Answer:
The probability is 70% that the sample mean amount of juice will be contained between 4.9168 ounces and 5.0832 ounces.
Step-by-step explanation:
To solve this question, the Normal probability distribution and the Central Limit Theorem are important.
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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 pvalue, 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 random variable X, with mean
and standard deviation
, a large sample size can be approximated to a normal distribution with mean
and standard deviation ![\frac{\sigma}{\sqrt{n}}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D)
In this problem, we have that:
![\mu = 5, \sigma = 0.4, n = 25, s = \frac{0.4}{\sqrt{25}} = 0.08](https://tex.z-dn.net/?f=%5Cmu%20%3D%205%2C%20%5Csigma%20%3D%200.4%2C%20n%20%3D%2025%2C%20s%20%3D%20%5Cfrac%7B0.4%7D%7B%5Csqrt%7B25%7D%7D%20%3D%200.08)
The probability is 70% that the sample mean amount of juice will be contained between what two values symmetrically distributed around the population mean?
The lower end of this interval is the value of X when Z has a pvalue of 0.5 - 0.7/2 = 0.15
The upper end of this interval is the value of X when Z has a pvalue of 0.5 + 0.7/2 = 0.85
Lower end
X when Z has a pvalue of 0.15. So X when
.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
Due to the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![-1.04 = \frac{X - 5}{0.08}](https://tex.z-dn.net/?f=-1.04%20%3D%20%5Cfrac%7BX%20-%205%7D%7B0.08%7D)
![X - 5 = -1.04*0.08](https://tex.z-dn.net/?f=X%20-%205%20%3D%20-1.04%2A0.08)
![X = 4.9168](https://tex.z-dn.net/?f=X%20%3D%204.9168)
Upper end
X when Z has a pvalue of 0.15. So X when
.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
Due to the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![1.04 = \frac{X - 5}{0.08}](https://tex.z-dn.net/?f=1.04%20%3D%20%5Cfrac%7BX%20-%205%7D%7B0.08%7D)
![X - 5 = 1.04*0.08](https://tex.z-dn.net/?f=X%20-%205%20%3D%201.04%2A0.08)
![X = 5.0832](https://tex.z-dn.net/?f=X%20%3D%205.0832)
The probability is 70% that the sample mean amount of juice will be contained between 4.9168 ounces and 5.0832 ounces.