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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:
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
In this problem, we have that:
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 .
Due to the Central Limit Theorem
Upper end
X when Z has a pvalue of 0.15. So X when .
Due to the Central Limit Theorem
The probability is 70% that the sample mean amount of juice will be contained between 4.9168 ounces and 5.0832 ounces.
The easiest way it to solve it.
You have
Thus
You have
Thus