Download Lab 10 - Normal Distributions

QUANTITATIVE METHODS
Martin Huard
Fall 2004
LAB 10
Normal Distributions
Graphing Normal Distributions
Make the usual heading (cells A1:A3) and labeling sheet 1 “Graphs”.
Let us graph the normal distribution with mean µ = 20 and σ = 4 .
By the theory seen in class, almost all of the graph lies between
µ − 3σ = 20 − 3 ⋅ 4 = 8 and µ + 3σ = 20 + 3 ⋅ 4 = 32
To graph the normal curve, we will take approximately 50 values of x between 8 and 32 and find a
the corresponding value of y on the normal curve.
In cell A5 write “x” and in cell B5 write “y”.
In cell A6 write 8 and in cell A7 write 8.5. Use your mouse to generate a list of numbers all the
way up to 32 (cells A6:A54).
Go to cell B6 and click on f x , STATISTICAL – NORMDIST. Fill in the relevant information,
where for X you enter A6, for MEAN 20, for STANDARD_DEV 4 and lastly for cumulative
False. (False is used for graphing and true to figure out probabilities. We will come back to
this later on.). Copy this in cells B7:B54 with your mouse.
Now you can sketch the graph, where you will choose as Chart – Type LINE and the Chart Subtype the one in the top left corner. The results should be:
Normal Distribution with Mean 20 and
St. Dev 4
0.12
0.1
0.06
0.04
0.02
x
32
30
28
26
24
22
20
18
16
14
12
10
0
8
y
0.08
QM
Lab 10
We can also graph two (or more) normal distributions on the same graph. For this, let us add a
second column of y-values. In cell C5, write y2(st.dev = 2). Repeat what you did for column B
to column C, except that you use for the standard deviation 2 instead of 4, and change the
column headings. You should have something like this:
x
y1(st.dev = 4)
y2(st.dev = 2)
8
0.00110796
3.03794E-09
8.5
0.00159953
1.31962E-08
9
0.00227339
5.38488E-08
M
M
M
To graph both of these curves, the procedure is the same as before, except that for the DATA
RANGE, you blacken B6:C54. Make your result look like this.
Normal Distriubtions with Mean = 20
and St.dev = 2 or 4
0.25
0.2
0.15
y
y1(st.dev = 4)
y2(st.dev = 2)
0.1
0.05
32
29
26
23
20
17
14
11
8
0
x
Fall 2004
Martin Huard
2
QM
Lab 10
Calculating Probabilities for the Normal Distribution
To find the probability P ( x ≤ x0 ) with Excel, we use the same function NORMDIST as we did for
graphing, except that for CUMULATIVE, we write True.
Go to Sheet 2, and label it “Probabilities”. Make the appropriate heading in cells A1:A3.
In cell A5 write “Normal distribution with mean 2.5 and standard deviation 0.7.
In cells A6:A9, write the following:
P ( x < 3)
P ( 2 < x < 3)
P ( 3 < x < 5)
P ( x > 3.5 )
In cells B6:B9, find the answer to these probabilities, assuming that µ = 2.5 and σ = 0.7 . The
results should be:
P ( x < 3)
0.7625
P ( 2 < x < 3)
0.5249
P ( 3 < x < 5)
0.2373
P ( x > 3.5 )
0.0766
Repeat the same thing as before, below the above work, for the following two problems.
1. Find the following areas under a normal distribution curve with µ = 20 and σ = 4 .
a) Area between x = 20 and x = 27
b) Area between x = 23 and x = 25
c) Area between x = 9.5 and x = 17
Answers: a) 0.4599
b) 0.1210
c) 0.2223
2. Determine the area of the normal distribution curve with µ = 55 and σ = 7 .
a) to the right of x = 58
b) to the right of x = 43
c) to the left of x = 67
d) to the left of x = 24
Answers: a) 0.3341
b) 0.9568
c) 0.9568
d) 0.000005
Note: If you use the table to do this problem, you would obtain the following answers:
a) 0.3336
b) 0.9564
c) 0.9564
d) 0.000
Fall 2004
Martin Huard
3
QM
Lab 10
Finding the x value
For this, we use the command NORMINV, where the probability P ( x ≤ x0 ) is entered, that is, the
area to the left of x0 .
Go to Sheet 3, rename it and give the appropriate headings.
Answer the following questions in cells B6:B11
Let x be a continuous random variable that follows a normal distribution with a mean of 200 and a
standard deviation of 25.
a) Find the value of x so that the area under the normal curve to the left of x is
approximately 0.6330.
b) Find the value of x so that the area under the normal curve to the right of x is
approximately 0.05.
c) Find the value of x so that the area under the normal curve to the right of x is
approximately 0.8051.
d) Find the value of x so that the area under the normal curve to the left of x is
approximately 0.015.
e) Find the value of x so that the area under the normal curve between µ and x is
approximately 0.4525 and the value of x is smaller than µ .
f) Find the value of x so that the area under the normal curve between µ and x is
approximately 0.48 and the value of x is greater than µ .
Answers: a) 208.50 b) 241.12
Fall 2004
c) 178.50
d) 145.75
Martin Huard
e) 158.26
f) 251.34
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