Probability And Statistics | Week 11

Probability And Statistics Week 11 Answers

Link : Probability And Statistics (nptel.ac.in)

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/4ⁿ
(B) 1-3n/4ⁿ
(C) 1-(1+3n)/4ⁿ
(D) 1+(1+3n)/4ⁿ

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

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

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

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

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

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

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) 0.2399

Q9.

(A) 1 – 1/2^A – 1/2^B
(B) 1 – 1/2^A + 1/2^B
(C) 1 – 1/2^A+1 – 1/2^B
(D) 1 – 1/2^A+1 + 1/2^B

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 H_o
(B) Accept H_o
(C) Critical point is 1.96
(D) The value of the test statistic is 1.645