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Answer:
The interval containing the middle-most 48% of sample means is between 218.59 to 221.41.
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 distributied random variable X, with mean and standard deviation
, the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
In this problem, we have that:
Find the interval containing the middle-most 48% of sample means:
50 - 48/2 = 26th percentile to 50 + 48/2 = 74th percentile. So
74th percentile
value of X when Z has a pvalue of 0.74. So X when Z = 0.643.
By the Central Limit Theorem
26th percentile
Value of X when Z has a pvalue of 0.26. So X when Z = -0.643
The interval containing the middle-most 48% of sample means is between 218.59 to 221.41.