A cricket ball manufacturing company conducts the following experiment. Every day, a bowling machine is set to one of three levels, "Medium", "Fast" or "Spin", and then bowls 100 balls towards the stumps. The number of times the ball hits the stumps and the average wind speed (in kilometres per hour) during the experiment are recorded, yielding the following data (abbreviated):

$\begin{array}{llll}\text { Day } & \text { Wind } & \text { Level } & \text { Stumps } \\ 1 & 10 & \text { Medium } & 26 \\ 2 & 8 & \text { Medium } & 37 \\ \vdots & \vdots & \vdots & \vdots \\ 50 & 12 & \text { Medium } & 32 \\ 51 & 7 & \text { Fast } & 31 \\ \vdots & \vdots & \vdots & \vdots \\ 120 & 3 & \text { Fast } & 28 \\ 121 & 5 & \text { Spin } & 35 \\ \vdots & \vdots & \vdots & \vdots \\ 150 & 6 & \text { Spin } & 31\end{array}$

Write down a reasonable model for $Y_{1}, \ldots, Y_{150}$, where $Y_{i}$ is the number of times the ball hits the stumps on the $i^{t h}$ day. Explain briefly why we might want to include interactions between the variables. Write $R$ code to fit your model.

The company's statistician fitted her own generalized linear model using $\mathrm{R}$, and obtained the following summary (abbreviated):

$\begin{array}{lrrrrr}\text { >summary(ball) } & & & & \\ \text { Coefficients: } & & & & & \\ & \text { Estimate } & \text { Std. Error } & \text { z value } & \operatorname{Pr}(>|z|) & \\ \text { (Intercept) } & -0.37258 & 0.05388 & -6.916 & 4.66 \mathrm{e}-12 & * * * \\ \text { Wind } & 0.09055 & 0.01595 & 5.676 & 1.38 \mathrm{e}-08 & * * * \\ \text { LevelFast } & -0.10005 & 0.08044 & -1.244 & 0.213570 & \\ \text { LevelSpin } & 0.29881 & 0.08268 & 3.614 & 0.000301 & * * * \\ \text { Wind: LevelFast } & 0.03666 & 0.02364 & 1.551 & 0.120933 & \\ \text { Wind: LevelSpin } & -0.07697 & 0.02845 & -2.705 & 0.006825 & * *\end{array}$

Why are LevelMedium and Wind: LevelMedium not listed?

Suppose that, on another day, the bowling machine is set to "Spin", and the wind speed is 5 kilometres per hour. What linear function of the parameters should the statistician use in constructing a predictor of the number of times the ball hits the stumps that day?

Based on the above output, how might you improve the model? How could you fit your new model in $R$ ?