# 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/4^{n}

(B) 1-3n/4^{n}**(C) 1-(1+3n)/4 ^{n}**

(D) 1+(1+3n)/4

^{n}

**Q2. Suppose that X is a random variable with the probability density functionf(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