computer chip manufacturer claims that at most 2% of the chips it produces are defective. To check the claim of the manufacturer, a researcher selects a sample of 250 of these chips. If there are eight defective chips among these 250, test the null hypothesis that more than 2% of the chips are defective at 5% level of significance. Does this disprove the manufacturer's claim. (Given that Z0.05 = 1.645)
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Under the null hypothesis, the probability that a randomly selected chip is defective is 2%=0.02, not 0.2 as you have written, which is 20%. Similarly, a 5% significance level is α=0.05, not α=0.5.
The probability that in a sample of n=250 chips, one observes 8 or more defective chips, is Pr[X≥8] if X∼Binomial(n=250,p=0.02). We notePr[X≥8]=1−Pr[X≤7]=1−7∑x=0(250x)(0.02)x(0.98)250−x.We can use a calculator or computer to obtain Pr[X≥8]≈0.131253. This means that if the claimed defect rate is at most 2%, the probability that 8 or more defects were observed is at most 0.131253; this suggests there is insufficient evidence at α=0.05 to conclude the defect rate is greater than 2%.
Under the null hypothesis, the probability that a randomly selected chip is defective is 2%=0.02, not 0.2 as you have written, which is 20%. Similarly, a 5% significance level is α=0.05, not α=0.5.
The probability that in a sample of n=250 chips, one observes 8 or more defective chips, is Pr[X≥8] if X∼Binomial(n=250,p=0.02). We notePr[X≥8]=1−Pr[X≤7]=1−7∑x=0(250x)(0.02)x(0.98)250−x.We can use a calculator or computer to obtain Pr[X≥8]≈0.131253. This means that if the claimed defect rate is at most 2%, the probability that 8 or more defects were observed is at most 0.131253; this suggests there is insufficient evidence at α=0.05 to conclude the defect rate is greater than 2%.
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