**QUESTION:
Find the probability that a single person ( sample size n = 1) gets a test value BETWEEN 74 and 78 for their exam score. Assume that the exams are normally distributed with a mean of 82 and a std dev of 4.**

NOTE: This question will use the โz tableโ

Here is a great z โ table:

http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf

There is also a z table in the back of any statistics book ๐

ANSWER:

First, letโs look at what we are given in the problem:

1) We are told that the mean = 82

2) We are told that the std dev = 4

3) We are told that our sample comes from a normally distributed population which means that we can use the z table to determine probabilities (percentages or also sometimes called area under the curve).

This image is what we are doing:

Anytime you are asked to find the probability, percentage, or area under the curve (these all mean the same thing) for a normal distribution, you will use the z table.

The z table is a table of probabilities for each z value (a z value is the number of standardized deviation you are from the mean).

To use the z table, you must first convert (standardize) the values in your question to z values.

Here is how:

Our first value here is 74

How many std dev from the mean is 74?

Visually, we can see that it is -2 std dev from the mean.

We can also use the formula for z to get this:

z = (x โ mean)/std dev

z = (74 โ 82) / 4 = – 8/4 = -2

Therefore, the value 74 in our problem is the same as z = -2

Next, we do the same thing for the value 78.

Why are we doing this for 74 and 78? Because the question is asking for the probability that a score is BETWEEN 74 and 78. We will use the z table to get this, but not until we convert 74 and 78 to their equivalent z values.

The z value for 78, visually is -1. You can see this by looking at the image above and noting that 78 is -1 deviation from the mean.

The z formula confirms this:

z = (x โ mean)/std dev

z = (78 โ 82)/ 4 = -4/4 = -1

So, the z value for 78 is -1

This tells us that

P( 74 < x < 78) is the same exact thing as P(-2 < z < -1)

Using the z table, we will need to do two things.

IMPORTANT NOTE: the z table is cumulative and always starts on the left.

So the P(z < -1) is .1587

**Next, we need the P(z < – 2)= .0228**

But this question is asking for the Probability that a value is BETWEEN 78 and 82 (which is the same as BETWEEN z = -1 and -2).

P(78 < x < 82) = P(-2 < z < -1) = .1587 – .0228 = .1359

**Final Solution:**

Therefore, the Probability that a single student will get a score between 4 and 78 is .1359 or about 13.5%. This makes sense because the mean is 82 and by the Empirical Rule, over 68% of the students will score between 78 and 86.