A distribution has a standard deviation of o=10. Find the z-score for each of the following locations in the distribution.

a. Above the mean by 5 points.

b. Above the mean by 2 points.

c. Below th mean by 20 points.

d. Below the mean by 15 points.

a) Using the z score formula we got:

b) Using the z score formula we got:

c) Using the z score formula we got:

d) Using the z score formula we got:

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

Where and

The best way to solve this problem is using the normal standard distribution and the z score given by:

Part a

Using the z score formula we got:

Part b

Using the z score formula we got:

Part c

Using the z score formula we got:

Part d

Using the z score formula we got:

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.