The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. a sample of 25 cans is to be selected. so, the middle 70% of all sample means will fall between what two values? a) 101.8 and 111.2 b) 104.8 and 115.2 c) 107.8 and 111.2 d) 101.8 and 119.2

B) 104.8 and 115.2

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 normally distributed samples are 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 normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

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

In this problem, we have that:

So, the middle 70% of all sample means will fall between what two values?

50 - (70/2) = 15th percentile

50 + (70/2) = 85th percentile

15th percentile

X when Z has a pvalue of 0.15. So X when Z = -1.037.

By the Central Limit Theorem

85th percentile

X when Z has a pvalue of 0.85. So X when Z = 1.037.

So the correct answer is:

B) 104.8 and 115.2

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