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Finding the Value of a Normal Random Variable

Example 1: The heights of a pediatrician’s 200 three-year-old females are normally distributed with mean 38.72 inches and standard deviation 3.17 inches.  Find the height of a three-year-old female at the 20^th percentile.  That is, find the height for a three-year-old female that separates the bottom 20% from the top 80%.

This is an inverse normal problem and the command invNorm( area,  is used.  In this type of problem, a percentage of the area under the normal curve is given and you are asked to find the corresponding X-value.  In this example, the percentage given is the top 80%.  The TI-83 always calculates probability from negative infinity up to the specified X-value.  To find the X-value corresponding to the top 80%, you must accumulate the bottom 20% of the area.  Press 2nd [DISTR] and select 3:invNorm( and type in  .20 , 38.72 , 3.17 ) and press ENTER.

Conclusion:  The height that separates the top 80% of three-year-old females from the bottom 20% is 36.05 inches.

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Example 2: The heights of a pediatrician’s 200 three-year-old females are normally distributed with mean 38.72 inches and standard deviation 3.17 inches.  The pediatrician wishes to determine the heights that separate the middle 98% of the distribution from the bottom and top 1%.  In other words, find the 1^st and 99^th percentiles.

This is an inverse normal problem and the command invNorm( area ,  is used.  In this problem the middle area is .98.  That leaves an area of .02 to be equally divided between the left and right tail areas.  Each of these areas are, therefore, equal to .01.  The height that separates the middle 98%  from the top 1 % is actually the height that separates the bottom 99% (the middle 98% plus the 1% in the left tail) from the top 1%.

To find the X-value corresponding to a left area of 0.99, press 2^nd [DISTR] and select 3:invNorm( and type in .99 , 38.72 , 3.17 ) and press ENTER.

The X-value that separates the top 1% of three-year-old females based on their heights) from the middle 98% is 46.09 inches.

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