Exam for the course Statistics for scientists. Thursday, January 19,

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1 STOCKHOLM UNIVERSITY EXAMINATION DEPARTMENT OF MATHEMATICS Statistics for scientists Div. of Mathematical Statistics Thursday, January 19, 2006 Exam for the course Statistics for scientists Thursday, January 19, Examinator: Anders Björkström, tel , You are permitted to use: Levine, Ramsey & Smidt: Applied Statistics for Engineers and Scientists. Your own lecture notes. Calculator. Note that some additional information is provided on the appendix Computer output... Solutions (in Swedish) will be available at from 2 p.m. today. Marked-up exams will be handed back: Thursday January 26, 2006 at 10 a.m. Room 312, house 6. If you want to know your result earlier, please write your address on the exam cover. Requirements: To pass the exam, 8 points are required (including those from the assignments). To pass with honors, the requirement is 8 points (not including points from the assignments). Do not treat more than one of the two alternative versions of Problem 5. Problem 1 The following are the weight losses of certain machine parts (in milligrams) due to friction, when three different lubricants were used under controlled conditions: Lubricant A: Lubricant B: Lubricant C: These data are illustrated graphically in the Appendix. a) Perform an analysis of variance and explore at 5 % significance level whether the lubricants are equivalent. (1 p)

2 Statistics for scientists, Thursday, January 19, b) What conditions must be fulfilled in order for this test to be justified? Comment as to whether the given data are satisfactory from this point of view. (1 p) Problem 2 The following are measurements of specific conductivity in a river in Virginia. Date Conductivity The average of the observations is 279,2 and the standard deviation is 69,9. The hydrologists believe that the long-term conductivity is a constant, and that any variation can be regarded as random deviations from this constant long-term average. A stubborn conservative professor wants to cling to the viewpoint that the long-term average is 350, since he read this in a textbook in his youth. a) Calculate a symmetric 95 % confidence interval for the average, assuming that fluctuations are independent and Gaussian. Is the professor s standpoint reasonable? (1 p) b) A conceivable objection against the approach in (a) is that data may not be Gaussian. The professor might argue that the population, although not Gaussian, has its median at 350 units. Show, using the given data, that this is hardly a tenable opinion. (Hint: Use the definition of median and show that the data are unlikely if the median actually is 350.) (1 p) Problem 3 A certain type of instrument for atmospheric chemistry measurements is designed to be posted in the open air at remote locations, operating without service for many weeks. Instruments of this type come in two different varieties, Alfa and Beta, and the researches want to explore whether these have on average equally long life span. To that end, five units of type Alfa

3 Statistics for scientists, Thursday, January 19, and five of type Beta are posted in the garden outside the Department, and their lifetime is measured. To save time, the chemists decide to stop the experiment when five of the ten units have ceased to operate. The result is that all five Alfa-units are first to give up. Undeniably, this is quite a strong counterargument against the hypothesis that the two types are equal. What is the p-value, if we assume a two-sided alternative hypothesis? (2 p) Problem 4 A group of hydrologists plan to start monitoring the flow rate in a river. Measuring flow rate (cubic meters per second) is complicated, so they consider the possibility to replace flow rate measurements with water-level measurements (meters), which can easily be obtained, reading off a graded stick. Ten careful measurements are taken in order to explore the relation between flow rate and water-level. The figures in the Appendix show the result. Figure 1 shows Flow rate versus water depth, figure 2 shows log(flow rate) versus log(depth). (The abbreviation Std Error means the same as Standard Error does in Excel and Minitab outputs). a) None of the figures seems to be perfect for a simple linear regression model. State one or a few weaknesses with each of them. (2 p) b) Calculate a 95% confidence interval for the slope of the regression line, for the model you prefer. (1 p) Problem 5, alternative I An experiment is carried out to explore how four factors A, B, C and D affect the outcome of a process. Each factor appears at two level, called plus and minus. For economic reasons, only the following eight of the sixteen possible experiments were carried out: A B C D

4 Statistics for scientists, Thursday, January 19, a) Is it possible to estimate the interaction effect AB? Alternatively, determine to which other interaction effect AB will be coupled. (1 p) b) Neglecting the D column and calculating as if the experiment had been a complete 2 3 -factorial experiment in the factors A, B, C, the researchers obtained the following effects: Effect Estimate A 5.3 B 8.1 C 3.3 AB 1.0 AC 2.2 BC 0.8 ABC 1.9 Assuming that high order interactions are negligible, compute an estimate of the main effect of D. (1 p) c) Suppose that is is necessary to split up the series of experiments between two days (Day 1 and Day 2) and we want to eliminate a possible confounding effect of the blocking factor Day (we assume that this factor does not interact with the other factors). Suggest a plan for the experiments, such that it would still be possible to estimate the main effects for A, B, C och D. Write your answer as a sequence of 1:s and 2:s, saying on which day each of the above eight experiments 1-8 should be carried out. Motivate. Problem 5, alternative II a) A laboratory produces devices that are supposed to become superconducting when cooled. Two production methods, called Beta and Gamma, are to be compared. Fifty devices are produced with each method, and it is examined how many of them actually become superconducting when cooled. The results are: Beta Gamma Superconductors Failures 8 28 Sum Test the hypothesis that method Beta has he same failure risk as method Gamma. (1 p)

5 Statistics for scientists, Thursday, January 19, b) Another laboratory has compared four production methods, Alfa, Beta, Gamma and Delta. Their results are: Alfa Beta Gamma Delta Superconductors Failures Sum Test the hypothesis that the failure risk is the same for all four methods. (2 p) Good luck!

6 STOCKHOLMS UNIVERSITET MATEMATISKA INSTITUTIONEN Avd. matematisk statistik Anders Björkström Computer output and other supplementary information for some of the problems (Bilaga till tentamen i Statistik för naturvetare ) Uppgift 1 Viktminskningen för de tre olika oljetyperna kan illustreras såhär: Loss of weight by oil type: Medelvärden och standardavvikelser är: Typ av smörjolja Oil type Medelvärde Sample average Standardavvikelse Sample standard deviation A (n=8) 9,35 3,59 B (n=8) 11,42 3,09 C (n=8) 16,74 4,16 Alla sammanslagna (n=24) 12,50 4,71

7 Uppgift 4 I figur 1 visas Flöde mot Djup, i figur 2 visas log(flöde) mot log(djup). Figur 1: Flux versus Depth 8 Plot 6 Flöde ,2 0,3 0,4 0,5 0,6 0,7 0,8 Djup Flöde Några beräkningsresultat för modellen Flöde mot Djup::

8 Figur 2: log(flux) versus log(depth) 2 Plot 1 Log(Flöde) ,50-1,25-1,00-0,75-0,50-0,25 Log(Djup) Log(Flöde) Några beräkningsresultat för modellen log(flöde) mot log(djup)::

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Lycka till! Avd. Matematisk statistik TENTAMEN I SF294 SANNOLIKHETSTEORI/EXAM IN SF294 PROBABILITY THE- ORY MONDAY THE 14 T H OF JANUARY 28 14. p.m. 19. p.m. Examinator : Timo Koski, tel. 79 71 34, e-post: timo@math.kth.se