The producer of HB pencils has 720 pencils randomly chosen from each day's productions. A defect rate of 15% is acceptable.
a) Assuming that 15% of all manufactured pencils are defective, what is the probability of finding at least 80 percent defective pencials in this sample?
b) If 120 pencils are found to be defective in today's sample, is it likely that the manufacturing process may need improvements? Explain....
I probably read this problem wrong, but wouldn't (a) be a trick question? If you take 15% defected pencils out of however many regular pencils, that particular sample will be all defected, wouldn't it? Like I said, I probably read it wrong, so plz wait for a different answer. :-)
Besides, did you realize you posted the same question within less than an hour? Please be patient with these people-- they're doing their best.
To find the probability of finding at least 80 percent defective pencils in the sample, we can use the binomial probability formula. Let's break down the process step by step:
a) Probability of finding at least 80 percent defective pencils in the sample:
Step 1: Determine the number of trials, which is the number of pencils randomly chosen from each day's production. In this case, it is 720.
Step 2: Determine the probability of success in one trial, which is the proportion of defective pencils in the overall production. As stated, the defect rate is 15%, which can be expressed as 0.15.
Step 3: Set up the binomial probability formula. The formula is as follows:
P(X ≥ k) = 1 - P(X < k)
Where P(X ≥ k) represents the probability of finding at least k defective pencils in the sample and P(X < k) represents the probability of finding less than k defective pencils in the sample.
Step 4: Calculate the probability using the formula. In this case, we want to find the probability of finding at least 80% defective pencils, so k = 0.8 * 720 = 576.
P(X ≥ 576) = 1 - P(X < 576)
P(X < 576) = P(X = 0) + P(X = 1) + ... + P(X = 575)
P(X < 576) = ∑ [720Ck * (0.15^k) * (0.85^(720-k))] for k = 0 to 575
P(X ≥ 576) = 1 - ∑ [720Ck * (0.15^k) * (0.85^(720-k))] for k = 0 to 575
This calculation involves summing up a large number of terms, so it may be more practical to use software, such as calculator or statistical software, to compute this value.
b) If 120 pencils are found to be defective in today's sample, we can evaluate whether the manufacturing process may need improvements by comparing this result to the defect rate that is considered acceptable (15%). To do this, we can calculate the probability of finding 120 or more defective pencils in the sample, given the acceptable defect rate.
P(X ≥ 120) = 1 - P(X < 120)
P(X < 120) = P(X = 0) + P(X = 1) + ... + P(X = 119)
P(X < 120) = ∑ [720Ck * (0.15^k) * (0.85^(720-k))] for k = 0 to 119
P(X ≥ 120) = 1 - ∑ [720Ck * (0.15^k) * (0.85^(720-k))] for k = 0 to 119
By calculating the probabilities above, you can determine if finding 120 defective pencils in today's sample is likely or unlikely based on the acceptable defect rate of 15%.