• Perform basic data manipulation/exploration in R and dplyr
  • load data from a csv file
  • generate random numbers using sample()
  • understand use of set.seed()
  • generate histograms
• Clone and contribute to the class Github repo
• Complete assignments using R Markdown
• Define random variables and distinguish them from non-random ones
• Recognize some important random distributions from their probability density plots:
  • Normal, Poisson, Negative Binomial, Binomial

Mean: \(\mu = \frac{1}{N}\sum_{i=1}^{N}x_i^2\)
Variance: \(\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2\)
N = # in population

• Identify the difference between populations and samples
  • give sampling strategies
• Identify properties of a Normal distribution

• Define the Central Limit Theorem and give examples of its application

• Book sections:
  • Chapter 1 - Inference, up to and including "Central Limit Theorem and t-distribution"

• The numbered population is the sampling frame
• Sampling probability of every individual in the population must be non-zero
• We must know what the sampling probability is for every individual in the population

• The population distribution is specified by parameters
• Individuals in the population are sometimes referred to as a record, on which you make an observation
• A sample is summarized by a statistic
• The statistic is used to make inference about the population

• Simple Random Sampling (SRS): each record sampled with equal probability

• Stratified Random Sampling: records are sampled in planned numbers from pre-defined strata (usually better than Simple Random Sampling)
• Cluster Random Sampling: clusters are randomly sampled from the population, then individuals randomly sampled from the clusters
• Complex Sampling: a general term for designs with unequal sampling probability
• Convenience Sampling: whatever records are easiest to observe

• A remarkable attribute of sampling:
  • as long as the population is large and you can obtain a random sample, it doesn't matter how big the population is. Only how big the sample is.
  • we normally treat the population as infinite

• Selection bias may be introduced by differences between:
  • theoretical and empirical target populations
  • assumed and actual sampling probabilities
  • widespread and not clear how to account for

It is the distribution of a statistic for many samples taken from one population.

1. Take a sample from a population
2. Calculate the sample statistic (e.g. mean)
3. Repeat.

• The values from (2) form a sampling distribution.

• Question: how is this different from a population distribution?

We observe 100 counts from a Poisson distribution (\(\lambda = 2\)):

Question: is this a population or a sampling distribution?

• We calculate the mean of those 100 counts, and do the same for 1,000 more samples of 100:

Question: is this a population or a sampling distribution?

The "CLT" relates the sampling distribution (of means) to the population distribution.

1. Mean of the population (\(\mu\)) and of the sampling distribution (\(\bar{X}\)) are identical
2. Standard deviation of the population (\(\sigma\)) is related to the standard deviation of the distribution of sample means (Standard Error or SE) by: \[ SE = \sigma / \sqrt{n} \]
3. For large n, the shape of the sampling distribution of means becomes normal

Recall Poisson distributed population and samples of n=30:

• Distributions are different, but means are the same

Standard deviation of the sampling distribution is \(SE = \sigma / \sqrt{n}\):

• The distribution of means of large samples is normal.
  • for large enough n, the population distribution doesn't matter. How large?
  • n < 30: population is normal or close to it
  • n >= 30: skew and outliers are OK
  • n > 500: even extreme population distributions

• Example: an extremely skewed (log-normal) distribution:

• Recall from the CLT the sampling distribution is normal, with standard deviation \(SE = \sigma / \sqrt{n}\):
  • For a normally distributed popluation, this holds for any sample size n
  • For non-normally distributed populations, it holds for large n
• But, this formula assumes we know \(\sigma\)
  • if we instead estimate standard deviation from the sample (\(s\)), the sampling distribution is not normal
  • it has wider tails than the normal distribution

Question: Why would such an apparently small difference from the normal distribution matter?

• Calculate the t-statistic for a sample: \(t = \frac{\bar{X} - \mu_0}{s}\)
• if the population distribution is normal, or you have large sample size,
• and you estimate standard deviation from a sample,
• THEN: the t-statistic is distributed as \(t_{df=n-1}\)

• this leads to the one-sample t-test
• note the difference in the means of two samples also turns out to be t distributed when \(s\) is estimated

• Hypothesis testing is not the only framework for inferential statistics, e.g.:
  • confidence intervals
  • posterior probabilities (Bayesian statistics)
  • read p-values are just the tip of the iceberg

1. Populations, Samples, and Estimates
2. Central Limit Theorem and t-distribution

• A built html version of this lecture is available.
• The source R Markdown is also available from Github.
• A recording of the lecture will be available on the class YouTube channel.