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. = (c) Let T ₁X₁ denote the number of defective items in the sample. Suppose that we decide to reject Ho if T ≤ 4. Let (p) denote the corresponding power function, i.e., T(p) = Pr (T ≤ 4|p). ine he value of (p) at the points p = 0, 0.1, 0.2, 0.3,...,0.9, and 1.0 and etch 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).)

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A: 1 over 8 means 1 divided by 8 So, dividing 1 by 8 , we get  18=0.125

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