JNTU
JNTU B.Tech II Year I Semester COMPUTER ORIENTED STATISTICAL METHODS Previous Question Paper (April/May - 2023 - R18)
Note: Part A is compulsory (25 Marks). In Part B, answer any one question from each unit (10 Marks each).
Part A
If the probability is 0.05 that a certain kind of measuring device will show excessive drift, what is the probability that the sixth measuring device tested will be the first to show excessive drift?
Answer
This problem follows a Geometric Distribution, which models the number of trials required to achieve the first success in a sequence of independent Bernoulli trials. Let p be the probability of success (excessive drift) and q be the probability of failure (no excessive drift).
- Given: Probability of success p = 0.05
- Probability of failure q = 1 - p = 0.95
- The probability that the first success occurs on the n-th trial is given by the formula: P(X = n) = qn-1p
- For the sixth device to be the first to show excessive drift, we set n = 6:
- P(X = 6) = (0.95)6-1 × 0.05
- P(X = 6) = (0.95)5 × 0.05
- P(X = 6) = 0.77378 × 0.05
- P(X = 6) = 0.038689
Thus, the probability that the sixth measuring device tested will be the first to show excessive drift is approximately 0.0387.
The probability density function of a continuous random variable X is given by f(x) = a*e^(-|x|), where -∞ < x < ∞. Prove or disprove that a = 1/2.
Answer
To determine if a = 1/2, we must verify the normalization condition for a valid probability density function (PDF), which states that the total area under the curve must equal unity:
∫-∞∞ f(x) dx = 1
Given f(x) = a e-|x|, we substitute this into the integral:
∫-∞∞ a e-|x| dx = 1
Since the function f(x) = a e-|x| is an even function (because |-x| = |x|), we can simplify the integral by evaluating it from 0 to ∞ and multiplying by 2:
- 2 ∫0∞ a e-x dx = 1
- 2a [-e-x]0∞ = 1
- 2a [-(0 - 1)] = 1
- 2a(1) = 1
- a = 1/2
Since the calculation yields a = 1/2, the statement is proven to be true. The function f(x) = (1/2)e-|x| satisfies the fundamental requirement for a probability density function.
When is the geometric distribution an appropriate model?
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If the mean of Binomial distribution is 3 and variance is 9/4, obtain the value of n.
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If z is normally distributed with mean 0 and variance 1, evaluate P(z ≤ 1.64).
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Obtain the s.d. of the sampling distribution of means of 300 random samples each of size n = 36 are drawn from a population of N = 1500 which is normally distributed with mean μ=22.4 and s.d. σ of 0.048, if sampling is done with replacement.
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Discuss the level of significance and type of errors.
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Explain the terms null and alternate hypotheses.
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Define Markov chain.
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Is the matrix [[0, 1, 0], [0.5, 0.25, 0.25], [?, ?, ?]] stochastic? (Note: matrix structure implied).
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Part B
Suppose colored balls are distributed in three indistinguishable boxes: Box I (2R, 3W, 5B), Box II (4R, 1W, 3B), Box III (3R, 4W, 3B). A box is selected at random from which a ball is selected at random. What is the probability that the ball is (a) red, (b) blue?
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A pair of fair dice is tossed. Let X denote the maximum of the numbers and Y denote the sum. Compute the mean, variance and standard deviation for X and Y.
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A population consists of {2, 3, 6, 8, 11}. Find (a) population mean, (b) population s.d., (c) mean of sampling distribution of means for sample size 2 (without replacement), (d) s.d. of sampling distribution of means.
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An efficiency expert tested 40 engineers (mean=12.73, s.d.=2.06). (a) Determine maximum error with 99% confidence, (b) Construct 98% confidence interval for true average time, (c) With what confidence can we assert that the sample mean does not differ from true mean by > 30 seconds.
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Weekly losses of hours in 10 plants before and after a safety program are given. Test whether the program is effective at 0.05 significance level.
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Urn A (5R, 3W, 8G), Urn B (3R, 5W). Toss a die: if 3 or 6 appears, choose from B, else from A. Find probability of choosing (a) red, (b) white, (c) green.
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Urn A has 2 white, Urn B has 4 red. At each step, interchange a marble from each. Xn is the number of red marbles in Urn A. (a) Find transition matrix P. (b) Probability of 2 red in Urn A after 3 steps.
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