JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II PROBABILITY AND STATISTICS Previous Question Paper (July/August - 2021 - R13)
Answer any five questions. All questions carry equal marks.
Main
A manufacturer of pins knows that 2% of his product is defective. If he sells pins in boxes of 100 and guarantees that not more than 4 pins will be defective. What is the probability that a box will fail to meet the guaranteed quality.
Answer
Given that the probability of a defective pin is p = 0.02 and the number of pins in a box is n = 100, we use the Poisson distribution as an approximation to the binomial distribution since n is large and p is small. The parameter λ = np = 100 × 0.02 = 2.
The probability of having x defective pins in a box is given by P(X = x) = (e^-λ × λ^x) / x!. The box fails to meet the guarantee if the number of defective pins X > 4. Thus, we need to calculate P(X > 4) = 1 - P(X ≤ 4).
Calculating each term:
- P(0) = e^-2 × 2^0 / 0! = 0.1353
- P(1) = e^-2 × 2^1 / 1! = 0.2707
- P(2) = e^-2 × 2^2 / 2! = 0.2707
- P(3) = e^-2 × 2^3 / 3! = 0.1804
- P(4) = e^-2 × 2^4 / 4! = 0.0902
Summing these, P(X ≤ 4) = 0.1353 + 0.2707 + 0.2707 + 0.1804 + 0.0902 = 0.9473. Therefore, the probability that the box fails the guarantee is P(X > 4) = 1 - 0.9473 = 0.0527.
Find the probability of getting 1 or 4 or 5 or 6 in throwing a die 5 to 7 times among 9 trials using normal distribution.
Answer
For a single die, the probability of rolling a 1, 4, 5, or 6 is p = 4/6 = 2/3. The probability of failure is q = 1 - 2/3 = 1/3. For n = 9 trials, the mean μ = np = 9 × (2/3) = 6 and the standard deviation σ = sqrt(npq) = sqrt(9 × 2/3 × 1/3) = sqrt(2) ≈ 1.414.
Using the normal approximation with continuity correction, we find P(5 ≤ X ≤ 7) = P(4.5 < X < 7.5). Converting to standard normal variable Z = (X - μ) / σ:
The probability is P(-1.06 < Z < 1.06) = Φ(1.06) - Φ(-1.06) = 2 × Φ(1.06) - 1. From standard normal tables, Φ(1.06) ≈ 0.8554.
Thus, the probability is 2 × 0.8554 - 1 = 0.7108.
Calculate the rank correlation coefficient for the following data:
| X | 65 | 66 | 67 | 67 | 68 | 69 | 70 | 72 |
|---|---|---|---|---|---|---|---|---|
| Y | 67 | 68 | 65 | 68 | 72 | 72 | 69 | 71 |
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The joint probability density function of the continuous random variables X and Y is given by:
0, elsewhere. Find f(x) and f(y).
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If x = 2y + 3 and y = kx + 6 are the regression lines of x on y and y on x respectively.
Show that 0 ≤ k ≤ 1/2
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If k = 1/8, find r and (x̅, y̅).
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Explain how a statistical hypothesis is tested.
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To test the effect of fertilizer on rice production 80 plots of land were chosen. Half of them were treated with fertilizer and half without it. Mean yield on untreated plots was 4.8 quintals with standard deviation of 0.4 quintals, while on treated plots it was 5.1 quintals with standard deviations of 0.36 quintals. Can we conclude that fertilizers provide significant improvement in production?
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What is meant by a statistical hypothesis? What are the two types of errors of decision that arise in testing a hypothesis?
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A random sample of boots worn by 40 combat soldiers in a desert region showed an average life of 1.08 years with a standard deviation of 0.05 years. Under standard conditions the boots are known to have an average life of 1.28 years. Is there reason to assert at a level of significance of 0.05 that use in the desert causes the mean life of such boots to decrease?
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A warehouse has only one loading dock manned by a three person crew. Trucks arrive at the loading dock at an average rate of 3 per hour and the arrival rate is Poisson distributed. The loading of a truck takes 10 minutes on an average and can be assumed to be exponentially distributed. The operating cost of a truck is Rs.25 per hour and the members of the loading crew are paid Rs.10 each per hour, would you advice the truck owner to add another crew of three persons more.
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The transition probability matrix of a Markov chain {Xn}; n = 1, 2, 3 having three states 1, 2 and 3 is P = [[0.1, 0.5, 0.4], [0.6, 0.2, 0.2], [0.3, 0.4, 0.3]] and the initial distribution is P(0) = (0.7, 0.2, 0.1) Find:
P{X2 = 3}
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P{X3 = 2, X2 = 3, X1 = 3, X0 = 2}.
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