Statistics How To

Word problems with normal distribution: “Between”

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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:

  1. The mean (average or μ).
  2. Standard deviation (σ).
  3. Number selected (i.e. “choose one at random” or “select ten at random”).
  4. 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:

    1. Sample size (i.e. 400 houses, 33 people, 99 factories, 378 plumbers etc.). OR
    2. 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:

word problems with normal distribution

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 :

z score formula

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

      1. subtract the mean from X
      2. divide by the standard deviation.

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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Word problems with normal distribution: “Between” was last modified: September 21st, 2015 by Stephanie

3 thoughts on “Word problems with normal distribution: “Between”

  1. Rebecca Gamble

    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”.

  2. Vanessa DuBarry

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

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