Download I. Getting To Know The Function

Math 243
In Class Worksheet - THE NORMAL DISTRIBUTION (do not turn in)
The normal distribution is an idealized mathematical concept. Many real populations are modeled by this
distribution. Your book shows you
the graph of this distribution. The
H e ig h ts o f M e n b e tw e e n th e A g e s o f 1 8 - 2 5 y rs o f
function that creates that graph is
σ 2π
e
-1 x - μ 


2 σ 
age
2
.
0.12
Notice, that this function is
determined by the population mean,
, and the population standard
deviation, . The numbers  and e
are the irrational numbers
approximately equal to 3.14 and
2.72 respectively.
Va lue of function
given by f(x) =
1
0.1
A re a re p re s e n t s t h e
0.08
fre q u e n c y o f m e n
0.06
betw een 64 and 68
0.04
in c h e s t a ll
0.02
0
50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90
H e i g h t i n i n c h e s (X )
xi
While the xi value is a particular
measurement taken from an
individual in the population. This
function creates a graph whose area underneath is one square unit. The area captured between any two
values on the horizontal axis will determine the frequency that we should find a number.
Let the variable X equal the height of a male in the state of Oregon between the ages of 18 to 25 years.
Then the variable xi would represent any value from that population.
If we asked the question, “approximately how many males are between 64 and 68 inches tall?”, then by
knowing the population parameters  and , it will allow us to create the graph and thus find the area
underneath the curve between those two values which will coincide with the frequency of this event.
I. Getting To Know The Function
If you have a graphing calculator you could graph the normal distribution function on your calculator, so
you can see how changing the population mean and population standard deviation alters the graph.
Suppose you want to graph a normally distributed population with a mean of 50 and a standard deviation of
4. Depending on the calculator you have, get to the screen that allows you to enter your function. Once
you are on the right screen, type the following on the appropriate place for your calculator.
1/(4 (2))*e^(-0.5*(x - 50)^2/4^2).
Now adjust the “window” on your calculator so you can see the graph. Set xMin = 38 (minimum value
for the x variable, independent), xMax = 62 (maximum value for the x-variable), yMin = 0 (minimum
value for the y variable, dependent), yMax = 0.2 (maximum value for the y variable). Depending on the
calculator you have the names described above can differ.
Now we will repeat the process except we will change the standard deviation to 2. Do not erase the
original function. I want to superimpose both functions. On another line type
1/(2 (2))*e^(-0.5*(x - 50)^2/2^2).
Graph the functions. Both graphs should be now in view. What occurred to the second graph with respect
to the first? At home explore changing the values for the mean and standard deviation to see how the graph
changes. Both graphs will have an area of 1 underneath the curve.
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II. The following questions refer to the histogram shown. Notice that the area for the rectangle
contained between $25,000 and $30,000 should be 32% of the total area with respect to all the
rectangles shown.
a) Approximately, what percentage of
the graduates earn a starting salary
below $25,000?
b) Approximately, what percentage of
the graduates earn a starting salary
between $30,000 and $45,000?
c) What is the general shape of the
distribution?
III. Random Number Generators and the Normal Distribution.
The purpose of this exercise is to let you see the relationship between the idealized normal
distribution and randomly acquired data. Excel is equipped to simulate sampling from a normal
population by creating a list of “random” numbers whose distribution is normal.
From Excel, go to Tools, and select Data Analysis. Scroll down until you can see Random
Number Generation. Select this option by clicking it with your mouse, and then click OK. A
window should have popped up. Make the following changes in that window.
Go to Distribution: and select Normal from the drop down menu.
The Number of Variables refers to how many samples do you want to create. Type the number 1
in the box provided.
For Number of Random Numbers this refers to your sample size. Type 10000.
Go to the Parameters and for mean type 50 and for standard deviation type 4.
The Random Seed category you can type any number between 1 and 32000. This number starts
the algorithm that creates the random list of numbers. If you leave it blank the computer will
choose a number.
Finally, click on the button Output, and click on the box next to it immediately. Type A1 in this
box. A1 refers to the location where the first number will be placed.
The other numbers will be placed underneath cell A1.
Now click on OK.
In another column type a bin range so you can create a histogram. For a
suitable bin range start at 36 and go to 68 in steps of 2. Create your
histogram. The Excel program should have also created a table for you
that states the number of data points that fall in that particular class.
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Next to the column entitled frequency, create a relative frequency column for your data.
a) Now answer the following question. For your distribution what percentage of the samples fall
at or below 42?
b) Now, go to your toolbar and click on the fx symbol (these steps are another way to calculate
the frequency instead of using the table in your book).
 Select Statistical from your function Category.
 And now, from function name select NormDist and click OK.
 For the X value type 42. For mean type 50 and for standard distribution type 4. Finally, type
true for Cumulative. Click OK.
c) Are your answers in part a and b close?
IV. Finding Frequencies for Situations Involving the Normal Distribution.
Example 1 - The length of life of an electric motor used in a ceiling fan is approximately
normally distributed with a mean of 6.4 years and a standard deviation of 1.1 years.




If the fan motor is guaranteed for five years, what is the probability that replacement under
the guarantee will be required?
If the manufacturer is willing to replace only 1% of the fan motors, what period of time
should be used for the guarantee period?
For the fans lasting beyond the 6.4 years, how many years do the longest functioning 5% of
the fans last?
How many fans would the manufacturer expect to malfunction after 6 years?
Example 2 – Over the years, the number of breakfast served per day in the company cafeteria
follows a normal distribution with mean of 150 meals served with a standard deviation of 15
meals. What is the probability that the number of breakfasts served on a randomly selected
workday will be between 165 and 180 meals?
Example 3 - The yearly income in a community is normally distributed, with a mean of $19,000
and a standard deviation of $2000.





What percentage of people earn less than $15000 per year?
What percentage of people earn more than $25,000 per year?
What percentage of people earn between $17,000 and $20,000 per year?
What minimum income does a member of this community have to earn in order to be in the
top 10%?
What is the maximum income one can have and still be in the middle 50%
Example 4 - A consumer reporter finds out that this month the current cost of a yellow onion is
on average $0.39/lb with a standard deviation of $0.03/lb. While white onions cost on average of
$0.67/lb with a standard deviation of $0.07/lb. The shopper goes to a particular store and finds
that the cost of the yellow onion costs $0.42/lb while the white onion costs $0.75/lb. Relatively
speaking, which onion is more expensive with respect to their perspective distributions?
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