Download SMAM 319 Computer Assignment 3 Use Minitab to do this problem

SMAM 319
Computer Assignment 3
Use Minitab to do this problem that demonstrates The Central Limit
Theorem
Do the computations in part A and answer the questions on page 2
stapled to the front of the computer output that should follow.
A.This part is to be done by hand. You will use the probability
mass(distribution) below with n is the number of letters in your last name.
1
, x = 1,2,3,…, n
n
For example if your last name is Doe your probability mass function is
f (x) =
1
1
1
f (1) = ,f (2) = ,f (3) =
3
3
3
For your value of the number n find the mean and the standard deviation
of the distribution.
For your value of n you should get an answer with numerical value.
n +1
n2 − 1
E( x =
sd(X) =
2
12
Thus, for John Doe
()
EX=2 sd(X)= 2 / 3
For me n = 6.
1
1
1
1
1
1 21 7
EX = + 2 ⋅ + 3⋅ + 4 ⋅ + 5⋅ + 6 ⋅ =
=
6
6
6
6
6
6 6 2
1
1
1
1
1
1 91
EX = + 22 ⋅ + 32 ⋅ + 42 ⋅ + 52 ⋅ + 62 ⋅ =
6
6
6
6
6
6 6
91 49 35
σ2 = −
=
6
4 12
σ = 1.708
You will now simulate your distribution 100 times.
calc>randomdata>integer
In dialog box
Minimum value 1
Maximum value 6
Generate 100 rows of data
Store in columns c1-c100
OK
Enable command language
enter the commands
rmeans c1-c100 c101
Make a stem and leaf display for c101 using the pull down menu or the
command
stem-and-leaf c101
describe c101
Welcome to Minitab, press F1 for help.
MTB > rmeans c1-c100 c101
MTB > stem-and-leaf c101
Stem-and-Leaf Display: C101
Stem-and-leaf of C101 N = 100
Leaf Unit = 0.010
1
1
2
10
25
45
(24)
31
15
6
1
29
30
31
32
33
34
35
36
37
38
39
0
6
23356799
011125567788889
00000123345666788888
000111111223344556677777
0000134455556899
022334578
03469
5
MTB > describe c101
Descriptive Statistics: C101
Variable
Q3
C101
3.6375
Variable
C101
N
N*
Mean
SE Mean
StDev
Minimum
Q1
Median
100
0
3.5105
0.0176
0.1761
2.9000
3.3925
3.5100
Maximum
3.9500
Make a normal probability plot c101.
Answer the following questions on the sheet stapled to the front of your
computer output.
1. Based on the stem and leaf display and the normal probability plot for
c101 does the data appear to be normally distributed? Explain your
answer.
The data appears to be normally distributed because the stem and leaf
display is bell shaped and the normal probability plot is close to a straight
line with the possible exception of one point.
2. What is the mean and the standard deviation obtained in the describe
command for c101?
Mean = 3.5105 Sd =.1761
3. What should the mean and standard deviation be in theory for c
101?[Hint: the mean is (n+1)/2 for your value of n .The standard
n 2 −1
/10
12n
The mean should be 3.5. The standard deviation should be .1708
4. Compare the mean and standard Deviation in questions 2 and 3 by
finding the percentage error?
deviation is
%error =
value in describe command - theory value
x100
theory value
3.5105 − 3.5
x100 = .3%
3.5
Standard deviation
Mean
%error =
%error =
.1761− .1708
x100 = 3.1%
.1708
5. State the Central Limit Theorem carefully and explain how the results
you obtain in c101 validate it for your problem.
The central limit theorem states that as n gets large the sample mean
tends to a normal distribution with the population mean and the
population sd divided by the square root of the number of observations.
For this data when n=100 the stem and leaf is bellshaped indicating
normality. Also the mean and standard deviations are reasonably close to
the values of the central limit theorem.
The solution will be different for students who work
independently but the conclusions should be similar. If
you get a large percentage error for your mean and
standard deviation check your calculation of the
theoretical mean and standard deviation. Chances are
pretty good that you made a mistake.