CH2-C CH3CH3D) CH3-CHz-CH2-OHandCHz-0-CH3CH:CHzOH and HC CH3
Which of the following 'pairs . compounds are isomers? CH3-CH2- 0-CH2-CH3 and CH2-0 - CH2 - CH3 CH3 CH3-CHz-C- CH3 and CH3 - CH2 - CH2- CH OH CH3-C CH2-€ CH-OH and CH3 - CH- CH2-C CH3 CH3 D) CH3-CHz-CH2-OH and CHz-0-CH3 CH:CHzOH and HC CH3...
5 answers
2;59Procedure Part € Data TableDownload JfroMass Pcrccnt and Molarity of Sodlum Chloride In a Salnnic S0lulionLullot imWeeennt CilhundeltareeM4is 6 51220m407di IrM0l EIttetA ul {L Muit pln routr duadoiandlduNA-i
2;59 Procedure Part € Data Table Download Jfro Mass Pcrccnt and Molarity of Sodlum Chloride In a Salnnic S0lulion Lullot im Weeennt Cilhundeltaree M4is 6 51220m407di Ir M0l EIttetA ul {L Muit pln routr duadoiandld uNA-i...
5 answers
What is general principle of microbial pathogenesis?
What is general principle of microbial pathogenesis?...

-- 0.022012--