It is known that IQ scores form a normal distribution with a mean of 100 and a standard deviation of 15. If a researcher obtains a sample of 16 students’ IQ scores from a statistics class at UT. What is the shape of this sampling distribution?

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:

the answer is c: divide the arc's degree measure by 360°, then multiply by the circumference of the circle.