A problem.

1. Customers arriving at a car dealership at a rate of 6 per hour.
2. Each customer has a 15% probability of making a purchase.
3. Purchases have

1. uniform profits over the interval $1000-$3000.
2. Normal profits that average 1500 with standard deviation 500

Putting it together. The tool is r, let’s try 1000.

1. Customers arriving at a car dealership at a rate of 6 per hour. C=Poisson(6)
2. Each customer has a 15% probability of making a purchase. P=Binomial(C,p=0.15)
3. Purchases have sum p of these

• Uniform profits over the interval $1000-$3000.
• Normal profits that average $1500 with standard deviation $500

Customers <- rpois(1000, 6) # Customers ~ Poisson(6)
Purchasers <- rbinom(1000, size=Customers, prob=0.15) # P ~ Binomial(Customers,0.15)
# Next part needs a coding trick.  For each row [of 1000], I want sum the Profits given Purchasers random draws.
Profits.U <- sapply(c(1:1000), function(x) 
  { sum(runif(Purchasers[[x]], 1000, 3000))} )
Profits.N <- sapply(c(1:1000), function(x) 
  { sum(rnorm(Purchasers[[x]], 1500, 500))} )

##    Profits.U       Profits.N      
##  Min.   :    0   Min.   : -36.92  
##  1st Qu.:    0   1st Qu.:   0.00  
##  Median : 1530   Median :1233.31  
##  Mean   : 1842   Mean   :1392.00  
##  3rd Qu.: 2901   3rd Qu.:2181.18  
##  Max.   :12372   Max.   :8849.13

Will play off of Monte Carlo Simulation such as this.

What values can x take? A large random sample gives the approximate min and max.

plot(density(data)) or hist(data) show samples.

The assignment operator or equals to save samples <- or =

poss.x <- seq(min.x, max.x, by=units)  # range from sim.
prob.x <- p<noun>(poss.x, verbs)  # probability of x

my.seq <- seq(-4,4,by=0.01)
plot(x=my.seq, y=pnorm(my.seq), main="The Standard Normal (0,1)", xlab="z", type="l", ylab="Probability")

The empirical rule(s).

pnorm(1)-pnorm(-1)

## [1] 0.6826895

pnorm(2)-pnorm(-2)

## [1] 0.9544997

pnorm(3)-pnorm(-3)

## [1] 0.9973002

On y: scale the data: \[\frac{x - \overline{x}}{s_{x}}\].

On x: generate the sorted quantiles given 1…n/(n+1).

Has an interval to give us an idea of how normal. An interval that, with a given probability, the observations should fall in the interval that frequently. The command requires library(car).

## Loading required package: carData

## [1] 6416 5059

In a world of data, Toulmin’s backing, rebuttals, and qualifiers come in two major forms:

• Assumption
• Implication

In the last lab and the coming, we posited a distribution for things and asked questions given that distribution. That was two things: a probability distribution and the parameters that complete that. The verb and the noun….

• Air Traffic Control incidents
• Filling grape jelly jars
• Scottish pounds
• The median…

In each of the aforementioned problems, there are two unknowns – the parameters and the distribution. This is because we are asking questions of the entire distribution. These are very hard questions and we have assumed answers to both questions: distribution and parameters.

install.packages("radiant")

installs a graphical user interface that, while limited, is very useful for probability distributions.

Interestingly, any given observation has a 50-50 chance of being over or under the median. Suppose that I have five datum.
1. What is the probability that all are under? 2. What is the probability that all are over? 3. What is the probability that the median is somewhere in between our smallest and largest sampled values?

Consider an example concerning the space shuttle Challenger and the O-rings. NASA posits that the probability of an accident is 1/60000. The Air Force posits an alternative, 1/35. Who is right?

The problem is that data cannot by themselves decide because we cannot continuously repeat the process of launching space shuttles and seeing whether or not they explode for reasons of, first and foremost morality, but also of expense measured in a host of ways.

Trials of the shuttle are not independent in any case. Poisson trials – damage to tiles and reentry may make the probability of accidents increasing in number of flights. But the core question is that we need to use the data that we have to say things about data that we do not have. We only know 1 in 25, 2 in 113, or 135 flights.

Each contains an assumption about what the data should look like. What should they look like?

• Find the bounds with a large sample using rpois.
  
• Plot the distribution within those bounds using p(q).
  
• Evaluate the data we have.

Four principle types of sampling

• Random
• Stratified
• Clustered
• Systematic Samples

Things are assumed to be random. Threats to this:

• Coverage error
• Nonresponse error
• Sampling error
• Nonresponse error
• Measurement error

• Means
• Variances
• \(\pi\) in a binomial
• Correlation

The mean is special. The process of averaging does many things. We have already seen that it guarantees the deviations above and below the mean cancel out. It also reduces variation.

Maintains that the population mean has a normal distribution, under general conditions, with standard error \(\frac{\sigma}{\sqrt{n}}\).

One specifies an interval given a level of probability.

The other specifies a decision rule for a statement given a level of probability. If the statement is at least \(p\) likely, then it is true.

They are dual to one another.

Type I and II

\[ 1 - (\frac{1}{k^2}) \]

is the probability that the data lie within \(k\) standard deviations of the mean where \(k\) is any positive integer greater than one. A bonus nugget from your homework

In statistics, as in nature, everything has variation. There is not a single answer; there is an interaction between probability and this range. If a normal holds, we have an empirical rule. For any distribution, we have Chebyshev’s theorem. There is one that is always correct; it is just not interesting.