Transcript Descriptive Statistics and Inferential Statistics

The TLSAMP Summer
Science Academic Program
Summer 2006
Descriptive Statistics and
Inferential Statistics
Population
Sample
Descriptive Statistics

Collecting Data
 Summarizing
 Describing the Characteristics
 True Statement
 Based on Sample Data
Inferential Statistics

Information from samples
 Conclusions
 Generalizations
 Estimate of the population
 Inference based on set of data
Physical Science Laboratory

Density
 Gravity
 Building’s Height
 Vector
 Friction
Density

By:
Amanda Linsey, Kenny Houston, & Jena` E. Murphy
Density
Theorem
 Density is the ratio of mass to size (volume).
D= Mass (M)
Volume (V)
 Volume for cylinders is found using the
formula:
V= Pie*r2 *h
 Mass is found by measuring the
object on a balance
Experiment

First you measure the length of
the rod and wire with a meter
stick, then you measure the
diameter of the rod and wire with
the venire caliper.

Next you calculate the mass and
volume of the objects using the
proper formulas (v=pie*r2*h)
(m=d / v)

Then you get the mass and
volume of the two objects, you
calculate the experimental
density of the objects using the
mass/volume formula.

After getting the experimental
density, find out the percent error
from the accepted density that
was already given.
Comparisons and Statistics
Cylinder of Tin
Height
Volume=
Pie*
Square
of R * H
Mass
Experimental
Density
Accepted
Density
Pb
g/(cm^3)
Diameter
Radius
Square
of Radius
1.85
0.925
0.856
3.14
3.1
8.33
64.5
7.744
7.28
6.38%
1.9
0.95
0.903
3.14
3.2
9.07
65
7.168
7.28
1.54%
1.8
0.9
0.810
3.14
3
7.63
64.6
8.466
7.28
16.30%
1.9
0.95
0.903
3.14
3.1
8.78
65
7.399
7.28
1.64%
1.89
0.945
0.893
3.14
3.14
8.80
63.6
7.223
7.28
0.78%
Pie
Percent
Cylinder of Lead
Height
Volume=
Pie*
Square
of R * H
Mass
Experimental
Density
Accepted
Density
Pb
g/(cm^3)
Diameter
Radius
Square
of Radius
1.9
0.95
0.903
3.14
2.03
5.752716
65.1
11.316
11.3
0.15%
1.9
0.95
0.903
3.14
2
5.6677
61.8
10.904
11.3
3.51%
1.8
0.9
0.810
3.14
1.9
4.83246
63.7
13.182
11.3
16.65%
1.9
0.95
0.903
3.14
2
5.6677
64.91
11.453
11.3
1.35%
1.89
0.945
0.893
3.14
2.01
5.636238
63.7
11.302
11.3
0.02%
Pie
Percent
gravity
The Scientific Method: The Simple Pendulum
theorem
For the Simple
Pendulum, the
formula is
and for small Angle of
Swing, the equation
reduces to
To solve for Gravity,
T  2
L 1 2  9

4
1  sin  sin  ...
g 4
2 64
2

T  2
L
g
4  2  L
Gravity 
T2
Experiment procedure
1.
2.
3.
4.
5.
Set up a simple pendulum
arrangement.
Measure the length of the
pendulum and record the
data onto a chart.
Slightly push the
pendulum.
Calculate the time that it
takes the pendulum to
make one full revolution (a
revolution is when the
pendulum moves from
point A to point B and
back to point A).
Record the number of
revolutions and the time
onto the chart.
Comparison & statistics
Number of
Revolutions
Time
Length
Period
Pi
Experimental
Gravity
Accepted
Gravity
Percent
Error
31
36.94
30.4
1.19
3.14
847.50
981 cm/s²
13.6%
26
29.9
30.4
1.15
3.14
907.48
981 cm/s²
7.5%
25
28.4
30.4
1.13
3.14
929.86
981 cm/s²
5.2%
10
13.2
30.3
1.32
3.14
686.52
981 cm/s²
30%
10
12.66
30.4
1.266
3.14
748.8
981 cm/s²
23.6%
20
22.1
30.3
1.10
3.14
979.66
981 cm/s²
0.1%
13
14.9
30.9
1.14
3.14
928.60
981 cm/s²
5.3%
8
10.1
30.9
1.26
3.14
765.34
981 cm/s²
22%
17
18.8
30.4
1.11
3.14
974.06
981 cm/s²
.71%
BUILDING’S HEIGHT
BY: Desmond Robertson
Fred Peete
Justin Vann
Desmond Robertson
*Theorem Guy
CBHS
TRACK AND FOOTBALL
Fred Peete
*Experiment Guy
G- Town
Track n Field
Justin Vann
*Comparison & Statistics Guy
G- Town
THE LADIES MAN!!!
Similar Triangles
Triangle AED is Similar to Triangle ABC
B
E
E
h
y
D
h
C
D
d
A
A
D
d
•Both Angle A’s are the same because they are the Angle of sight.
•Angle D and C are equal because they are both 90 degree angle’s.
Two triangles will be similar if two of their Angles are equal
The ratio of corresponding sides of two similar triangles are equal
y
Dd

h
d
Buildings Height Experiment
1.
2.
3.
4.
First you have to get an
obstacle that you can start
the height of the little
triangle, which is the O
Second, you get someone to
squat down in front of the
obstacle until they are
perfectly squared with the
building you are measuring,
which is P. Then you would
measure from the person’s
eye distance, height, and
foot distance from the
obstacle too
Third measure the distance
from the building to the
obstacle, which is D
Fourth find the height of the
building by making an
imaginary line from the
obstacle to the building,
which is H
y Dd
( D  d )h

y
h
d
d
H  y p
Using the concept of similar triangles to find the
height of "GOH" building from Parking Lot
Percent Error
Height of Building H = y + P
36
227.6
24.4
55.2
570.10
606.10
588
3.08%
77
44
362
18.3
33
685.79
729.79
588
24.11%
92.8
57.5
191.9
11
35.3
651.12
708.62
588
20.51%
Height of Obstacle O
Height of Observer P
Distance from Building to Obstacle
D
Distance from Observer to Obstacle
d
Difference between Heights of
Obstacle and Observer h = O - P
Height of building from Mr. Ali's
Measurement
y = ((D + d ) / d ) * h
91.2
The Addition and Resolution
of vectors: The Force Table
Gregory Cowan
Kalishia Tidwell
Candace Kirk
Theorem
To find the sum of two or more vectors:
1.
Find all the X components of the vectors
2.
Find all the Y components of the vectors
3. Use Pythagorean theorem to find the hypothesis
which is the magnitude of the resulting vectors
4.
To find the direction, we divide the sum of all Y
components by the sum of all X components and
find the Arc tan of that result
Actual Resultant Force
 F    F 
F 

  arct an

F
 
R
2
2
x
y
y
x
Two or more concurrent forces can be replaced by a
single resultant force that is statically equivalent to these
forces.
Experimental
1. We set up a force table with two threads.
One is hanging 150 units and the angle is
0 degree. The other force is 250 units
and the angle is 90 degrees.
2. After putting two threads on the desired
positions, the third thread is added to
balance the force table.
3. Once the force table is balanced, record
the angle and the weight of the third
thread.
4. Repeat the above methods on the
positions of 70o & 200o and 150o & 290o
Comparison & Statistics
RACT =
sqrt((ΣFx)2+
(ΣFy)2)
ӨACT =
ARCTAN
(ΣFy / Fx)
291.55
59.04
350
50
20.05%
15.31%
191.81
163.2
250
160
30.34%
1.96%
165.97
254.48
79
259
52.40%
1.77%
291.55
59.04
300
62
2.90%
5.02%
191.81
163.2
190
163
0.94%
0.12%
165.97
254.48
120
222
27.70%
12.76%
291.55
59.04
300
60
2.90%
1.63%
191.81
163.2
200
164
4.27%
0.49%
165.97
254.48
160
258
3.60%
1.38%
291.55
59.04
300
61
2.90%
3.33%
191.81
163.2
200
163
4.27%
0.12%
165.97
254.48
150
255
9.62%
0.20%
Өexp
Rexp
% Error of R
% Error of Ө
Friction Theorem
• Friction is the force that opposes the
relative motion or tendency of such motion
of two surfaces in contact.
• Friction is the force applied in the opposite
direction of an object’s velocity.
• Friction is independent of the surface
contact area.
Theorem cont.
Friction Facts
• When two smooth surfaces rub together there is
very little friction.
• When two rough surfaces rub together there is
more friction.
• There is less friction when there is a liquid
between the two surfaces.
• There is more friction if the two surfaces are
forced against each other.
Experiment
The main objective of our experiment was to
determine µs.
µs is what we wanted to calculate which is the coefficient of static
friction.
Formulas used
• N = (Mb + Mw)g
N is the normal force
Mb is the mass of the block
Mw is the added mass
g is gravity
• Fs = F = Mg
• µs = M/(Mb + Mw)
Experimental Results
Mass of Block Mb
217
217
217
217
217
Added Mass Mw
0
100
200
400
500
N / g = Mb + Mw
217
317
417
617
717
fs / g =F / g = M
50
67
85
130
150
µs=M / (Mb+Mw)
0.2304
0.2114
0.2038
0.2107
0.2092
Mass of Block Mb
321
421
521
721
821
Added Mass Mw
0
100
200
400
500
N / g = Mb + Mw
321
521
721
1121
1321
fs / g =F / g = M
60
80
100
130
150
µs=M / (Mb+Mw)
0.1869
0.1536
0.1387
0.1160
0.1136
Mass of Block Mb
272.3
272.3
272.3
272.3
272.3
Added Mass Mw
0
100
200
400
500
N / g = Mb + Mw
272.3
372.3
472.3
672.3
772.3
fs / g =F / g = M
60
80
100
150
170
µs=M / (Mb+Mw)
0.2203
0.2149
0.2117
0.2231
0.2201
Experimental
Coefficient
of Static
Friction
Expected
Coefficient of
Static Friction
Percent
Error
0.230415
0.2
15.21%
0.211356
0.2
5.68%
0.203837
0.2
1.92%
0.210697
0.2
5.35%
0.209205
0.2
4.60%
0.186916
0.2
6.54%
0.153551
0.2
23.22%
0.138696
0.2
30.65%
0.115968
0.2
42.02%
0.11355
0.2
43.23%
0.220345
0.2
10.17%
0.21488
0.2
7.44%
0.21173
0.2
5.87%
0.223115
0.2
11.56%
0.220122
0.2
10.06%
Descriptive Statistics:
•
•
•
•
Using Excel’s arithmetical operations to
do calculations
--------Jeremy Neville
Using Excel’s functions to find summary
statistics
--------Terrell Hudson
Using data analysis package to find
summary statistics ------Geneo Fleming
Calculating mean and standard deviation
of quiz scores
------Kimberly Johnson
Finding the outliers
Original
Data
Sorted
Data
884.71
686.52
1
median
9.5
954.83
907.48
748.80
2
Q1
4.75
857.38
929.99
765.34
3
Q3
14.25
1026.68
686.52
845.21
4
IQR
1025.55
861.44
5
979.67
884.71
6
928.61
907.48
7
861.44
928.61
8
765.34
929.99
9
2318.77
979.67
10
1942.87
981.33
11
1678.53
985.66
12
985.66
1008.47
13
1008.47
1025.55
14
1030.07
1030.07
15
845.21
1678.53
16
981.33
1942.87
17
748.80
2318.77
18
position
Position
value
169.30
Lower fence
603.44
upper fence
1280.62
Lower fence=Q1-1.5*IQR
Upper fence=Q3+1.5*IQR
Box Plot
1030.07
686.52
603.44
857.38
954.83 1026.68
1280.62
Hypothesis Testing


The Null Hypothesis is the theory that is
assumed to be true unless proven
otherwise.
The Alternative Hypothesis can only be
believed valid if the null hypothesis can be
disproved.
Hypothesis Testing – Conti.



In order to reject or fail to reject the null
hypothesis you must find the test statistic (tstatistic) from your sample and calculate your pvalue.
If your p-value is less than your significance level,
then you reject your null hypothesis. If it is greater
,then you fail to reject it.
If you reject the null hypothesis, then you accept
the alternative hypothesis.
Gravity Experiment Results
884.7
907.5
930.0
686.5
1025.5
979.7
928.6
861.4
765.3
2318.8
1942.9
1678.5
985.7
1008.5
1030.1
845.2
981.3
748.8
Mean
904.6
Standard Error
Median
Mode
Standard Deviation
Sample Variance
27.4
928.6
#N/A
106.06
11249.57
Kurtosis
-0.36
Skewness
-0.75
Range
343.55
Minimum
686.52
Maximum
1030.07
Sum
Count
13568.84
15
H 0 :   981
H1 :   981
t-stat = -2.79
p-value = 0.014
p-value < 0.05, reject H0
Need carefully counting time
Need more experiments
Experimental Coefficients of Static Friction are
0.2304
0.2114
0.2038
0.2107
0.2092
0.1869
0.1536
0.1160
0.1136
0.2203
0.2149
0.2117
0.2231
0.2201
H 0 :   0.2
H a :   0.2
t-stat
p-value
-0.8759
0.3959
Fail to reject H0 since p-value
> 0.05.
0.1387
Coefficient of static friction
Mean
0.191
Standard Error
0.010
Median
0.211
Mode
#N/A
Standard Deviation
0.040
Sample Variance
0.002
Kurtosis
-0.203
Skewness
-1.146
Range
0.117
Minimum
0.114
Maximum
0.230
Sum
2.864
Count
Therefore, the coefficients of static friction is 0.2.
15
Linear Regression


Is to determine the relationship
between two quantities.
For example: in the gravity experiment
the time was dependant on the number
of revolutions.
Definition


Linear regression is defined as the
process of collecting data , identifying
relationships within the data, and
predicting future information.
We discussed linear regression in two
major parts; the correlation coefficient
and the line of regression.
The Correlation Coefficient




The strength and direction of the
relationship between the X and Y
factors of the data.
 1  r  1 r is the correlation
coefficient
If:r  0 there is a weak relationship.
If: r  1or r  1 there is a strong
relationship
Relationship Between Time
and number of revolutions
# of
Revol
utions
Time
Coeffic
ients
Standa
rd
Error
t Stat
P-value
25.5
29.7
Intercept
0.719
0.653
1.102
0.291
26
29.9
# of Revolutions
1.110
0.033
33.962
0.000
25
28.4
10
13.2
15
16.2
20
22.1
13
14.9
10
11.9
Regression Statistics
8
10.1
Multiple R
0.9944
21
23.4
R Square
0.9889
31
34.15
Adjusted R Square
0.9880
10
15
16.35
Standard Error
0.9539
0
31
36.94
Observations
17
18.8
10
12.66
Time  1.110 numbersof revolutions  0.719
# of Revolutions Line Fit Plot
40
15
Time
30
Time
20
Predicted Time
0
20
# of Revolutions
40
Conclusion
Learned valuable information

Physical Science- different experiments:
–
–
–
–
–

Density – mass/volume (g/cm3)
Gravity – 981 cm/s2
Similar Triangles – ratio of corresponding sides are equal
Vectors – sum of two vectors
Friction – finding coefficient of the static friction
Statistics :
– Using Excel Program to do complex math, Box Plots,
Variance, Standard Deviation, Linear Regression, t-statistic
and p-value.

Algebra – Fractions, Decimals, Percents, Exponents