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Practice exam 1

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David Wright

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ADM2303 Final Exam 2005 Fall Semester Q1. Because cheating has been on the increase, the Rector has set up a special investigation. As part of this, a private investigation company has been engaged, and they arrange for one of their agents to "register" for a particular statistics course. The agent finds out that a group of 10 students met at a bar and bought a copy of the mid-term quiz and its solution in advance. Unfortunately for them, the professor who wrote the exam made a rather unusual mistake in the solution to question 2 of the quiz, so it is pretty easy to spot who was cheating if their answer matches this mistake. There are 65 students registered for the course. 4 a) [ 4 ] If the marker checks 5 papers as a sample, what is the chance none of them belongs to one of the cheaters? Be sure to name any methods or models you use and show how you get your answer. b) In a separate investigation, professors belonging to the activist group Cascade Rounding Angers Professors want to correct the improper practice of "Cascade Rounding" where 1.23456 is rounded to 1.24 by first rounding to 1.2346, then 1.235, and finally 1.24. (THIS IS WRONG, for those who do it The correct answer is 1.23). First, they want to know how many students do this, so set up a calculation where the mean of seventeen numbers is $1.23456, then ask for it to be rounded to the nearest cent. The believe fully 12% of students are CR users. 4 1b1) [ 2 ] If they sample 7 papers from a very large collection of exams, what is the probability no student cascade rounds? Be sure to name any methods or models you use and show how you get your answer. 3 1b2) [ 3 ] If they sample 7 papers from a very large collection of exams, what is the probability at least 2 students of the seven cascade rounds? 5 1c) [ 5 ] A nationwide study is conducted on cascade rounding and there are many thousands of high school graduates asked to round 1.23456 to 2 decimals. What is the probability that a random sample of 198 such graduates returns less than 12 "cascade rounders"? Bonus Question: 2 1d) [ 2 ] Suppose you had conducted the study in ( c ) only for the U of Ottawa where 546 students were in the population. What happens to your answer in ( c )? Specify your answer with the new value of the resulting probability. Q2. BuzOff is a cosmetics maker that gets its revenue from specialty products that combine cosmetics with insecticides and insect repellents. In particular, their two key products are a lice-killing shampoo (Niet!) and a mosquito-repellent skin cream. (Zeroz). Due to the specialty nature of their business, Buzoff gets mixed prices for their products. However, they have fairly good evidence that the average profit per case of Niet is $23.45 with standard deviation $1.29, and for Zeroz is $31.23 with standard deviation $2.45. Because Niet tends to sell better during schooltime when children take too literally their parents' advice to share (at least where lice are concerned), while Zeroz gets sold mainly in the summertime when mosquitos are active, the correlation between profits of these two products is -0.4. mu(N)= 23.45 sigma(N) = 1.29 mu(Z)= 31.23 sigma(Z)= 2.45 4 2a) [ 4 ] If Buzoff sells 1225 cases of Niet and 3543 cases of Zeros, what is the coefficient of variation of the resulting profit? 2 2b) [ 2 ] Suppose the profit on a case of Zeroz is Gaussian (Normally) distributed. What is the probability a case earns a profit of more than $35? 2 2c) [ 2 ] Suppose the profit on a case of Niet is exponentially distributed with mean $23.45. What is the probability the profit on a case is less than $20? 2 2d) [ 2 ] Why does the information about Niet in part ( c ) contradict information in the introduction to this question?(Hint: Compare basic parameters.) Q3. Blotz-a-Lotz makes ink cartridges for printers. The cartridges have to satisfy many specifications, but the label says "50 ml". Blotz decides that it is unlikely consumers can measure to better than 1ml, so specifies that the cartridges must contain between 49 and 51 ml.of ink. Note that the specification refers to measurements on individual cartridges rather than their means. 2 3a) [ 2 ] What is a reasonable value to use for the standard deviation of "fill" (i.e., the number of ml. of ink in the cartridge) in order to meet the "specification"? Explain briefly. 2 3b) [ 2 ] Blotz looks at the histograms of ink content for each batch of 475 from 2 machines. Figures 3-2 present the histograms and Figures 3-3 are the corresponding normal probability plots for these two machines labelled AA and DD.. Can you presume that ink fill is Gaussian distributed? Justify your answer BRIEFLY. 3c) Two Xbar/s charts are given for each of these two machines using samples of 5 cartridges at a time. The commands that drew them are given as well. 2 i)State which charts (by name of chart) you would use and why. 2 ii)Then explain whether Blotz can use these machines for filling cartridges AA has too much variability -- uncontrollable DD is OK on variability -- controllable, but level too low (out of control) 1 + 1 MUST mention variability or controllability properly and if so, 1 iii) how they need to proceed to get into production. Use DD but adjust fill level. 5 3d) For a particular machine, Blotz is convinced that ink fill is Gaussian distributed. However, the colour and consistency of the ink affects the variability of fill. If it is assumed that the mean fill over the whole production is 49.6 ml., what is the probability, possibly approximate, of observing the following sample or one with a mean as or more different from the given production value. 49.2, 49.9, 49.7, 50.1, 50.2, 49.8, 49.6 (ml.) 2 3e) If the standard deviation of fill in (d) is reliably known to be .35 ml over production in general, work out the probability requested in part (e). Q 4. Electonic communication errors are considered to happen at a rate per time. Combined with the transmission capacity of the line, this can be translated into an error rate per volume of data. On your Internet connection, it therefore turns out that data-transmission errors happen at the rate of 1 error per 350,000 kBytes of data transmitted. (If you know about such things, please ignore the fact that we usually measure in bits rather than bytes.) You want to transfer three large files. File: Size in kBytes AA: 600,000 600000 BB: 60,000 60000 CC: 120,000 120000 2 4a) [ 2 ] What model should you use for computing probabilities of errors in transmission. Justify briefly. 4 4b) [ 4 ] What are the probabilities of transfering each of the three files separately without error? 4 4c) [ 4 ] What is the probability exactly one of the three files has an error (i.e., at least one error) when transmitted separately and in the order given? 2 4d) [ 2 ] What is the probability the transfer of file AA occurs with two or more errors? Q 5. Bankruptcy is a s
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