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SECTION–I
Q. 2
(a)What is meant by a frequency distribution? Describe briefly the main steps in the preparation of a frequency table from raw data. [6]
(b)A man travels from A to B at average speed of 30 miles per hour and returns from B to A along the same route at an average speed of 60 miles per hour. Find the average speed of the entire journey. [6]
(c)Define mean-deviation and its co-efficient. Discuss its advantages and uses. Estimate the mean deviation from the arithmetic mean of the following set of examination marks. Marks 0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 No. of students 2 3 8 24 27 40 11 5 [8]
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Q. 3
(a)Define mutually exclusive events. State and prove the theorem of addition of probabilities concerning mutually exclusive events. [6]
(b)Show that the multiplication law P(A ∩ B)=P(A|B)P(B), established for two events, may be generalized to three events as follows; P(A ∩ B ∩ C)=P(A|B ∩ C) P(B|C) P(C) [6]
(c)There are three coins, identical in appearance, one of which is ideal and the other two biased with probabilities 1/3 and 2/3 respectively for a head. One coin is taken at random and tossed twice. If a head appears both the times, what is the probability that the ideal coin was chosen? [8]
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Q. 4
(a)(i) Explain briefly how the principle of least squares is used to find a regression line based on a sample of size n. Illustrate on a rough sketch the distance whose squares are minimized, taking care to distinguish the dependent and independent variables. (ii) Find the least square estimates of parameters in a simple linear regression model Yᵢ=α+βXᵢ+eᵢ where eᵢ s are distributed independently with mean zero and constant variance. (iii) What are the properties of least square regression line? [6]
(b)The following means, standard deviations and correlations are found for X₁ = Seed-hay crops in owts. Per acre X₂ = Spring rainfall in inches X₃ = Accumulated temperature above 42° F in spring in a certain district in England during 20 years. X ˉ 1 = 28.02 , S 1 = 4.42 , r 12 = 0.80 , \bar{X}_{1} = 28.02,\quad S_{1} = 4.42,\quad r_{12} = 0.80, X ˉ 1 = 28.02 , S 1 = 4.42 , r 12 = 0.80 , X ˉ 2 = 4.91 , S 2 = 1.10 , r 13 = − 0.40 , \bar{X}_{2} = 4.91,\quad S_{2} = 1.10,\quad r_{13} = -0.40, X ˉ 2 = 4.91 , S 2 = 1.10 , r 13 = − 0.40 , X ˉ 3 = 594 , S 3 = 85 , r 23 = − 0.56 , \bar{X}_{3} = 594,\quad S_{3} = 85,\quad r_{23} = -0.56, X ˉ 3 = 594 , S 3 = 85 , r 23 = − 0.56 , Find the partial correlation and the regression equation for hay-crop on spring rainfall and accumulated temperature. [14]
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Section II
Q. 5
(a)Explain what you understand by the probability sampling and non probability sampling. What are their relative advantages and disadvantages? [6]
(b)What is a sampling distribution? Describe the properties of the sampling distribution of the means. [6]
(c)A finite population consists of the numbers 2, 4 and 6. Form a sampling distribution of sample mean, when random samples of size 4 is drawn with replacement. Also verify its properties. [8]
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Q. 6
(a)Under what condition is the sampling distribution of σ2s2 an F-distribution? Explain the relationship between the F and t distributions, between the F and Chi-Square distributions. [6]
(b)The proportion of families buying milk from company A in a certain city is believed to be p=0.6. If a random sample of 10 families shows that 3 or less buy milk from company A, we shall reject the hypothesis that p=0.6 in favour of the alternative p<0.6. Evaluate α if p=0.6, evaluate β for the alternatives p=0.3, p=0.4 and p=0.5. [6]
(c)Define a Chi-square random variable and its density function. Discuss the important properties of Chi-square distribution. Show that the Chi-square distribution tends to normal distribution for large degrees of freedom. [8]
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Q. 7
(a)Describe the Randomized Complete Block Design, its model and analysis. What are its advantages and disadvantages? [6]
(b)Compare Randomized Complete Block experiments with Completely Randomized experiments, comparing their respective advantages and relative efficiency, with illustrations. [6]
(c)Three varieties A, B and C of a crop are tested in a randomized block design with four replications, the layout being given below. The plot yields in pounds are also indicated therein. Analyze the experimental yields and state your conclusions. Replications 1 2 3 4 Variety A 32.1 17.0 40.8 26.8 Variety B 31.7 32.7 25.3 47.9 Variety C 34.2 30.7 48.2 59.6 [8]
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Q. 8
(a)Define gross and net production rates. Explain how would you compute the net production rate and what interpretations can be made if it is 1, less than 1 or greater than 1. [6]
(b)Explain with suitable illustrations the object of standardizing various vital statistics relating to births, deaths and marriages. [6]
(c)Compute the gross and net reproduction rates for the following data: Age-group (years) Female Population (000) Female births Probability of survival 15-19 1558 18900 0.914 20-24 1112 71100 0.899 25-29 1595 96900 0.884 30-34 1629 64200 0.868 35-39 1627 34900 0.852 40-44 1522 10800 0.834 45-49 1401 800 0.813 [8]
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