Probability and Statistics > Normal Distributions > Word problems with normal distribution: “Between”

## Word problems with normal distribution: “Between”

This how-to covers solving problems that contain the phrase “between” and includes an upper and lower limit (i.e. “find the number of houses priced between $50K and 200K”. Note that this is different from finding the “middle percentage” of something. For other problem types, see the Normal Distribution Word Problems Index.

## Word problems with normal distribution: “Between”: Steps

**Step 1:** *Identify the parts of the word problem*. The word problem will identify:

- The mean (average or μ).
- Standard deviation (σ).
- Number selected (i.e. “choose one at random” or “select ten at random”).
- X: the numbers associated with “between” (i.e. “between $5,000 and $10,000” would have X as 5,000 and as $10,000).

In addition, you will be given EITHER:

- Sample size (i.e. 400 houses, 33 people, 99 factories, 378 plumbers etc.). OR
- You might be asked for a probability (in which case your sample size will most likely be everyone, i.e. “Journeyman plumbers” or “First year pilots.”

**Step 2:** *Draw a graph*. Put the mean you identified in Step 1 in the center. Put the number associated with “between” on the graph (take a guess at where the numbers would fall–it doesn’t have to be exact). For example, if your mean was $100, and you were asked for “hourly wages between $75 and $125”) your graph will look something like this:

**Step 3:***Figure out the z-scores*. Plug the first X value (in my graph above, it’s 75) into the z value formula and solve. The μ (the mean), is 100 from the sample graph. You can get these figures (including σ, the standard deviation) from your answers in step 1 :

- subtract the mean from X
- divide by the standard deviation.

- *Note: if the formula confuses you, all this formula is asking you to do is:

* Step 4: Repeat step 3 for the second X*.

**Step 5:** *Take the numbers from step 3 and 4 and use them to find the area in the **z-table**.* (If you don’t remember how to find an area you can find instructions for finding an area under a normal distribution curve here).

If you were asked to find a probability in your question, go to step 6a. If you were asked to find a number from a specific given sample size, go to step 6b.

**Step 6a:**

*Convert the answer from step 5 into a percentage. *

- For example, 0.1293 is 12.93%.

That’s it–skip step 6b!

**Step 6b**

*Multiply the sample size (found in step 1) by the z-value you found in step 4*. For example, 0.300 * 100 = 30.

That’s it — you’ve solved *Word problems with normal distribution: “Between”*

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This will really come in handy when I take the exam. I have found that that the word “exact” helps key in on the “X”.

I wish I would have seen this example earlier but I know it will be a great help on the final!