# Probability And Statistics | Week 12

## Probability And Statistics Week 12 Answers

Q1. Let X1,….Xn, be a random sample from a N(u,o2)population where o2 = o20 is known. A test for H1: u = u0 vs. K1: u = u1, where u0 > u1 is

(B)

Q2. A random sample of size 10 is taken from a normal population and the values are:
9.7 1267 11.81 7.89 11.18 10.83 1298 1256 13.05 8.43
For testing the hypothesis that the mean of the population is 10 against the hypothesis that it is not equal, the value of the test statistic and the critical value at 5% level of significance are

(A) 1.86 and 2.262
(B) 1.51 and 2.228
(C) 1.68 and 2.262
(D) 1.15 and 2.228

Q3. A random sample of size 12 is taken from a normal population with mean u and variance o2. Let X and S2 be the sample mean and sample variance respectively and T = Vn (X-u0/S). For testing H0: u = u0 -versus H1: u # u0 at 5% level of significance, the test is to
(A) Reject H0 if T >= 2.179
(B) Reject H0 if T <= 2.179
(C) Reject H0 if | T | <= 2.201
(D) Reject H0 if | T | >= 2.201

Q4. Two machines are utilized to fill 12-ounce net volume plastic bottles. The filling processes can be assumed to be independently normally distributed with means u1,u2 and standard deviations 01 = 0.018 and 02 = 0.021 respectively. An experiment is performed by taking random samples of size 5 from the output of each machine. For testing the hypothesis H0:u1 H0: u1 not equals to u2, find the value of the test statistic.

(A) 24.761
(B) 38.159
(C) 10.846
(D) 32.766

Q5. The burning times (in minutes) of chemical flares of two different formulations are given in the following table. Assume that these follow normal distributions

For test the hypothesis that the two variances are equal, the value of test statistic is
(A) 1.018
(B) 1.037
(C) 1.965
(D) 1.982

Q6.

(A)

Q7. The diameters of a metal disc as measured by two distinct calipers follow independent normal distributions with unknown means and unknown variances. Two random samples of ten metal discs are taken to measure diameters with two calipers, and the corresponding outcomes are obtained as

For testing the equality of means of measurements using two calipers, the value of the test statistic and the critical value at 5% level of significance are
(A) 2.012 and 2.11
(B) 1.211 and 2.11
(C) 1.642 and 2.05
(D) 1.739 and 2.43

Q8. A random sample of size n from a N(μ₁, 225) population has mean 50. Another independent random sample of size n from a N(μ₂,225) has mean 45. In order to test Ho : μ1 = μ₂ vs. H₁ : μ1μ value of n so that H, is accepted at 5% level of ₂ the maximum significance is
(A) 48
(B) 69
(C) 96
(D) 84

Q9. Let X₁,…, X15 be a random sample from N(μ, 15). To test H₁ : μ = 10 vs. K₁ : μ≠10, consider the following critical region : C = {x̄ : |x̄-10| ≥ a}. Find the value of a so that the size of the critical region is 0.05.
(A) 0.94
(B) 1.96
(C) 2.33
(D) 2.54

Q10. A researcher claims that at least 15% of all football helmets contain manufacturing faults that could result in damage. Twenty helmets out of a sample of 200 contained these flaws. Find the value of the test statistic to test the claim of the researcher.
(A) -0.73
(B) -0.96
(C) -1.65
(D) -1.98 0 