MAT 158 PROJECT 3
Objectives: At
the end of this project you should know the following.
1.
The Normal Distribution
2.
Drawing a Normal Distribution Curve
3.
Evaluating Probabilities Using Normal Distribution
4.
Shifting a Normal Curve by Changing its Mean and Standard Deviation
5.
The Central Limit Theorem
1. Generate a 100 random data from a normal distribution with mean 75 and standard deviation 7; and store the data in C1.
Ø
Calc®Random data®Normal
Ø
Generate:
100
Ø
Store
in: C1
Ø
Mean:
75
Ø
Standard
deviation: 7
Ø
Click
OK
2.
Standardize
your data and store the z-values in C4.
Ø
Calc®Standardize
Ø
Input
Column: C1
Ø
Store
results in: C4
Ø
Click
on: Subtract mean and divide by standard deviation
Ø
OK
3. Calculate the probability density function values and store them in C2 (this gives you the y-values stored in C2 paired with the x-values that are already stored in C1, from a normal distribution with mean 75 and standard deviation 7).
Ø
Calc
® Prob dist ® Normal
Ø
Select:
Probability density
Ø
Mean:
75
Ø
Standard
deviation: 7
Ø
Input
Column: C1
Ø
Optional
Storage: C2
Ø
OK
Ø
Print
4. Draw the graph of a normal probability curve with x-values in C1 and y-values in C2.
Ø
Graph®Plot
Ø
Graph
variables: Y:C2, X:C1
Ø
Data
display: Display: Connect
Ø
OK
Ø
Print
Graph
5. Find the Cumulative probabilities (store them in C3) paired with the x-values in C1 from a normal distribution with mean 75 and standard deviation 7.
Ø
Calc®Prob dist®Normal
Ø
Select:
Cumulative Probabilities
Ø
Mean:75
Ø
Standard
deviation: 7
Ø
Input
Column: C1
Ø
Optional
storage: C3
Ø
OK
Ø
Print
6. Find P(x < 75), P(54 < x < 96), and P(x > 90), using the following commands.
Ø
Calc®Prob dist®Normal
Ø
Click
on: Cumulative Probabilities
Ø
Click
on : Input Constant and enter value for x
Ø
Enter
the value for x : 75, 54, 96, and 90
Ø
OK
Ø
Print
Important
Note: You need to work out your answers because the values you will get from
Minitab are cumulative probabilities.
7. Generate a 100 random data from a normal distribution with mean =80 and same standard deviation as the previous distribution (standard deviation=7).
Ø
Calc®Random Data®Normal
Ø
Generate:
100
Ø
Store
in: C5
Ø
Mean:
80
Ø
Standard
deviation: 7
8. Calculate the probability density function values of C5 and store them in C6.
Ø
Calc
® Probability distributions ®
Normal
Ø
Select:
Probability density
Ø
Mean:
80
Ø
Standard
deviation: 7
Ø
Input
Column: C5
Ø
Optional
Storage: C6
Ø
Print
9. Plot two normal curves on the same graph (different means, same standard deviation).
Ø
Graph
® Plot
Ø
For
Graph 1: Enter C2 for Y and C1 for X
Ø
For
Graph 2: Enter C6 for Y and C5 for X
Ø
Click
on arrow next to Frame
Ø
Select
Multiple Graphs
Ø
In
the Multiple Graph dialogue box Select: Overlay Graphs on same page
Ø
Click
OK, twice
Ø
Print
Question: Look at the graph you just printed out and draw your conclusion about
the shape of the graphs.
10. Generate a 100 random data from a normal distribution with mean = 75 and standard deviation = 3
Ø
Calc®Random Data®Normal
Ø
Generate:
100
Ø
Store
in: C7
Ø
Mean:
75
Ø
Standard
deviation: 3
11. Calculate the probability density function values of C7 and store them in C8.
Ø
Calc
® Probability distributions ®
Normal
Ø
Select:
Probability density
Ø
Mean:
75
Ø
Standard
deviation: 3
Ø
Input
Column: C7
Ø
Optional
Storage: C8
Ø
Print
12. Plot two normal curves on the same graph (Same mean but Different standard deviations).
Ø
Graph
® Plot
Ø
For
Graph 1: Enter C2 for Y and C1 for X
Ø
For
Graph 2: Enter C8 for Y and C7 for X
Ø
Click
on arrow next to Frame
Ø
Select
Multiple Graphs
Ø
In
the Multiple Graph dialogue box Select: Overlay Graphs on same page
Ø
Click
OK, twice
Ø
Print
Question: Look at the graph you just printed out and draw your conclusion about
the shape of the graphs.
1.
Randomly
generate 100 samples of size 6 (each) from a Normal Distribution of mean 20 and
standard deviation 4.5.
Ø
Calc
® Random data ® Normal
Ø
Generate:
100 rows of data
Ø
Store
in Columns: C1-C6
Ø
Mean:
20
Ø
Standard
deviation: 4.5
Ø
OK
2. Find the mean for each of the 100 samples and store the result in column C7.
Ø
Calc
® Row Statistics
Ø
Under
Row Statistics: Click on Mean
Ø
Input
variables: C1-C6
Ø
Store
result in: C7
Ø
OK
Ø
Print
3. Using the 100 sample means in Column C7, Construct a Histogram; also, find the mean and standard deviation of the distribution of the 100 sample means.
Ø
Graph
® Histogram
Ø
Under
Graph 1 Enter : C7
Ø
Choose:
Options
Ø
Click
on : Cutpoint
Ø
Click
OK, twice
Ø
Print
Graph
Ø
Calc
® Column statistics ® Mean ® C7 ® OK
Ø
Calc
® Column statistics ® Standard
deviation ® C7 ® OK
Ø
Print
Session
4.
Compare the above results with the Central Limit Theorem. State you Conclusion