# Z-score, MA121 Introduction to statistics unit 2 test, q 11

#1

Hi. I just took the test for unit 2 of MA121 Introduction to Statistics.

The last question goes like this:

Suppose that the average income in the population in City X is \$50,000 with a standard deviation of \$10,000. Calculate the z-score for a person that earns less than \$45,000, assuming that income is normally distributed.

It should be very straightforward:

• Z = -0.5
• Go to the table and check the value for -0.5, which is 0.3085.

But I got the answer wrong, and instead the correct solution is much lower.
I cannot find what I am doing wrong. I have also tried with a Ti-36X and I also get 0.3085.

Thank you for hour help.

A.

#2

A normal distribution is just the graph of a function that we may call f(x).

For a definition of f(x) see: http://statweb.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.html
The equation z = (x-m)/s transforms a normal distribution with mean m and standard deviation s into a normal distribution with mean 0 and standard deviation 1.

A table of z scores gives the portion of the area under the standard normal distribution (the bell curve) that lies to the left of the z-score. That is, events in the distribution that have a probability of happening that is less than the z-score will lie to the left of the z-score.

Because of the definite article ‘the’ in the statement “Calculate the z-score for a person that earns less than \$45,000…” we need to know how much the specific person referred to by the definite article earns. But we aren’t given that information.

If we substitute the indefinite article ‘a’ for ‘the’, so that the statement reads “Calculate a z-score for a person that earns less than \$45,000…”, there are infinitely many choices for a salary less than \$45,000.

What this means is that stated either way, the sentence “Calculate …” is essentially meaningless.

In any case, your calculation of the z-score is correct.

#3

I was frustrated as well. I did the same calculations and used the Normal Distribution Graphing calculator provided through the course and obtained the same z-score answer 0.3085. Same is listed in the Appendix under conditional probabilities. I was very confused as to why my answer was marked in correct… using all three methods, the formula, the table and the graphical display I obtained the same answer