Welcome to Statistics using Catapults
September 25, 2015
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Linear Equation What do m and b represent in the equation y=mx+b? 20 18
y = 2x + 12 16 14
12
y
10 8 6 4 2 0
-5
-4
-3
-2
-1
0
1
2
3
4
5
x Will you ever use this in the “real” world? September 25, 2015
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Statistics ▶ The science of collecting, analyzing, interpreting, and
presenting data. ▶ Applied mathematics ▶ Utilizes the scientific method 1.
State the Problem
2.
Research Your Topic
3.
Develop a Hypothesis
4.
Perform an Experiment
5.
Gather and Record Data
6.
Analyze Your Results
7.
Draw Conclusions
8.
Report Your Results and Conclusion September 25, 2015
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Statistically Designed Experiments ▶ A designed experiment is a series of tests Change input variables in an organized method Measure the effect on the output variables
▶ A statistically designed experiment
Limits the number of experimental combinations (variables)
Provides a more organized experiment
Run fewer number of experiments (less resources, less time, less chaos)
Provides a statistical model showing the relationship between the input variables (x factors) and output variable (y response) y= mx + b
y = b + m1x1 + m2x2 + … + mixi
September 25, 2015
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The Catapult Experiment ▶ Learn about the catapult How many variables are on the Catapult? What other variables exist? We will ‘ignore’ some variables
▶ Launch and collect data Each team (4-5 students) has its own Catapult Record data
▶ Analyze & share results Numbers & Graphs September 25, 2015
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Catapult Variables How many variables/factors?
How many settings per variable/factor?
How many possible combinations of settings?
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Catapult Variables How many variables/factors?
5 How many settings per variable/factor?
3 How many possible combinations of settings?
35 =243 September 25, 2015
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Catapult Variables Make TWO of the variables constant - Arm Tension - Upright Tension
Now there are just THREE variables – how many combinations?
Do we need to use all 3 settings in our first experiment?
September 25, 2015
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Catapult Variables Make TWO of the variables constant - Arm Tension - Upright Tension
Now there are just THREE variables – how many combinations? 3
3 =27
Do we need to use all 3 settings in our first experiment?
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Statistical Hypothesis ▶ Two complementary hypotheses statements: Null Hypothesis: None of the 3 factors affect the
distance the ball travels Alternative Hypothesis: At least one factor affects
the distance the ball travels
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Catapult Experiment Worksheet ▶ 3 variables at 2 levels each = 2 x 2 x 2 = 23 = 8 combos ▶ Perform the launch of each combination 3 times each Combo Elevator Ball Seat Turntable 1 HIGH HIGH HIGH 2 HIGH HIGH low 3 HIGH low HIGH 4 HIGH low low 5 low HIGH HIGH 6 low HIGH low 7 low low HIGH 8 low low low
September 25, 2015
Distance Shot 1 2 3
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The Catapult Experiment ▶ Team roles: LAUNCHER – Launches the ping pong balls from the catapult STABILIZER – Holds the catapult launch pad in place PROCESS LEAD – Guide the team through process SPOTTER – Note where ball first hits DATA COLLECTOR – record distance on paper in centimeters (OPTIONAL E-RECORDER – Record data on iPad ‘Numbers’ App)
▶ Rotate roles or not? What is the benefit to keeping people in the same role?
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Let’s Start the Experiment ▶ Following the Scientific Method: Step #1 – State your problem • Identify if there is a specific combination of settings that will launch
the ping pong ball the longest distance Step #2 – Research your topic • Launch each of the 3 ping pong balls at different settings • Note how each of the settings [Elevator Height, Turntable Height,
Ball Seat] influence how the ping pong balls travel • Record your observations on your team Data Sheet • DO NOT record measurement data YET
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Continuing with the Scientific Method Step #3 – Develop a Hypothesis • You have been given the two Hypotheses, but can you make your
own Hypothesis about which setting will send the ping pong ball the farthest? • Note that on your datasheet by putting a star next to the combo
setting in the table (1-8) Identify who will take each role for the team • Step #4 – Perform an Experiment • Step #5 – Gather and Record Data
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Quickly decide who will perform each team role LAUNCHER – Launches the ping pong balls from the
catapult STABILIZER – Holds the catapult launch pad in place PROCESS LEAD – Guide the team through process SPOTTER – Note where ball first hits
DATA COLLECTOR – record distance on paper in
centimeters (OPTIONAL E-RECORDER – Record data on iPad
‘Numbers’ App) September 25, 2015
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Analyze the Results
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Questions 1. Which combination gave you the longest distance?
Class A Seat L/H B * TurnCTable L/H D Elevator L/H * Ball Session
E
F
Team 1 Team 2 Team 3 Team 4 Team 5
2. Did the Elevator, Ball Seat OR Turn Table have the largest
effect on the distances the ping pong ball flew?
How much of an effect? Can you quantify it? September 25, 2015
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Can we model the results? ▶ Example ONE Small differences in the 3 trials
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▶ Example TWO 140 120 100
Trial 1 Trial 2
80
Trial 3
60
Large differences in the 3 trials
40 20 0 1 - HHH 2 - HHL 3 - HLH 4 - HLL 5 - LHH 6 - LHL 7 - LLH 8 - LLL 140
Catapult Experiment
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Distance
Distance
Can we model the results?
100 80 Trial 1 60
Trial 2
40
Trial 3
20 0 1 - HHH 2 - HHL 3 - HLH 4 - HLL 5 - LHH 6 - LHL 7 - LLH September 25, 2015
8 - LLL
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How much did one factor affect the distance? ▶ Changing the turntable from Low to High reduced the distance
by 27.4 inches on average!
Distance (inches)
100.00 80.00 60.00
40.00 20.00 0.00 TT = Low
September 25, 2015
TT = High
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Interaction Effect ▶ Distance depends on the combination of different factors
100.00
80.00 60.00
TT = Low
40.00
TT = High
20.00 0.00 EH = Low
Distance (inches)
Distance (inches)
100.00
80.00
60.00 TT = Low
40.00
TT = High
20.00 0.00
EH = High
BS = Low
September 25, 2015
BS = High
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Conclusion ▶ You performed a statistically designed experiment Was this a better method than randomly trying different settings?
▶ Were we able to disprove the statistical hypothesis based on
the experimental results? ▶ We perform statistically designed experiments at Micron to
optimize our products and processes ▶ If you become a scientist, engineer, etc., you will use algebra!
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More Details
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Can we model the results? ▶ Example – 1 rubber band; Josie launch; DaRon measure ▶ Effect = average change in distance when going from L to H
setting (negative values indicate lost distance)
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Can we model the results? Yes!!! ▶ The overall mean (intercept) and effects (1/2 of each effect) are
used to build the linear equation
L or Low setting = -1 H or High setting = +1
y b m1 x1 m2 x2 m3 x3 m12 x1 x2 m13x1 x3 m23x2 x3 m123x1 x2 x3 y 69.04 1.8751 x1 2.042 x2 13.708x3 3.542 x1 x2 1.708x1 x3 12.958x2 x3 7.042 x1 x2 x3 September 25, 2015
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Model Predictions for MMM Setting ▶ Elevator Height = M (middle), Ball Seat = M, and Turntable =M
L or Low setting = -1 H or High setting = +1 M or Middle setting = 0
▶ Actual distance values: Distance Shot Combo Elevator Ball Seat Turntable Trial 1 Trial 2 Trial 3 Average MMM 0 0 0 70 68 72 70
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What about physics versus statistics? ▶ Angles? Trajectory? Velocity?
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