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Answer:
Where and
We select a sample of n=16 and we are interested on the distribution of , since the distribution for X is normal then we can conclude that the distribution for
is also normal and given by:
Because by definition:
And for this case we have this:
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the IQ scores of a population, and for this case we know the distribution for X is given by:
Where and
We select a sample of n=16 and we are interested on the distribution of , since the distribution for X is normal then we can conclude that the distribution for
is also normal and given by:
Because by definition:
And for this case we have this: