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Resources for this lesson:

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Here is the distribution of the normal curve you are familiar with.

The graph shows a normal curve. The horizontal axis is labeled ‘standard deviations’ and extends from negative infinity to infinity (not shown), counting in increments of one. There is no vertical axis shown The curve starts just above where negative 4 would be, rises to a maximum point when x = 0, and then decreases to just above where 4 would be. The area under the curve is divided into eight sections. The first section extends from negative infinity to negative 3, and is labeled 0.13%. The second section extends from negative 3 to negative 2, and is labeled 2.15%. The third section extends from negative 2 to negative 1, and is labeled 13.59%. The fourth section extends from negative 1 to 0, and is labeled 34.13%. The fifth section extends from 0 to 1, and is labeled 34.13%. The sixth section extends from 1 to 2, and is labeled 13.59%. The seventh section extends from 2 to 3, and is labeled 2.15%. The eighth section extends from 3 to infinity and is labeled .13%. Above the curve, different sections of the area under the curve are detailed. The section that extends from negative 1 to 1 is labeled 68.26%. The section that extends from negative 2 to 2 is labeled 95.44%. The section that extends from negative 3 to 3 is labeled 99.74%

The graphing calculator can graph normal distributions and it will provide the different percentiles to which Justyce refers. (This, of course, makes our friend Khalid very happy.)

Let’s use the data from the beginning of this lesson, which states that the ages of the presidents at inauguration form a normal distribution with a mean of μ = 54.659 and a standard deviation of sigma = 6.186.

To obtain a graph of a normal distribution, follow these steps using the graphing calculator:

Check Your Understanding


A screenshot from the graphing calculator is shown. The Window menu is displayed. The x minimum is set to 35, the x maximum is set to 75, the x scale is set to one, the y minimum is set to zero, the y maximum is set to one tenth, the y scale is set to one, and x resolution is left at oneBefore we continue with analyzing the data, let’s discuss why the window should be set like this. The Xmin and Xmax correspond to the presidents’ minimum and maximum ages.

The normal curve is always above the x-axis, so the Ymin is always 0. The y-axis is the frequency, or number of times, the different ages showed up in the data. This is now a percentage, and not a whole number. A formula that can be used to determine the Ymax value for the Window is

y max equal zero point four over sigma

A screenshot from the graphing calculator is shown. A bell shaped normal curve is shown. The curve is being traced, with the value x equals 53 and y equals 0.06221313 shown.For example, press TRACE after graphing the normal curve. The x-values that appear are the various ages of the presidents at inauguration. The corresponding y-values are the percentage of presidents that were at that age at inauguration.

Type in 53 and hit ENTER. The y-value you get is 0.062. This means that (approximately) 6.2% of the presidents were 53 years old at inauguration.

Real-Life Scenarios


> Text version for animation opens in new window

Check Your Understanding

 

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