Respuesta :
Answer:
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
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 [tex]\mu = 50, \sigma = 0.6[/tex]
Sample of 100:
This means that [tex]n = 100, s = \frac{0.6}{\sqrt{100}} = 0.06[/tex]
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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{50.12 - 50}{0.06}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] has a pvalue of 0.9772
X = 49.88
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{49.88 - 50}{0.06}[/tex]
[tex]Z = -2[/tex]
[tex]Z = -2[/tex] 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.
