Sma2110:Real Analysis Ii Question Paper
Sma2110:Real Analysis Ii
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2013
QUESTION ONE (30 MARKS)
a) What is a design experiment? (2 Marks) b) State and explain the three basic principles of an experimental design. (3 Marks) c) Complete the following table for the analysis of variance and expected mean average of a fixed effects randomized block design. To test the hypothesis that i. The treatment effects and ii. The block effects are equal to zero use = 0.05 level of significance and explain all your steps. (4 Marks)
Source of variation
Sum of squares
Degrees of freedom
Mean of squares
f-value
blocks 0.05 3 ? ? Treatment ? 9 0.0194 ? Error 0.0048 ? ? Total 0.2294 39
d) For the model
= + + +
= 1,2, = 1,2,
Where is ith treatment effect (Random) is the jth block effect (Random) ~ Nidd 0, 2 Show that under the null hypothesis, mean sum of squares due to blocking is unbiased estimator of variation due to error 2. (8 Marks)
2
e) To study the performance of 3 different detergents at 3 water temperatures three loads of washing at each combination of detergents and water temperatures were tested. The following results were obtained. Detergent A B C Water: Cold 45,39,46 43,46,41 55,48, 53 Warm 37,32,43 40,37,46 56, 51, 53 Hot 42,42,46 44,45,38 46, 49, 42 i. Construct the ANOVA table. (3 Marks) ii. Use the level of significance = 0.01 to test (i) The difference between the detergents (2 Marks) (ii) Difference due to water temperatures. (2 Marks) (iii)Interactions (2 Marks)
QUESTION TWO (20 MARKS)
a) Consider the following Latin square design Driver 25 35 50 60 70 1 B(19.5) E(21.7) A(18.1) D(14.8) C(13.7) 2 D(16.2) B(19.0) C(16.3) A(17.9) E(17.5) 3 A(20.6) D(16.5) E(19.5) C(15.2) B14.1) 4 E(22.5) C(18.5) D915.7) B(16.7) A(16.0) 5 C(20.5) A(18.5) B(15.6) E(18.7) D(12.7)
A,B,C,D and E denotes 5 makes of cars. Do the data provide sufficient evidence to indicate a difference in i. Speed in miles/hour ii. Drivers iii. Makes of the car
Use = 0.05 level of significance. (9 Marks)
b) A development engineer is interested in determining if varying the cotton content in a synthetic fiber effects the tensile strength. He runs a completely randomized experiment with five levels of cotton percentage and five replicates. The data is tabulated as below Observed tensile strength (lbs/in2)
Percentage of cotton
15 7 7 15 11 9 20 12 17 12 18 18 25 14 18 18 19 19 30 19 25 22 19 23 35 7 10 11 15 11
Analyse the data and draw conclusions using = 0.01 level of significance. (6 Marks)
3
c) Describe i. The fixed effects model (2 Marks) ii. The random effect model (2 Marks) iii. The mixed effect model (1 Mark)
QUESTION THREE (20 MARKS)
a) Describe the following designs i. Latin square design (3 Marks) ii. Graeco latin square design. (3 Marks) b) The following data is obtained from a farm where the harvest of certain maize seeds was analyzed after applying different types of fertilizers. Letting tij to denote the ith fertilizer applied to block j with their corresponding yield effect per plot. Analyze the data using = 0.05 given that each treatment is replicated four times 179(114) 71(1222) 52(1323) 65(1424) 107(1223) 122(1124) 59(1421) 147(1322) 31(1324) 95(1423) 105(1122) 99(1221) 164(1422) 73(1321) 62(1224) 102(1123) (6 Marks) c) A plant manager has decided to run an experiment to determine whether the material received at different times has the same tensile strength. Five randomly driven time periods are to be considered and five randomly driven men work on the material.
Table below results of the expt Men Time 1 2 3 4 5 1 7.5 11.8 17.6 8.8 17.9 2 21.4 12.9 12.4 15.0 20.6 3 16 9.7 7.4 18.4 16.6 4 16 18.3 23.6 27.4 25.2 5 23.3 25.8 25.8 24.4 26.6
i. Test at 1% for homogeneity of time periods and that of men. (3 Marks) ii. Estimate the time variability (3 Marks) iii. What is the efficiency of the design compared with a CRD? (2 Marks)
QUESTION FOUR (20 MARKS)
a) In an experiment carried out to determine the effects of nitrogen (N), phosphorous (P) and potassium (K) on potato yields with four replicates, the treatments totals were given as follows 1 2 3 4 I 101 106 87 181 N 106 89 128 103 K 265 272 279 302 NK 291 306 334 272
4
P 312 324 324 361 NP 373 338 324 361 KP 391 407 423 425 NPK 450 449 421 437
i. Calculate the main effects (4 Marks) ii. Calculate the interaction effects. (3 Marks) iii. Which effects are significant on potato yields (use = 0.05. (2 Marks) b) An experiment was conducted to compare the effects of v=4 chemical substance on the skin of male rats. The area of experimentation on animal skin was confined to a region which was known to be relatively homogerous but this restricted the experimenter to 2 experimental units per animal. Hence to estimate the rat to rat variability for two comparison of treatments the experiment was divided into blocks on rats using the BIBD shown below. The blocks correspond to 6 rats. Do the data present sufficient evidence to indicate a real difference in the effect of chemical substance on the skin? Use = 0.05 level of significance. (10 Marks)
QUESTION FIVE (20 MARKS)
a) Define the following terms as used in design of experiments. i. Experimental unit (2 Marks) ii. Treatmentus (2 Marks) iii. Block (2 Marks) b) An agricultural station conducted an experiment to determine whether there was any difference in the yield of 5 variates of maize. The design adopted was 5 randomized block of 5 plots each. The yield in ha obtained in the experiment were Variates Blocks 1 2 3 4 5 Total 1 30 23 34 35 20 142 2 39 22 28 25 28 142 3 56 43 43 31 49 222 4 38 45 36 35 32 186 5 44 51 23 58 40 216 Total 207 184 164 184 169
i. Analyze the design and comment on your findings. (8 Marks) ii. Obtain the efficiency of this design relative to the layour of CRD. (6 Marks)
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