Brewing

• Low cost, high quality, and unoffensive to the masses?

  • That’s a German beer

• Barley, hops, and water

• Barley and hop flavor notes are a lie, focus on yield

• Water is the only thing that matters for quality

Cost Reduction

To develop this beer we have to run an experiment.

• Samples can get very expensive

In as few samples as possible we need to determine:

• The highest yield crop varieties

• The location with the highest quality water source

In-class Activity

Take 5 minutes and pair up into groups.

• Think about what skills your group members will need

• Each group at least needs someone ready to speak up at the end

Challenge 1

• You have \(\$85000\) of company funds at your disposal

• Every sample has a cost to it:

  • \(\$500\) per sample for Barley experiments

  • \(\$800\) per sample for Hop experiments

  • \(\$2000\) per sample for water testing

• The head brewer will only use the recipe you suggest if the data AND your explanation justify its use

Challenge 1

• For each “correct” ingredient, you’ll see a \(150\%\) return on investment

  • “Second best” ingredients see \(80\%\) return

  • “Third best” see \(20\%\)

  • Anything else results in a \(30\%\) loss

• You’ll have 2 minutes with each data source to make your decisions

• You’ll have 5 minutes to develop your “defense” of your recipe

Barley

Hops

Water

Student’s \(t\)-distribution

Student’s \(t\)-distribution

Central Limit Theorem

• I’ll attempt this explanation

• I am not as good as this guy

  • I highly suggest you watch this, but it’s 31 minutes so that’s a you problem

Central Limit Theorem

• Pull cards from a deck of 52 playing cards

  • Pull cards until you hit the queen of hearts

  • Record how many cards you pulled to get there, shuffle the deck, and repeat

• If you shuffle perfectly:

  • All possible positions are equally likely

  • Thus the chance of hitting a “center” queen is equally fair

Central Limit Theorem

• The proper representation of reality would be:

How often do people pull the queen in the center of the deck, on average?

• Zoom out and look at average results, i.e.

  • Each person pulls cards until the queen of hearts 20 times

  • That number is averaged and plotted out

Central Limit Theorem

Why does this matter?

Who actually won our little competition?

• The answer feels obvious, but it isn’t

• The CLT holds as \(n \rightarrow \infty\)

• At smaller samples we need the \(t\)-distribution

  • That’s not very convenient though

Distribution of Statistics

In practice, the convergence can happen much quicker

• For any sample mean \(\bar x\):

\[ \text{When } n>30, \quad \bar x \sim N(\mu_{\bar x}, \sigma^2_{\bar x}) \]

Where:

\[ \mu_{\bar x} = \mu \quad \text{and} \quad \sigma^2_{\bar x} = \frac{\sigma^2}{n} \]

Challenge 2

The “correct” ingredient can be described as the one that has the highest probability of the best outcome

• You’ll be given the mean and variance of each ingredient

• With the sample sizes you chose:

  • Find the probability that you achieve the “best” ingredients \(\text{Q}_3\) (noted with *)

• The team with the highest probabilities for each wins

Barley

\[ \begin{array}{|c|} \hline \text{Ingredient} & \text{Mean} & \text{Variance} & \text{Q}_3\\ \hline \text{Manchuria} & 103 & 674 & 115 \\ \hline \text{Peatland} & 110 & 455 & 120 \\ \hline \text{Svantosa} & 102 & 677 & 138 \\ \hline \text{Trebi*} & 127 & 1345 & 138 \\ \hline \text{Velvet} & 103 & 1066 & 123\\ \hline \end{array} \]

Hops

\[ \begin{array}{|c|} \hline \text{Ingredient} & \text{Mean} & \text{Variance} & \text{Q}_3\\ \hline \text{Cascade} & 31.7 & 42.7 & 33.0 \\ \hline \text{Millennium} & 37.8 & 31.4 & 42.4 \\ \hline \text{Mt. Hood} & 28.4 & 10.4 & 31.2 \\ \hline \text{Nugget*} & 37.5 & 29.2 & 41.1 \\ \hline \end{array} \]

Water

\[ \begin{array}{|c|} \hline \text{Ingredient} & \text{Mean} & \text{Variance} & \text{Q}_3\\ \hline \text{Zürich} & 2.00 & 0.469 & 1.5 \\ \hline \text{Ontario} & 2.32 & 0.416 & 2 \\ \hline \text{Poland} & 1.93 & 0.178 & 1.6 \\ \hline \text{Marinique*} & 1.70 & 0.317 & 1.3 \\ \hline \end{array} \]