At a factory plant, washing powder is manufactured and packed into 500g boxes by production lines A and B. To maintain quality control, the actual weight of the powder is recorded. Washing powder filled by both lines is normally distributed. Line A packs powder with a mean weight of 498g and a standard deviation of 5g, while line B packs powder with a mean weight of 495g but a larger standard deviation of 15g. If a box exceeds 510g, it is considered overfilled and removed from the line. Use Excel to answer this question.What is the probability that a randomly selected box from line A is overfilled? a.0.8413b.0.0159c.0.9918d.0.0082
Question
At a factory plant, washing powder is manufactured and packed into 500g boxes by production lines A and B. To maintain quality control, the actual weight of the powder is recorded. Washing powder filled by both lines is normally distributed. Line A packs powder with a mean weight of 498g and a standard deviation of 5g, while line B packs powder with a mean weight of 495g but a larger standard deviation of 15g. If a box exceeds 510g, it is considered overfilled and removed from the line. Use Excel to answer this question.What is the probability that a randomly selected box from line A is overfilled? a.0.8413b.0.0159c.0.9918d.0.0082
Solution
To find the probability that a randomly selected box from line A is overfilled, we need to find the probability that a box weighs more than 510g given that the mean weight is 498g and the standard deviation is 5g.
This is a problem of finding the probability of a value in a normal distribution, which can be solved using the Z-score formula:
Z = (X - μ) / σ
where:
- X is the value we are interested in (510g in this case),
- μ is the mean (498g for line A), and
- σ is the standard deviation (5g for line A).
First, we calculate the Z-score:
Z = (510 - 498) / 5 = 2.4
Next, we use a Z-table or Excel's standard normal distribution function (NORM.S.DIST) to find the probability that Z is less than 2.4. However, since we want the probability that a box is overfilled (i.e., weighs more than 510g), we need to find the probability that Z is greater than 2.4, which is 1 - P(Z < 2.4).
In Excel, you can calculate this as follows:
=1 - NORM.S.DIST(2.4, TRUE)
The result is approximately 0.0082.
Therefore, the answer is d. 0.0082.
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