A manufacturer of electronic components claims that their production process produces resistors that are 90% defect-free.However, a customer suspects that the actual proportion of defective resistors may be much higher than that.To investigate this, a customer takes a random sample of 20 resistors from the manufacturer's production line and inspects them for defects.If three or more resistors are defective, the customer will not purchase the electronic components.Define the Null and Alternative Hypothesis;Define the Critical Value and Critical Region;Find the probability of committing a type I error assuming that manufacturer claim is True.Find the probability of committing a type II error for the alternative proportion of defected items of p=0.2.Hint: Use the Statistical Table A1 for the Binomial Probability Sums.
Question
A manufacturer of electronic components claims that their production process produces resistors that are 90% defect-free.However, a customer suspects that the actual proportion of defective resistors may be much higher than that.To investigate this, a customer takes a random sample of 20 resistors from the manufacturer's production line and inspects them for defects.If three or more resistors are defective, the customer will not purchase the electronic components.Define the Null and Alternative Hypothesis;Define the Critical Value and Critical Region;Find the probability of committing a type I error assuming that manufacturer claim is True.Find the probability of committing a type II error for the alternative proportion of defected items of p=0.2.Hint: Use the Statistical Table A1 for the Binomial Probability Sums.
Solution
The Null Hypothesis (H0) and Alternative Hypothesis (H1) are defined as follows:
H0: p = 0.9 (The manufacturer's claim that 90% of the resistors are defect-free is true) H1: p < 0.9 (The actual proportion of defect-free resistors is less than 90%)
The critical value is the number of defective resistors that would lead the customer to reject the manufacturer's claim. In this case, the critical value is 3, as the customer will not purchase the components if three or more resistors are defective.
The critical region is the set of values that would lead to the rejection of the null hypothesis. Here, the critical region is {3, 4, 5, ..., 20}.
The probability of committing a type I error (rejecting the null hypothesis when it is true) is the probability of finding three or more defective resistors given that the manufacturer's claim is true. This can be found using the binomial probability formula or a binomial probability table.
The probability of committing a type II error (accepting the null hypothesis when it is false) is the probability of finding two or fewer defective resistors given that the actual proportion of defective resistors is 0.2. This can also be found using the binomial probability formula or a binomial probability table.
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