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2 answers:
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Answer:
P(35.3 < M < 35.4) = 0.0040.
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 random variable X, with mean and standard deviation
, the sample means with size n of at least 30 can be approximated to a normal distribution with mean
and standard deviation
In this problem, we have that:
Find the probability that a single randomly selected value is between 35.3 and 35.4
This is the pvalue of Z when X = 35.4 subtracted by the pvalue of Z when X = 35.3. So
X = 35.4
By the Central Limit Theorem
has a pvalue of 0.9927
X = 35.3
has a pvalue of 0.9887
0.9927 - 0.9887 = 0.0040
So the answer is:
P(35.3 < M < 35.4) = 0.0040.