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:
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.
Convert the answer from step 5 into a percentage.
- For example, 0.1293 is 12.93%.
That’s it–skip 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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