The population of scores on a nationally standardized test forms a normal distribution with μ = 300 and σ = 50. If you take a random sample of n = 25 students, what is the probability that the sample mean will be less than M = 280?

And using a calculator, excel or the normal standard table we have that:

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 scores of a population, and for this case we know the distribution for X is given by:

 

Where and

Since the distribution for X is normal then we know that the distribution for the sample mean is given by:

We can find the individual probability like this:

And using a calculator, excel or the normal standard table we have that:

Given that and are common external tangents of circle G and circle D.

We recall that the lengths of the tangents to a circle drawn from the same point are equal.
i.e. and

Thus, we find the length of which is equal to the length of .

But .

Given that , to find we recall that the tangent of any given circle makes a right angle with the radius (or diameter) of the circle at the point of tangency.

We then use the pythagora rule to find the length .

Notice that triangle AFG is a right triangle with the side being the hypothenus.

Recall that the pythagoras theorem states that the square of the hypothenus is equal to the sum of the square of the lengths of the other two legs of a right triangle.

Thus,


Thus, and hence .