denotes the proportion of defective products in the population. It is desirable to the the proportion of defective products is below 30%. te the null hypothesis Ho and the alternative hypothesis Ha. cribe the events of Type I error and Type II error in the procedure of making statis isions. 20 T = Σ²₁X₂ denote the number of defective items in the sample. Suppose tha ide to reject Ho if T ≤ 4. Let (p) denote the corresponding power function, i.e., T(P) = Pr (T ≤ 4|p). ermine the value of (p) at the points p = 0, 0.1, 0.2, 0.3,..., 0.9, and 1.0 and sk power function. (Hint: Use the statistical table for Binomial distribution.) at is the size of the test procedure in part (c), i.e., the maximum probability of ma no Lorror?
denotes the proportion of defective products in the population. It is desirable to the the proportion of defective products is below 30%. te the null hypothesis Ho and the alternative hypothesis Ha. cribe the events of Type I error and Type II error in the procedure of making statis isions. 20 T = Σ²₁X₂ denote the number of defective items in the sample. Suppose tha ide to reject Ho if T ≤ 4. Let (p) denote the corresponding power function, i.e., T(P) = Pr (T ≤ 4|p). ermine the value of (p) at the points p = 0, 0.1, 0.2, 0.3,..., 0.9, and 1.0 and sk power function. (Hint: Use the statistical table for Binomial distribution.) at is the size of the test procedure in part (c), i.e., the maximum probability of ma no Lorror?
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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![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 \( \pi(p) \).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc58b2eee-5b79-4c9d-923b-d4e913d44dbe%2F6bd013bf-3a62-4d81-a62c-023e15c8ab07%2F90i2dpq_processed.png&w=3840&q=75)
Transcribed Image Text: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 \( \pi(p) \).)
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