Respuesta :
Answer:
The probability that at most 10 of the next 150 items produced are unacceptable is 0.8315.
Step-by-step explanation:
Let X = number of items with unacceptable quality.
The probability of an item being unacceptable is, P (X) = p = 0.05.
The sample of items selected is of size, n = 150.
The random variable X follows a Binomial distribution with parameters n = 150 and p = 0.05.
According to the Central limit theorem, if a sample of large size (n > 30) is selected from an unknown population then the sampling distribution of sample mean can be approximated by the Normal distribution.
The mean of this sampling distribution is: [tex]\mu_{\hat p}= p=0.05[/tex]
The standard deviation of this sampling distribution is: [tex]\sigma_{\hat p}=\sqrt{\frac{ p(1-p)}{n}}=\sqrt{\frac{0.05(1-.0.05)}{150} }=0.0178[/tex]
If 10 of the 150 items produced are unacceptable then the probability of this event is:
[tex]\hat p=\frac{10}{150}=0.067[/tex]
Compute the value of [tex]P(\hat p\leq 0.067)[/tex] as follows:
[tex]P(\hat p\leq 0.067)=P(\frac{\hat p-\mu_{p}}{\sigma_{p}} \leq\frac{0.067-0.05}{0.0178})=P(Z\leq 0.96)=0.8315[/tex]
*Use a z-table for the probability.
Thus, the probability that at most 10 of the next 150 items produced are unacceptable is 0.8315.
