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MA50259: Statistical Design of Investigations Battery life experiment   An engineer is designing a battery for use in a device that will be subject to extreme variations in temperature

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MA50259: Statistical Design of Investigations

Battery life experiment

 

An engineer is designing a battery for use in a device that will be subject to extreme variations in temperature. From experience, the engineer knows that the temperature will affect the battery life. A key design element is the plate material for the batteries and there were three possible choices. The temperature can be controlled in the laboratory and the engineer decided to test all three plate materials at three temperature levels: $15^\circ$F, $70^\circ$F and $125^\circ$F because these levels are consistent with the product end-use environment. Four batteries were tested at each combination of plate material and temperature and all 36 tests were run in random order. The resulting observed battery life  data (in hours) are shown in the following table.

 

 

\begin{center}

\begin{tabular}{c|cccccc}\hline

& \multicolumn{6}{c}{\textbf{Temperature} }\\

\textbf{Material} & &&&&&\\

 \textbf{Type}  & \multicolumn{2}{c}{\textbf{15}$^\circ$} & \multicolumn{2}{c}{\textbf{70}$^\circ$} & \multicolumn{2}{c}{\textbf{125}$^\circ$}\\ \hline

\textbf{1}& 130 & 155 & 34 & 40 & 20 &  70 \\

& 74 & 180 & 80 & 75 & 82 & 58 \\\hline

\textbf{2}& 150 & 188 & 136 & 122 & 25 & 70  \\

& 159 & 126 & 106 & 115 & 58 & 45 \\\hline

\textbf{3} & 138 & 110 & 174 & 120 & 96 & 104 \\

& 168 & 160 & 150 & 139 & 82 & 60 \\

\hline

\end{tabular}

\end{center}

 

a) What are the experimental units and what is the experimental design type?

 

 

b) Create a data frame "battery" in R in such a way that it can be used by the \texttt{lm} function without any modification. Your data frame should have 3 variables and 36 rows. The variables should be called \texttt{Type}, \texttt{Temperature} and \texttt{Life}.

```{r}

Type <- rep(c(1, 2, 3), each = 12)

Temperature <- rep(rep(c(15, 70, 125), each = 4), 3)

Life <- c(130, 74, 155, 180, 34, 80, 20, 82, 70, 58, 150, 159, 188, 126, 136, 106, 25, 58, 70, 45, 138, 168, 110, 160, 174, 150, 96, 82, 104, 60)

 

battery <- data.frame(Type, Temperature, Life)

 

 

```

```{r}

Type <- rep(c(1, 2, 3), each = 8)

Temperature <- rep(c(15, 15, 70, 70, 125, 125), times = 4)

Life <- c(130, 74, 155, 180, 34, 80, 40, 75, 20, 82, 70, 58,

          150, 159, 188, 126, 136, 106, 122, 115, 25, 58, 70, 45,

          138, 168, 110, 160, 174, 150, 120, 139, 96, 82, 104, 60)

 

battery <- data.frame(Type = Type, Temperature = Temperature, Life = Life)

 

```

 

 

 

c) Construct in R the complete (less than  full rank) design matrix $\boldsymbol{X}$ corresponding to the model:

$$y_{ijk}=\mu+\tau_i+\alpha_j+\gamma_{ij}+\epsilon_{ijk}$$

where

 

- $\tau_{15}, \tau_{70}$ and $\tau_{125}$ are the effects of the levels of the temperature

 

- $\alpha_1, \alpha_2$ and $\alpha_3$ are the effects of the three types of materials and

 

- $\gamma_{15,1},\gamma_{15,2},\ldots,\gamma_{125,2},\gamma_{125,3}$ are the effects of the interactions.

 

- All $\epsilon_{ijk}\sim N(0,\sigma^2)$ and are mutually independent.

```{r}

X <- model.matrix(~Type + Temperature + Type:Temperature, data = battery)

 

```

 

 

d) How many unknown parameters are there in the model?

 

e) What is the rank of the design matrix $\boldsymbol{X}$? Explain why we do obtain this rank.

 

 f) Find a solution to the normal equations using the generalised inverse given by the R command \texttt{ginv}. Compare your solution to the estimates  given by the command

```{r,eval=FALSE}

model.fit<-lm(Life~Temperature*Type,battery)

coefficients(model.fit)

```

and explain why they are different. What is the specific choice of generalised inverse taken by R in order to produce those estimates?

 

 g) Let $\boldsymbol{\beta}^T=(\mu,\tau_{15},\dots,\alpha_1,\ldots,\gamma_{15:1},\ldots)$. Determine if each of the following linear combinations  of the form $\boldsymbol{\lambda}^T\boldsymbol{\beta}$ is estimable and if so, find corresponding unbiased estimates using the battery life data.

 

- The mean battery life at 70$^\circ$ for a battery made with plate material 2

 

- $\tau_{125}-\tau_{70}$

 

 

h) If we let $\mu_{ij}$ denote the cell means, that is

$$\mu_{ij}=\mu+\tau_i+\alpha_j+\gamma_{ij}$$

then the marginal means for the first factor are defined as

$$\bar{\mu}_{i\cdot}:=\frac{1}{s}\sum_{j=1}^s \mu_{ij}=\mu+\tau_i+\frac{1}{s}\sum_{j=1}^s \alpha_j+\frac{1}{s}\sum_{j=1}^s \gamma_{ij}$$

and the marginal means for the second factor are defined as

$$\bar{\mu}_{\cdot j}=\frac{1}{t}\sum_{i=1}^s \mu_{ij}=\mu+\frac{1}{t}\sum_{i=1}^t\tau_i+ \alpha_j+\frac{1}{t}\sum_{i=1}^t \gamma_{ij}$$

Show that all the marginal means are estimable.

 

i) What is the dimension of the vector subspace of linear combinations of the form $\boldsymbol{\lambda}^T\boldsymbol{\beta}$?

 

 

j) Produce an interaction plot  and interpret it in the context of the battery life experiment. You may want to have a look at the help file \texttt{?interaction.plot}

 

 

k) Test the null hypothesis that all interaction effects are zero, that is

$$H_0\,:\,\gamma_{15,1}=\gamma_{15,2}=\gamma_{15,3}=\cdots=\gamma_{125,1}=\gamma_{125,2}=\gamma_{125,3}=0$$

 at the significance level $\alpha=0.05$. You should provide the value of the F ratio statistic, the corresponding degrees of freedom and the associated p-value. You should do the computations using the theory given in the lectures and not simply use the command \texttt{aov}.

 

 

# Part 2: Distance travelled by paper airplanes

 

In this problem you will design a factorial experiment, collect the data and then analyse it! You will carry out this experiment to identify how different design factors influence the total  distance travelled by a paper airplane.

 

## Paper airplane aerodynamics

 

First we discuss some simple aerodynamics. There are four different physical forces that make a paper airplane fly:

 

- **Thrust:** When you throw a paper plane in the air, you are giving the plane a push to move forward. That push is a type of force called thrust.

 

<!--

paper clip photo

https://www.instructables.com/id/how-to-make-the-fastest-paper-airplane/

-->

 

- **Lift:**  Airplane wings are shaped to make air move faster over the top of the wing. When air moves faster, the pressure of the air decreases. So the pressure on the top of the wing is less than the pressure on the bottom of the wing. The difference in pressure creates a force on the wing that lifts the wing up into the air. Bigger wings increase lift.

 

<!-- NASA webpage:

https://www.grc.nasa.gov/WWW/K-12/UEET/StudentSite/dynamicsofflight.html#forces

 

https://www.cheapflights.com/news/science-of-flight-paper-airplanes

-->

 

 

- **Drag:** As a paper plane moves through the air, the air pushes against the plane, slowing it down. This force is called drag.

 

 

- **Weight:**  The weight of the paper airplane affects its flight and brings it to a landing. Weight is the force of Earth's gravity acting on the paper plane.

 

The figure  below shows how all four of these forces, thrust, lift, drag, and weight, act upon a paper airplane.

 

```{r, out.width = "60%",echo=FALSE}

# Bigger fig.width

include_graphics("directions.png")

```

 

We can also see these forces act orthogonal to each other! These same forces apply to real airplanes, too. If the paper plane has enough thrust and the wings are properly designed, the plane will have a nice long flight!

 

<!--Basic dart design

http://www.amazingpaperairplanes.com/simple-basicdart.html

need to do a folding diagram in PPT

-->

 

 

 

## Experimental units

 

In this experiment each experimental unit will correspond to a single paper airplane. Replicates will mean you will have to fold as many planes as needed.

 

## Instructions

 

In this experiment you will need to throw many paper airplanes into the air! Please follow the following instructions:

 

- Use white printer paper from the university printers to fold the planes.

 

-  The paper airplane design you will use is the so-called Nakamura Lock (see folding instructions below)

 

-  Label your planes according to the factor variables (described below) and the replicates number.

 

- Perform the experiment in the foyer of the 4W (Maths) building which is the open space in level 1. This is a nice large open space, mostly isolated from wind.

 

-  Always throw the planes in the same way, straight, and parallel to the ground, not up nor down angle as the following chart shows

 

```{r, out.width = "40%",echo=FALSE}

# Bigger fig.width

include_graphics("chart.png")

```

 

- You should try to get the plane as far as possible in every throw.

 

- Before each throw, make sure that the wings of the airplane  point upwards so that your plane looks like this from the back 

Wings angled upwards are called \textit{dihedral} and give your plane stability.

 

- Throw you paper airplanes holding from the fuselage. The fuselage is the bottom part of the plane where the fold separates the two sides.

 

 

- You can do some practice throws before starting the main experiment, but I recommend to use different planes for this. The practice planes can be used more than once.

 

- Do not do all the experiment in one go. Take a few breaks to make sure your arm does not get tired and alawys use the same arm for the throws!

 

- Recycle all the paper after finishing the experiment.

 

 

 

## Response

 

Flight length in meters.  To measure the flight distance, follow the following instructions:

 

- You need to establish an origin for the flight. You can mark the edge of a floor tile as the beginning of the flight and then multiply by the number of fixed length floor tiles travelled to obtain the total distance. Use the farthest most part of the plane to record the distance measurement. Measure up to 2 decimal places, that is, up to the nearest centimeter.

 

- Record the distances in a spreadsheet making reference to each plane label.

 

- If anything goes wrong, for example  if the plane goes behind you, repeat that particular throw with a different (pristine) plane.

 

- By the way, the  world record for the longest flight of a paper airplane is by John Collins who made a paper airplane that flew 69.14 meters!

 

## Factor variables

 

 

 

- **Factor 1 (Size):** (2 levels) A4, A5 paper.

 

- **Factor 2 (Wings aspect ratio)** (2 levels) standard versus longer and wider wings (in folding details below)

 

<!-- **Factor 4 (Thrust:)** (2 levels) Standard throw vs fast throw.  -->

     

<!--

https://paper-design.wonderhowto.com/forum/make-easy-paper-airplanes-nakamura-lock-spy-plane-0147763/

-->

     

<!--   Harrier   https://www.youtube.com/watch?v=7JbDGsXA2oY -->

 

<!-- From 9 types of paper airplanes

http://www.paperaeroplanes.com

-->

 

<!-- Harrier

https://www.instructables.com/id/To-Fold-The-Harrier-Paper-Airplane/

-->

 

 

<!--it would be nice to have orthogonality!!! Factor 5: Add paper clips: in the back, the front, the middle, or the wings. one paper clip or several? -->

 

 

## The randomization

 

Remember to  assign the factor level combinations to the experimental units (the paper airplanes) completely at random. You can do this in R using the \texttt{sample} function. For example, for a balanced design with $r=3$ replicates, you can assign the level combinations to the total of 12 planes using the following command

```{r}

f<-rep(c("A3,Std","A4,Std","A3,long","A4,long"),each=3)

sample(f,12)

```

## Folding instructions

 

### Nakamura Lock

 


 

 


 

In the above figure wing span level 1 is referred as "std" and wing span level 2 is referred as "long"