Question

Each day a manufacturing plant receives a large shipment of drums of Chemical ZX-900. These drums are supposed to have a mean fill of 50 gallons, while the fills have a standard deviation known to be 0.6 gallon. Suppose the mean fill for the shipment is actually 50 gallons. If we draw a random sample of 100 drums from the shipment, what is the probability that the average fill for the 100 drums is between 49.88 and 50.12 gallons

Answer 1

To find the probability that the average fill for 100 drums is between 49.88 and 50.12 gallons, use the Central Limit Theorem. Calculate the z-scores for the given sample means, then find the probabilities corresponding to those z-scores using the standard normal distribution table or a calculator.

Step-by-step explanation:

To find the probability that the average fill for 100 drums is between 49.88 and 50.12 gallons, we need to use the Central Limit Theorem. According to the Central Limit Theorem, the sampling distribution of the sample means will be approximately normally distributed, regardless of the shape of the population distribution, as long as the sample size is sufficiently large. In this case, the sample size is 100, which satisfies the condition for the Central Limit Theorem.

To calculate the probability, we need to find the z-scores corresponding to the given sample means. The z-score formula is z = (x - μ) / (σ / √n), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Given that the population mean is 50 gallons, the population standard deviation is 0.6 gallon, and the sample size is 100, we can calculate the z-scores for the lower and upper limits: z1 = (49.88 - 50) / (0.6 / √100) and z2 = (50.12 - 50) / (0.6 / √100).

Once we have the z-scores, we can use the standard normal distribution table or a calculator to find the probabilities corresponding to those z-scores. The probability that the average fill for the 100 drums is between 49.88 and 50.12 gallons can be calculated as the difference between the two corresponding probabilities: P(49.88 ≤ x ≤ 50.12) = P(z1 ≤ Z ≤ z2).

Answer 2

0.9544 = 95.44% probability that the average fill for the 100 drums is between 49.88 and 50.12 gallons.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Suppose the mean fill for the shipment is actually 50 gallons. Standard deviation of 0.6 gallons.

This means that
`\mu = 50, \sigma = 0.6`

Sample of 100:

This means that
`n = 100, s = (0.6)/(√(100)) = 0.06`

What is the probability that the average fill for the 100 drums is between 49.88 and 50.12 gallons?

This is the pvalue of Z when X = 50.12 subtracted by the pvalue of Z when X = 49.88. So

X = 50.12

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (50.12 - 50)/(0.06)`

`Z = 2`

`Z = 2` has a pvalue of 0.9772

X = 49.88

`Z = (X - \mu)/(s)`

`Z = (49.88 - 50)/(0.06)`

`Z = -2`

`Z = -2` has a pvalue of 0.0228

0.9772 - 0.0228 = 0.9544

0.9544 = 95.44% probability that the average fill for the 100 drums is between 49.88 and 50.12 gallons.