(a preview)
2025-10-09
We calculated summaries and linear regressions using whatever data we have on hand.
These descriptions could be insightful for what they are but what exactly do we learn from these descriptions?
We have to think hard about these summaries and linear regressions before doing any computations.
What you have seen before are empirical quantities.
To truly understand what these empirical quantities are actually all about, we cannot avoid abstraction.
We create theoretical (idealized) counterparts of these empirical quantities.
Pretend that “the dog chooses the correct cup” is like “getting Heads in a coin toss”.
Pretend we are in the long run.
# Number of times to repeat the process of tossing a fair coin 10 times
nsim <- 2
# Repeat for nsim times "tossing of a fair coin 10 times"
a <- replicate(nsim, rbinom(10, 1, 0.5))
a [,1] [,2]
[1,] 0 0
[2,] 1 0
[3,] 1 0
[4,] 1 0
[5,] 0 0
[6,] 1 1
[7,] 1 0
[8,] 1 0
[9,] 1 1
[10,] 0 0# Calculate the relative frequency or sample proportion for every repetition
props <- colMeans(a)
props[1] 0.7 0.2If you take the position that Harley cannot understand human gestures, how often can you hypothetically observe something more extreme than what you actually observed in real data?
Lay out these ingredients for the examples so far:
Several fish deaths were reported just downstream of a large swale through which runoff from a commercial hog facility had escaped.
A pollution control agency decided to closely monitor dissolved oxygen downstream of the swale once every 11 days. The plan is to do the monitoring 10 times.
The agency wanted to see how often the dissolved oxygen was less than the 5.0 mg/L standard. A measurement with a lower level was considered “noncompliant”.
Water quality guidelines suggest that remedial action would be taken if the river was noncompliant more than 10% of the time.
Here is the dataset. An S indicates that the river was not in compliance that day.
| Mar 8 | Mar 19 | Mar 30 | Apr 10 | Apr 21 |
| S | F | S | F | S |
| May 2 | May 13 | May 24 | Jun 4 | Jun 15 |
| F | F | F | F | S |
One consultant tried to attract patients by noting the historical complication rate for liver donor surgeries in the US is about 10%, but her clients have had only 3 complications in the 62 liver donor surgeries she has facilitated.
She claims this is strong evidence that her work meaningfully contributes to reducing complications.
Do you think there is evidence to support this claim?
Heart transplants at a London hospital were suspended because of concern that more patients were dying than usual.
Newspapers reported that the 80% mortality rate in the last ten cases was of particular concern because it was over 5 times the national average.
Although there was not an officially reported mortality rate, researchers have determined that 15% was a reasonable benchmark for comparison.
What do you think of the evidence?
To what we did in class
Harder in these cases because we have to think about how to generate fake data under the condition that the net weight “in the long run” is equal to the advertised weight
A company wants to determine if they should enter a market. But there are other potential competitors. We have the following internal projections:
Profits depend on how many competitors enter the market. The 100 million will be shared equally among those in the market.
Let \(X\) be the number of other entrants in the market. Let \(Y\) be the profit of the company. How would you express uncertainty in this situation?
Let \(X\) be the number of other entrants in the market. This number is uncertain:
| Outcome of X | How likely? |
|---|---|
| 1 | 0.1 |
| 2 | 0.25 |
| 3 | 0.3 |
| 4 | 0.25 |
| 5 | 0.1 |
How about \(Y\)?
entrants <- 1:5
profits <- 100/(1+entrants)-26
a <- sample(entrants, 10^4, prob = c(0.1, 0.25, 0.3, 0.25, 0.1), replace = TRUE)
b <- sample(profits, 10^4, prob = c(0.1, 0.25, 0.3, 0.25, 0.1), replace = TRUE)
table(a)a
1 2 3 4 5
1019 2468 2960 2520 1033 b
-9.33333333333333 -6 -1 7.33333333333334
953 2509 2964 2521
24
1053 [1] 3.008000 1.148768 1.684667 9.434589Suppose the fixed costs of entry and the total market value are uncertain in the following sense:
| Outcome of D | How likely? | Outcome of V | How likely? |
|---|---|---|---|
| 20 | 0.2 | 70 | 0.3 |
| 26 | 0.5 | 100 | 0.4 |
| 30 | 0.2 | 120 | 0.2 |
| 34 | 0.1 | 150 | 0.1 |
entrants <- sample(1:5, 10^4,
prob = c(0.1, 0.25, 0.3, 0.25, 0.1), replace = TRUE)
values <- sample(c(70, 100, 120, 150), 10^4,
prob = c(0.3, 0.4, 0.2, 0.1), replace = TRUE)
costs <- sample(c(20, 26, 30, 34), 10^4,
prob = c(0.2, 0.5, 0.2, 0.1), replace = TRUE)
# Create a profit function
profit <- function(v, c, e)
{
return(v/(1+e)-c)
}
# Apply profit function to V, C, E
res <- mapply(profit, values, costs, entrants)
c(mean(res), sd(res))[1] 1.11750 12.52011