Just C please 1. A random sample of $ n = 20 $ products are taken from the production lines of a factory. Let $ X_i \overset{\text{IID}}{\sim} \text{Bernoulli}(p) $ be the indicator of whether the $ i $-th product is defective for $ i = 1, \ldots, 20 $, where $ p $ denotes the proportion of defective products in the population. It is desirable to test whether the proportion of defective products is below 30%. (a) State the null hypothesis $ H_0 $ and the alternative hypothesis $ H_a $. (b) Describe the events of Type I error and Type II error in the procedure of making statistical decisions. (c) Let $ T = \sum_{i=1}^{20} X_i $ denote the number of defective items in the sample. Suppose that we decide to reject $ H_0 $ if $ T \leq 4 $. Let $ \pi(p) $ denote the corresponding power function, i.e., \[ \pi(p) = \Pr(T \leq 4 \mid p). \] Determine the value of $ \pi(p) $ at the points $ p = 0, 0.1, 0.2, 0.3, \ldots, 0.9, $ and 1.0 and sketch the power function. (Hint: Use the statistical table for Binomial distribution.) (d) What is the size of the test procedure in part (c), i.e., the maximum probability of making a Type I error? (e) What is the power of the test procedure in part (c) when $ p = 0.1 $? (Hint: Both part (d) and part (e) can be obtained immediately from the power function $ \pi(p) $.)

The defined no. of defective samples is: T=∑i=120Xi. And the claim under H0 if T≤4.

1. A random sample of n = 20 products are taken from the production lines of a factory. Let = 1,..., 20, IID Xi Bernoulli(p) be the indicator of whether the i-th product defective for i where Р denotes the proportion of defective products in the population. It is desirable to test whether the the proportion of defective products is below 30%. (a) State the null hypothesis Ho and the alternative hypothesis Ha. (b) Describe the events of Type I error and Type II error in the procedure of making statistical decisions. 20 (c) Let T i=1 ₁X₁ denote the number of defective items in the sample. Suppose that we decide to reject Hỏ if T ≤ 4. Let π(p) denote the corresponding power function, i.e., π(p) = Pr (T ≤ 4|p). Determine the value of (p) at the points p = 0, 0.1, 0.2, 0.3,..., 0.9, and 1.0 and sketch the power function. (Hint: Use the statistical table for Binomial distribution.) = (d) What is the size of the test procedure in part (c), i.e., the maximum probability of making Type I error? (e) What is the power of the test procedure in part (c) when p = 0.1? (Hint: Both part (d) and part (e) can be obtained immediately from the power function (p).)

1. A random sample of n = 20 products are taken from the production lines of a factory. Let = 1,..., 20, IID Xi Bernoulli(p) be the indicator of whether the i-th product defective for i where Р denotes the proportion of defective products in the population. It is desirable to test whether the the proportion of defective products is below 30%. (a) State the null hypothesis Ho and the alternative hypothesis Ha. (b) Describe the events of Type I error and Type II error in the procedure of making statistical decisions. 20 (c) Let T i=1 ₁X₁ denote the number of defective items in the sample. Suppose that we decide to reject Hỏ if T ≤ 4. Let π(p) denote the corresponding power function, i.e., π(p) = Pr (T ≤ 4|p). Determine the value of (p) at the points p = 0, 0.1, 0.2, 0.3,..., 0.9, and 1.0 and sketch the power function. (Hint: Use the statistical table for Binomial distribution.) = (d) What is the size of the test procedure in part (c), i.e., the maximum probability of making Type I error? (e) What is the power of the test procedure in part (c) when p = 0.1? (Hint: Both part (d) and part (e) can be obtained immediately from the power function (p).)

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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