During testing, she always hangs a sign on the door communicating that she does not wish to be disturbed. She reads the instructions verbatim and asks her students if they have any questions prior tc beginning the test. She tries to keep the testing situation as low-key as possible for her students. To this end, she generally plays music quietly while the students are taking the tests. Test results are routinely distributed to parents with student report cards. Parent-teacher conferences are conducted in the week following distribution. Parents inevitably have many questions about standardized test scores, such as: ? M. Carter, what does this grade equivalent mean? It says here that Emily has a grade equivalent of 4.3. Does that mean we should ask to have her placed in fourth grade?" ? "Ms. Carter, John scored at the 90th percentile on language, but he's getting Cs in your class. I just don't understand." ? "Ms. Carter, how can it be that my daughter scored in the 60th percentile on the ability test, the 70th on the achievement test, and doesn't meet state standards in math?" ? "Ms. Carter, how can my son score at the 40th percentile on the ability test and the 80th percentile on the achievement test? That just doesn't make any sense!" 1. What are the issues regarding standardized tests in this case study? 2. Examine Ms. Carter's testing procedures. What does she do incorrectly? How might this reduce the validity of the students' scores? 3. How would you answer each of the parents questions?

4. What does scoring at the 90' percentile on the language portion of the standardized test mean? A. The student got 90% of the items correct. B. The student scored as well or better than 90% of the students in the norm group. C. The student scored as well as or worse than 90% of the students in the norm group. D. The student scored as well or better than 90% of the students in his class. 5. If a student scored at the 70" percentile on the math portion of an achievement test, and doesn't meet state standards in math, which of the following would be true? A. The achievement test has poor predictive validity for performance on the state math test. B. The achievement test has good predictive validity for performance on the state math test. C. The achievement test is unreliable D. The state math test is easier than the achievement testQuestion 1 .a] What is Baye's theorem? What is its practical significance in risk probabilistic risk assessment and management? stp . b) A safety system consists of3 monitors. A plant abnormal state requiring shutdown occurs with probability of 3.2. Ifthe safety system fails to shutdown, 3113,1303 is lost. Each spurious {or false} shutdown costs S4000. Determine the optimal moutofthree safety system using as data: [tZPrimonitor shutdown failure | abnormal plant} = CLIEIl Pr[monitor spurious signal | normal plant] 0.05 Etylssume statistically independent failures. lab} Question 2 A machine to detect improper welds in a fabricating shop detects 80% of all improper welds but it also incorrectly indicates an improper weld on 5% of satisfactory welds. Past experience indicates that 13% of all welds are improper. What is the probability that a weld which the machine indicates to be defective is in fact satisfactory? and i would like you to solve these three question from this textbook "Probabilistic Risk ssessment and Management for Engineers and Scientists 2nd Edition, by Hiromitsu Kumamoto and Ernest J. Henley." and here is the three questions Problem 4.2, page 223 of textbook 2. Problem 4.3, page 224 of textbook Two pairs of functions y1 and y2 are given as follows: (1). y1(t) = t, y2(t) = te^(2t) on (-00,20) (2). y1(x) = cos^2 x, y2(x) = 1 + cos(2x) on [0, nt/2) Use the following 2 approaches to determine whether y1 and y2 are linearly independent: 1. by definition (i.e., definition 1 of the Textbook, page 161) 2. by computing the Wronskian W(y1, y2)(t). (Answer: 1st pair linearly independent, and 2nd one linearly dependent ) page 161 uploaded belowSection 4.2 Homogeneous Linear Equations: The General Solution 161 Existence and Uniqueness: Homogeneous Case Theorem 1. For any real numbers a(#0), b, c. fo. K. and Yi, there exists a unique solu- tion to the initial value problem (10) ay" + by' + cy = 0: y() = Yo. y' (ra) = 1, . The solution is valid for all r in (=co. +co). Note in particular that if a solution y(t) and its derivative vanish simultaneously at a point to (ie., Yo = Y, = 0), then y(t) must be the identically zero solution. In this section and the next, we will construct explicit solutions to (10), so the question of existence of a solution is not really an issue. It is extremely valuable to know, however, that the solution is unique. The proof of uniqueness is rather different from anything else in this chapter, so we defer it to Chapter 13. Now we want to use this theorem to show that, given two solutions y, () and ya() to equa- tion (2), we can always find values of c, and cy so that con() + czyz() meets specified initial conditions in (10) and therefore is the (unique) solution to the initial value problem, But we need to be a little more precise; if, for example, y,(1) is simply the identically zero solution, then cu() + czyz() = co() actually has only one constant and cannot be expected to sat- isfy two conditions. Furthermore, if yz(f) is simply a constant multiple of y,()-say. yz(1) = ky, (1)-then again coy(1) + ezz(1) = (e, + kez)y(1) = Cy,() actually has only one constant. The condition we need is linear independence. Linear Independence of Two Functions Definition 1. A pair of functions y, () and yz(1) is said to be linearly independent on the interval / if and only if neither of them is a constant multiple of the other on all of 1." We say that y, and ya are linearly dependent on / if one of them is a constant multiple of the other on all of I.1 2. Max is trying to develop a more reliable method for detecting possible developmental disorders in two-year-old children, Currently, many children are misdiagnosed at this age and many children who go on to develop severe problems are not identified at an early enough age to make intervention effective. To make progress on this task, Max first identifies five distinct behavioral types: A, B, C. D, and E. He writes a manual to tell observers how to determine which type a particular child exemplifies. He then asks two research assistants to help him classify each of 100 children at a clinic according to the type of problem the child seems to display. Max would like to know how to proceed. Please answer each of the following questions. What type of reliability statistic should Max compute to see if his two research assistants can apply his scheme for classifying developmental disorders in a reliable fashion? b How should Max set up an SPSS data file in Max should created columns (r than ifhe is order to get this statistic from SPSS? (Be sure to discuss what the data file will look like in terms also dassitying the kick ). He will have 100 rows, of columns and rows and what data will be with the identified behavioral type of each stored in each cell of the data file.) chronter coded to represent A-E (15). Each observer will gour their own rating which can them be used to test for inter- roter ritiability. This helps measure the level of agreement between raters, which is beefed in store home welf the meatpies What steps should Max follow in SPSS in order Analyze > Descriptive Statities > crosstabs > to get the statistic he needs if he wants to report inter-rater reliability? Stat miss kappa What value of the statistic would be considered Mature of agreement - Kappa acceptable as a measure of inter-rater reliability? 13. The U.S. Olympic Committee wants to know whether the 6 judges for the figure skating competition give consistent (i.c., similar) ratings of the skaters' artistic merit. Each judge gives each skater a score between 1 and 10 for their performance, I indicates very low performance whereas a 10 indicates very high performance. Which statistic should the Olympic Committee use to measure the reliability across the 6 judges? Explain your choice. 14. Emic and Bert are developing a measure of Open-mindedness. They plan to use a questionnaire with 15 questions cach of which will ask about open-mindedness in a somewhat different way. Examples of the questions are "I like to keep my mind open about new foods until I taste them on at least three occasions" and "I will read a book even if the cover does not look very interesting." Participants will circle a number from 1=This statement is never true of me to 5-This statement is always true of me for each of the 15 questions. Which statistic should Ernie and Bert use to2. Let n = 10 and now consider A to be a discrete uniform random variable on 1, 2,3. .. ., 10. (3 points each) (a) Use R to graph the probability histogram for the distribution of X. You can use the function that I have posted, or write your own code. Use some other color for the probability histogram. Does it look as you expect? (b) Make a box of the integers 1 through 10. Now sample 1000 times (with replacement) from this box and plot the values in a his- togram. Since this is a histogram obtained by sampling, make it a different color than the one you had earlier for the theoretical probability histogram. (c) Does this histogram look as you expect? If not, check and adjust the breaks of the histogram. Remember, it shouldn't look too different from the probability histogram that you plotted in (a). (d) Now, we are going to look at the average of the draws. First, you are going to sample 5 times from the box and compute the average of your draws. Then repeat this process 50 times and plot the histogram of these 50 numbers, where each number is the average of 5 draws. You will find the command replicate useful here. Comment on this histogram. Does it look like the histograms in parts (a) and (b)? Does it look as you expected? (e) Again, record the mean of 5 draws with replacement, and repeat this process 1000 times. Plot the histograms of these 1000 means. (f) Do the same process, but record the average of 100 draws, and repeat the process 50 times. Plot the histogram of these 50 values. (g) Repeat part (f) - record the average of 100 draws and repeat this 1000 times. Plot the histogram of these averages.WE. Carter is a third grade teacher in a district that uses more than one Standardized test to measure student achievement and ability. It uses the test required by its state to assess the degree to which students have met or exceeded state standards in math, science, reading, writing and social sciences. This test yields individual, school, and district scores, and compares. these to state averages. In addition, the district uses a nationally normed test to assess both achievement and cognitive ability. The achievement test yields individual scores as they relate to national norms. These scores are reported as percentile rank scores and grade equivalent scores. The cognitive ability test yields percentile rank scores and IGtype scores. In addition, the testing service provides a narrative that discusses and compares the achievement and cognitive ability scores for each student. i'vis. Carter is not thrilled about giving her students so many standardized tests. She says, "Sometimes it seems all we do is prepare forthese tests and take them." She makes sure she has taught her students appropriate testtaking strategies. She also tries to give her students some experiences that mirror the standardized testssuch as filling in bubbles on answer sheets, and having limited time in which to complete tests. She also sends notes home to parents asking them to ensure that their children get adequate sleep and eat breakfast during testing weeks. During testing, she always hangs a sign on the door communicating that she does not wish to be disturbed. She reads the instructions verbatim and asks her students if they have any questions prior to beginning the test. She tries to keep the testing situation as low-key as possible for her students. To this end, she generally plays music quietly while the students are taking the tests. Test results are routinely distributed to parents with student report cards. Parent-teacher conferences are conducted in the week following distribution. Parents inevitably have many questions about standardized test scores, such as: . "Ms. Carter, what does this grade equivalent mean? It says here that Emily has a grade equivalent of 4.3. Does that mean we should ask to have her placed in fourth grade?" . "Ms. Carter, John scored at the 90' percentile on language, but he's getting Cs in your class. I just don't understand." . "Ms. Carter, how can it be that my daughter scored in the 60" percentile on the ability test, the 70" on the achievement test, and doesn't meet state standards in math?" . "Ms. Carter, how can my son score at the 40' percentile on the ability test and the 80th percentile on the achievement test? That just doesn't make any sense!" 1. What are the issues regarding standardized tests in this case study? 2. Examine Ms. Carter's testing procedures. What does she do incorrectly? How might this reduce the validity of the students' scores? 3. How would you answer each of the parents' questions