A 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)
I have so far done some steps but not getting the correct answer:
2% of chips are defective = 0.2 Sample = 250 Chips No.Of Defectives = 8 5% level of confidence
Let us test the claim at 5% level of significance . To see if rejection is called for, we need to compute the probability that the sample of 250 chips would have resulted in 8 or more defectives when p = 0.2. If the probability is less than or equal to 0.5, then the manufacturer's claim should be rejected.
p^o.2 [x>=8] = 1-P^0.2 [x<8]
I have solved this much and now i am not getting the answer . Can someone help me out ?
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 $\alpha = 0.05$, not $\alpha = 0.5$.
The probability that in a sample of $n = 250$ chips, one observes $8$ or more defective chips, is $\Pr[X \ge 8]$ if $X \sim \operatorname{Binomial}(n = 250, p = 0.02)$. We note $$\Pr[X \ge 8] = 1 - \Pr[X \le 7] = 1 - \sum_{x=0}^7 \binom{250}{x} (0.02)^x (0.98)^{250-x}.$$ We can use a calculator or computer to obtain $\Pr[X \ge 8] \approx 0.131253$. 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 $\alpha = 0.05$ to conclude the defect rate is greater than $2\%$.