Normal distribution word problems: finding a lower cut off point

Normal Distribution 2 Comments

Sometimes on a normal distribution word problem you’ll be asked to find a “lower limit of an upper percentage” of something (i.e. “find the cut-off point to pass a certain exam where the upper 40% of test takers pass”). For a full list of normal distribution word problem types, see the Normal Distribution Word Problem Index.

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. the “upper percentage” amount (i.e 40%, 10%…)

Step 2: Convert your percentage to a decimal. For example, the top 10% would be 0.100.

Step 3:Subtract your answer in Step 4 from 0.500. For example, 0.500-0.100=0.400.

Step 4: Look in the Z-table for the answer you calculated in step 4: You are probably not going to find the exact answer, so find the closest value in the center portion of the table. Example: the closest value I could find to .3000 is .2995, highlighted in the table below. The z-score for that value is .84 (0.8 in the left hand column and 0.04 at the top).

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549
0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133

Step 5: Substitute into the equation X = z σ + μ.* Get the mean ( μ) and standard deviation (σ) from step 1. Insert the z-score you found in step 4 (for X) into the ‘z’ spot in the equation.

If formulas confuse you, all the formula is asking you to do is:

  1. multiply the z score you found in step 4 by the standard deviation from step 1
  2. add that result to the mean

That’s it–you’re done!

*This equation isn’t new–it’s just a rearrangment of this formula:

z value formula

Related posts:

  1. Normal distribution Word Problems: finding an upper cut off point
  2. Normal Distribution Word Problems: Finding upper and lower cut off points
  3. Normal Distribution Word Problems on the TI-89: Finding Cut Off Points For a Top Percentage
  4. Normal Distribution Word Problems: “less than”
  5. Normal Distribution Word Problems: “Above”
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2 Responses to “Normal distribution word problems: finding a lower cut off point”

  1. How to Do Everything Statistics » Normal distribution word problems Index said:

    Sep 04, 09 at 10:24 am

    [...] “upper percentage” (click here) [...]

  2. Vanessa DuBarry said:

    Dec 10, 09 at 8:48 pm

    I always had a difficulty with this kind of problems even though this example is very helpful. i always get confused with the numbers.

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