Probability And Statistics | Week 11

Course Name: Probability And Statistics

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These are Probability And Statistics Week 11 Assignment 11 Answers

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Q1. Let X ~ Bin(n,p), where n is known and 0 < p < 1. In order to test H : p = 1/2 vs K : p = 3/4, a test is “Reject H if X 22”. Find the power of the test.
(A) 1+3n/4 n
(B) 1-3n/4n
(C) 1-(1+3n)/4n
(D) 1+(1+3n)/4n

Answer: (C) 1-(1+3n)/4n

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Q2. Suppose that X is a random variable with the probability density function
f(x,0) = 0x-1,0 < x < 1.
In order to test the null hypothesis Ho : 0 = 2 against H₁ : 0 = 3, the following test is used : “Reject H₁ if X₁ ≥ ½”, where X₁ is a random sample of size 1 drawn from the above distribution. Then the power of the test is
(A) 0.875
(B) 0.5
(C) 0.33
(D) 0.75

Answer: (A) 0.875

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These are Probability And Statistics Week 11 Assignment 11 Answers

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Q3. A random sample of 500 registered voters in Phoenix is asked if they favor the use of oxygenated fuels year-round to reduce air pollution. If more than 400 voters respond positively, we will conclude that more than 60% of the voters favor the use of these fuels, i.e., we are testing Ho : p = 0.6 vs H₁ : p = 0.6 What is the type II error probability if 75% of the voters favor this action? (use normal approximation to the binomial).
(A) 0.004911
(B) 0.9951
(C) 0.00589
(D) 0.99647

Answer: (A) 0.004911

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Q4. A textile fiber manufacturer is investigating a new drapery yarn, which the company claims follows a normal distribution having mean thread elongation of 12 cms with a standard deviation of 0.5cm. The company wishes to test the hypothesis Ho : μ = 12 against H₁ : μ< 12, using a random sample of four specimens. What is the type I error probability if the critical region is defined as x < 11.5 cms.
(A) 0.0227
(B) 0.1569
(C) 0.0358
(D) 0.0587

Answer: (A) 0.0227

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These are Probability And Statistics Week 11 Assignment 11 Answers

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Q5. The probability density function of the random variable X is f(x) = ¹e,x > 0, λ > 0. For testing the hypothesis Ho : λ = 3 vs HA : λ = 5, a test is given as “Reject Ho if X ≥ 4.5”. The probability of Type I error and power of this test are respectively
(A) 0.135 and 0.497
(B) 0.183 and 0.379
(C) 0.202 and 0.449
(D) 0.223 and 0.407

Answer: (D) 0.223 and 0.407

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Q6. The proportion of adults living in Tempe, Arizona, who are college graduates is estimated to be p = 0.4. To test this hypothesis, a random sample of 20 Tempe adults is selected. If the number of college graduates is between 4 and 8 (endpoints included), the hypothesis will be accepted; otherwise we will conclude that p = 0.4. Find the type I error probability for this procedure assuming p = 0.4.
(A) 0.0339
(B) 0.5
(C) 0.5339
(D) 0.4225

Answer: (C) 0.5339

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These are Probability And Statistics Week 11 Assignment 11 Answers

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Q7. Let X be a single observation from the population f(x,0) = 0e-8x, x > 0,0 > 0. If X > 1 is a critical region for testing H : 0 = 1 vs K : 0 = 2, find the Type I error and power of the test
(A) e and 1-1/e^2
(B) e – 1 and 2/e^2
(C) 1 – e and 1/e
(D) 1/e and 1/e^2

Answer: (D) 1/e and 1/e^2

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Q8. A manufacturer is interested in the output voltage of a power supply used in a PC. Output voltage is assumed to be normally distributed with standard deviation 0.2 volt and the manufacturer wishes to test Ho : μ = 5 volts against H₁ : μ # 5 volts using n = 8 units. If the acceptance region is 4.85 ≤ x ≤ 5.15. Find the power of the test for detecting a true mean output voltage of 5.1 volts.
(A) 0.1896
(B) 0.548
(C) 0.12558
(D) 0.23995

Answer: (D) 0.23995

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These are Probability And Statistics Week 11 Assignment 11 Answers

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Q9.

Answer: D

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These are Probability And Statistics Week 11 Assignment 11 Answers

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Q10. Let X₁,.., Xn be a random sample from a N(u, 1) population. Consider the hypothesis Ho : μ = 0 vs H₁ : μ> 0. A random sample of size five from this population is 1.4, 2.4, 4.2, -3.4 and 1.2. Based on this sample which of the following statements is valid for a uniformly most powerful test of size 0.05?
(A) Reject H0
(B) Accept H0
(C) Critical point is 1.96
(D) The value of the test statistic is 1.645

Answer: (B) Accept H0

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These are Probability And Statistics Week 11 Assignment 11 Answers

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These are Probability And Statistics Week 11 Assignment 11 Answers

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