MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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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 \( \pi(p) \).)
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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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