A manufacturing company measures the weight of boxes before shipping them to customers. If the box weights have a population mean of 90 lbs. and a standard deviation of 24 lbs., then based on a sample size of 36 boxes, what is the probability that the average weight of the boxes will be less than 84 lbs.?

Question

14-05-2024
Mathematics

College

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A manufacturing company measures the weight of boxes before shipping them to customers. If the box weights have a population mean of 90 lbs. and a standard deviation of 24 lbs., then based on a sample size of 36 boxes, what is the probability that the average weight of the boxes will be less than 84 lbs.?

Answer :

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qwfish qwfish · Answer 1 · 2024-11-23T20:40:30-05:00

Final answer:

The probability that the average weight of the boxes will be less than 84 lbs is approximately 6.68%.

Explanation:

To find the probability that the average weight of the boxes will be less than 84 lbs, we need to use the normal distribution. The population mean is 90 lbs and the standard deviation is 24 lbs. Since the sample size is 36, we can use the Central Limit Theorem to approximate the distribution of the sample mean as normal. The mean of the sample mean distribution will still be 90 lbs, but the standard deviation will be the population standard deviation divided by the square root of the sample size, which is 4 lbs. Now we can calculate the z-score for the value 84 lbs using the formula z = (x - μ) / (σ / √n).

Plugging in the values, we get z = (84 - 90) / (4) = -1.5. Using a standard normal distribution table or a calculator, we can find the probability that a standard normal random variable is less than -1.5, which is approximately 0.0668, or 6.68%.

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