PROJECT:  NORMAL PROBABILITY DISTRIBUTION

Objective:  Use the Normal Probability Distribution to make decisions about a population.

I.  The female division of the company

Female Data in inches

66.4

68.1

66.7

67.9

63.1

67.8

66.1

68.9

66.1

69.2

64.9

67.6

57.6

65.1

66.7

67.8

66.8

63.6

67.5

60.2

69.4

68.4

62.2

67.2

64.7

66.3

64.2

62.2

64.3

67.2

63.2

58.1

 

ti-83.gif (1031 bytes)

Using the STAT functions to enter data and calculate the sample mean 
and the sample standard deviation
  1. Press the "STAT" button.
  2. With "EDIT" highlighted select "1:Edit" by pressing "ENTER".
  3. If there is data in List 1 (L1) clear it by using the up arrow to highlight "L1".  Then press "CLEAR" and "ENTER".
  4. Enter your data into List 1 (L1) by entering each value and pressing "ENTER" after each value.
  5. Press the buttons "2nd" and "MODE" (QUIT) to signify the end of the data.
  6. Press the "STAT" button.
  7. Use the right arrow to highlight the "CALC" selection.
  8. Choose "1:1-Var Stats" by pressing "ENTER".
  9. Press "ENTER" again.
  10. Use the down arrow to scroll the remaining statistics on the screen.
  11. Press "CLEAR" to clear the screen.

A.  The sample mean is 65.5 inches and the sample standard deviation is 3 inches.  These values will be used as 
      estimates for the population mean and population standard deviation when calculating the standard (z) scores.

ti-83.gif (1031 bytes)

Using the library function "normalcdf" on your TI-83 calculator.
The cumulative probability distribution function for the Normal Distribution.

1.  To access the "Distributions" on the TI-83, press the "2nd" button followed by the "VARS" button.
2.  Scroll down and highlight "2:normalcdf(".  Then press "ENTER".  This selects the  normal cumulative probability
    distribution.  The normalcdf function calculates area under the Normal curve between two data points.  The syntax 
    for the normalcdf command is:  normalcdf(x1, x2, , )."
    Where:  x1 is on the left and x2 is on the right.
                   is the mean of the distribution
                  is the standard deviation of the distribution

Note:  For an x score of negative infinity (less than problems), use 10 standard deviations to the left of the mean.
            In this case use 65.5 - 10(3) = 35.5.

            For an x score of positive infinity (greater than problems), use 10 standard deviations to the right of the mean.
            In this case use 65.6 + 10(3) = 95.5.

    B1.  What percent of female adults are taller than 6 feet?

         The percent of female adults taller than 6 feet is about 1.5%.  This matches the answer using the table.

     B2.  What percent of female adults are taller than 5 feet?

           About 96.7% of females are taller than 5 feet.  This answer differs from the answer using the table by 0.1%.

     B3.  What percent of female adult heights are between 60 inches and 72 inches?

           The percent of female adult heights between 60 inches and 72 inches is  about 95.1%.
           This answer matches the answer using the table.

ti-83.gif (1031 bytes)

Using the library function "invNorm" on your TI-83 calculator.
The inverse cumulative probability distribution function for the Normal Distribution.

An alternate solution to finding the solution to Example 3 is to use the inverse cumulative normal  probability function on the TI-83.

1.  To access the "Distributions" on the TI-83, press the "2nd" button followed by the "VARS" button.
2.  Scroll down and highlight "3:invNorm(".  Then press "ENTER".  This selects the  normal cumulative 
     probability distribution.  The invNorm function calculates data value, x, for a given area under the Normal 
     curve to the left of the x value.
     The syntax for the invNorm command is:  invNorm(area, , ).

    C1.  The bottom 8%

             The shortest female height for leather pants is approximately 5 ft 1 in.
             This answer agrees with the answer using the table.

     C2.  The upper 6%.
             Since the "InvNorm" program works to the left, use 0.94 in the program.

             The tallest female height for leather pants is approximately 5 ft 10 in.
             This answer agrees with the answer using the table.

II.  The male division of the company

Male Data in inches

68

65.5

68.1

72.5

65.4

71.2

67.7

73.5

67.7

65.1

65.3

65.5

72

73.2

62.5

77.2

70.5

66.7

67.5

70.2

67.4

71.8

65.1

67.2

66.3

69.3

67.7

67

73.8

66.5

66.1

68.6

 

ti-83.gif (1031 bytes)

Using the STAT functions to enter data and calculate the sample mean 
and the sample standard deviation
  1. Press the "STAT" button.
  2. With "EDIT" highlighted select "1:Edit" by pressing "ENTER".
  3. If there is data in List 1 (L1) clear it by using the up arrow to highlight "L1".  Then press "CLEAR" and "ENTER".
  4. Enter your data into List 1 (L1) by entering each value and pressing "ENTER" after each value.
  5. Press the buttons "2nd" and "MODE" (QUIT) to signify the end of the data.
  6. Press the "STAT" button.
  7. Use the right arrow to highlight the "CALC" selection.
  8. Choose "1:1-Var Stats" by pressing "ENTER".
  9. Press "ENTER" again.
  10. Use the down arrow to scroll the remaining statistics on the screen.
  11. Press "CLEAR" to clear the screen.

A.  The sample mean is 68.5 inches and the sample standard deviation is 3.25 inches.  These values will be used as 
      estimates for the population mean and population standard deviation when calculating the standard (z) scores.

ti-83.gif (1031 bytes)

Using the library function "normalcdf" on your TI-83 calculator.
The cumulative probability distribution function for the Normal Distribution.

1.  To access the "Distributions" on the TI-83, press the "2nd" button followed by the "VARS" button.
2.  Scroll down and highlight "2:normalcdf(".  Then press "ENTER".  This selects the  normal cumulative probability
    distribution.  The normalcdf function calculates area under the Normal curve between two data points.  The syntax 
    for the normalcdf command is:  normalcdf(x1, x2, , )."
    Where:  x1 is on the left and x2 is on the right.
                   is the mean of the distribution
                  is the standard deviation of the distribution

Note:  For an x score of negative infinity (less than problems), use 10 standard deviations to the left of the mean.
            In this case use 68.5 - 10(3.25) = 36.

            For an x score of positive infinity (greater than problems), use 10 standard deviations to the right of the mean.
            In this case use 68.5 + 10(3.25) = 101.

   B1.  What percent of male adults are shorter than 6 feet?

             The percent of male adults shorter than 6 feet is about 85.9%.  
             This answer differs from the answer using the table by 0.1%.

      B2.  What percent of male adults are shorter than 5 feet?

            The percent of male adults shorter than 5 feet is about 0.4%.
            This answer agrees with the answer using the table.

      B3.  What percent of male adult heights are between 60 inches and 72 inches?

            The percent of male adult heights between 60 inches and 72 inches is about 85.5%.
            This answer differs from the answer using the table by 0.1%.

ti-83.gif (1031 bytes)

Using the library function "invNorm" on your TI-83 calculator.
The inverse cumulative probability distribution function for the Normal Distribution.

An alternate solution to finding the solution to Example 3 is to use the inverse cumulative normal  probability function on the TI-83.

1.  To access the "Distributions" on the TI-83, press the "2nd" button followed by the "VARS" button.
2.  Scroll down and highlight "3:invNorm(".  Then press "ENTER".  This selects the  normal cumulative 
     probability distribution.  The invNorm function calculates data value, x, for a given area under the Normal 
     curve to the left of the x value.
     The syntax for the invNorm command is:  invNorm(area, , ).

     C1.  The bottom 9%.

              The shortest male height for leather pants is approximately 5 feet 4 inches.
              This answer agrees with the answer using the table.

     C2.  The upper 7%.
              Since the "InvNorm" program works to the left, use 0.93 in the program.

             The tallest male height for leather pants is approximately 6 feet 1 inch.
             This answer agrees with the answer using the table.

Click on the hand to return to the "It's Real!", Seniors Exclaim  projects.  

Click on the hand to return to the Gen. Ed. Math. projects.