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....
a) To find the probability of finding at least 80% defective pencils in the sample, we can use a binomial distribution.
First, we need to determine the parameters of the binomial distribution. The number of trials (n) is 720, and the probability of success (p) is 0.15 (15% of pencils being defective).
We are interested in finding the probability of at least 80% of the pencils being defective, which means we want to find the probability of getting 576 or more defective pencils.
Using a binomial probability calculator or a statistical software, we can calculate the probability as follows:
P(X ≥ 576) = 1 - P(X < 576)
Alternatively, we can use the cumulative distribution function (CDF) of the binomial distribution to find the probability directly.
b) If 120 pencils were found to be defective in today's sample, we can use statistical inference to assess whether the manufacturing process may need improvements.
To determine whether the observed number of defective pencils is statistically significant, we can perform a hypothesis test.
The null hypothesis (H0) is that the proportion of defective pencils in the sample is equal to the defect rate of 15%. The alternative hypothesis (Ha) is that the proportion of defective pencils in the sample is greater than 15%.
We can use the binomial test or the normal approximation to the binomial distribution to perform the hypothesis test.
Using the binomial test, we compare the observed number of defective pencils to the expected number under the null hypothesis. If the observed number of defective pencils is significantly higher than the expected number, we can reject the null hypothesis and conclude that the manufacturing process may need improvements.
Using the normal approximation to the binomial distribution, we calculate the z-score and compare it to the critical value for a given significance level. If the z-score is greater than the critical value, we can reject the null hypothesis.
In both cases, the decision whether the manufacturing process needs improvements depends on the significance level chosen and the corresponding critical value.
It's important to note that statistical inference provides a measure of confidence or probability rather than definitive proof. Further investigations and analysis may be required to determine the exact cause of the high number of defective pencils and identify potential improvements in the manufacturing process.