A manufacturing company measures the weight of boxes before shipping them to the customers. If the box weights have a population mean and standard deviation of 90 lbs. and 24 lbs., respectively, then based on a sample size of 36 boxes, what is the probability that the average weight of the boxes will exceed 94 lbs.

The probability that the average weight of the boxes will exceed 94 lbs is 0.1587.

Step-by-step explanation:

Let X = weight of the boxes shipped.

The mean weight is, μ = 90 lbs and the standard deviation, σ = 24 lbs.

A sample of n = 36 boxes is selected.

According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and an appropriately huge random samples (n ≥ 30) is selected from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

Then, the mean of the sample means is given by,

And the standard deviation of the sample means (also known as the standard error) is given by,

Hence the sampling distribution of the sample mean weight of boxes is Normally distributed.

Compute the probability that the average weight of the boxes will exceed 94 lbs as follows:

Thus, the probability that the average weight of the boxes will exceed 94 lbs is 0.1587.

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