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Answer:
50% probability of a sample mean being less than 12,751 or greater than 12,754
Step-by-step explanation:
To solve this question, we have 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 random variable X, with mean and standard deviation
, a large sample size can be approximated to a normal distribution with mean \mu and standard deviation, which is also called standard error
In this problem, we have that:
Find the probability of a sample mean being less than 12,751 or greater than 12,754
Less than 12,751
pvalue of Z when X = 12751.
By the Central Limit Theorem
has a pvalue of 0.5.
50% probability of the sample mean being less than 12,751.
Greater than 12,754
1 subtracted by the pvalue of Z when X = 12,754.
has a pvalue of 1
1 - 1 = 0
0% probability of the sample mean being greater than 12754
Less than 12,751 or greater than 12,754
50 + 0 =50
50% probability of a sample mean being less than 12,751 or greater than 12,754