b) Two dice are thrown. Let X assign to each outcome (a, b) the maximum of the two numbers appearing on the dice. Obtain the probability distribution of X. Find the mean and variance of the distribution.
(OR)
a) Define probability mass function and probability density function. Give one example of each.
b) Let X be a continuous random variable with probability density function . Find the value of k and determine P(1<X<2).
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UNIT - II
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a) Find moment generating function of binomial distribution and hence find its mean and variance.
b) Fit a Poisson distribution to the following data and test the goodness of fit at 5% level of significance.
X 0 1 2 3 4 f 123 59 14 3 1 (OR)
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a) Define normal distribution and explain its properties.
b) In a distribution exactly normal, 7% of the items are under 35 and 89% are under 63. What are the mean and standard deviation of the distribution?
UNIT - III
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a) Define the terms population, sample, parameter and statistic. Explain the types of sampling.
b) A random sample of size 100 is taken from a population with σ = 8. Given that the sample mean is x̄ = 82, construct a 95% confidence interval for the population mean μ.
(OR)
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a) What is the procedure generally followed in testing of hypothesis?
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b) The means of two large samples of sizes 1000 and 2000 members are 67.5 inches and 68.0 inches respectively. Can the samples be regarded as drawn from the same population of standard deviation 2.5 inches?
UNIT - IV
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a) Explain types of errors in testing of hypothesis.
b) The following are the average weekly losses of worker hours due to accidents in 10 industrial plants before and after a certain safety program was put into operation:
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Before 45 73 46 124 33 57 83 34 26 17 After 36 60 40 119 35 51 77 29 24 11 Use a 0.05 level of significance to test whether the safety program is effective in reducing the average weekly losses of worker hours due to accidents.
(OR)
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a) Explain χ2 test for goodness of fit.
b) The number of automobile accidents per week in a certain city were as follows: 12, 8, 20, 14, 10, 15, 12, 6, 14, 12. Are these frequencies in agreement with the belief that accident conditions were the same during this 10 week period?
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UNIT - V
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a) Explain the terms correlation and regression.
b) The following data is obtained concerning two variables X and Y:
ΣX = 120, ΣY = 450, ΣX2 = 1660, ΣY2 = 17100, ΣXY = 6300, n = 10
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Find the equations of the regression lines. Also find the correlation coefficient between the variables X and Y.
(OR)
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a) Explain the terms time series and forecasting.
b) Fit a straight line trend to the following data.
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Year 2015 2016 2017 2018 2019 2020 2021 Production (in tons) 70 75 69 83 92 87 98 Also estimate the production for the year 2024.
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