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According to a USPS study, the weight of a randomly selected shipment box is normally distributed with a mean of 18 lbs and a standard deviation of 5.6 lbs. Let [tex]X[/tex] be the weight of a randomly selected shipment box, and let [tex]S[/tex] be the total weight of a random sample of size 12.

1) Describe the probability distribution of [tex]X[/tex] and state its parameters [tex]\mu[/tex] and [tex]\sigma[/tex]. Find the probability that the weight of a randomly selected shipment box is less than 13 lbs.

2) Use the Central Limit Theorem to describe the probability distribution of [tex]S[/tex] and state its parameters [tex]\mu_S[/tex] and [tex]\sigma_S[/tex]. (Round the answers to 1 decimal place.) Find the probability that the total weight of a sample of 12 randomly selected shipment boxes is less than 197 lbs.

Answer :

1) The probability distribution of X, the weight of a randomly selected shipment box, is normally distributed with parameters μ = 18 lbs and σ = 5.6 lbs. The probability that the weight of a randomly selected shipment box is less than 13 lbs is approximately 0.1515.

2) Using the Central Limit Theorem, the probability distribution of S, the total weight of a sample of 12 randomly selected shipment boxes, is approximately normally distributed with parameters μs = 12 * 18 = 216 lbs and σs = √[tex](12 * (5.6)^2) = 30.768 lbs[/tex]. The probability that the total weight of a sample of 12 randomly selected shipment boxes is less than 197 lbs is approximately 0.1217.

1) For the weight of a randomly selected shipment box X, the probability distribution is given as X ~ N(μ, σ), where μ = 18 lbs and σ = 5.6 lbs.

The probability that X is less than 13 lbs can be found using the standard normal distribution formula:

P(X < 13) = Φ((13 - μ) / σ)

Substituting the given values:

P(X < 13) = Φ((13 - 18) / 5.6)

P(X < 13) = Φ(-0.8929)

Using a standard normal distribution table or a calculator, we find that Φ(-0.8929) ≈ 0.1841.

So, the probability that the weight of a randomly selected shipment box is less than 13 lbs is approximately 0.1841.

2) According to the Central Limit Theorem, the distribution of the sample mean of a large sample size from any population will be approximately normally distributed.

For S, the total weight of a sample of 12 randomly selected shipment boxes, the parameters are:

μs = n * μ = 12 * 18 = 216 lbs

σs = √[tex](n * σ^2)[/tex] = √(12 * [tex](5.6)^2)[/tex] = 30.768 lbs

Now, to find the probability that the total weight of a sample of 12 randomly selected shipment boxes is less than 197 lbs, we standardize the variable S:

Z = (197 - μs) / σs

Z = (197 - 216) / 30.768

Z = -0.6164

Using a standard normal distribution table or a calculator, we find that Φ(-0.6164) ≈ 0.2708.

So, the probability that the total weight of a sample of 12 randomly selected shipment boxes is less than 197 lbs is approximately 0.2708.

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