• Understand fundamental concepts of statistical inference
  • Population vs Sample
  • Parameter vs Statistic
  • Sampling Variability & Bias
• Apply basic sampling strategies
  • Simple and Stratified Random Sampling
• Readings
  • ISRS: ch. 1.3-1.4

• Goal: answer questions about a population
  • Population: collection of all objects of interest (actual or notional)
  • E.g. What is average age of people living in Canada?
• Precise answer for entire population requires census
  • E.g. Ask everyone living in Canada (cost ~$700mil!)
  • Typically either impracticable or impossible
• Alternatively sample the population

• Sample is subset of population that helps answer question "approximately"
  

• Answer quality depends critically on how sample is collected

• Answers typically expressed as statements about summary measure of interest
  • Parameter: summary of entire population
  • (Sample) Statistic: summary of sample

• Continuous Variables

• Discrete Variables

• Dinesafe data: interested in average number of inspections per establishment

pop = dinesafe %>% 
  group_by(ESTABLISHMENT_ID) %>% 
  distinct(INSPECTION_ID) %>% 
  summarise( N_INSPECTIONS = n() ) 
pop %>% summarise( mean(N_INSPECTIONS) ) 
## # A tibble: 1 x 1
##   `mean(N_INSPECTIONS)`
##                   <dbl>
## 1                  3.41
pop %>% sample_n(100) %>% summarise( mean(N_INSPECTIONS) )
## # A tibble: 1 x 1
##   `mean(N_INSPECTIONS)`
##                   <dbl>
## 1                  3.38

• Statistic value varies with different samples
  • Sampling variability is extent to which statistics diverge from their mean
• Sampling variability can be controlled by the sample size (\(n\))
  • Larger \(n\) \(\rightarrow\) lower variability (higher accuracy)

• Distribution of statistics (avg # of inspections) of different sample sizes

• Statistic changes with different samples, so how do we pick our sample?

• Regularities in sampling can lead to bias, i.e. systematic deviation of statistic from parameter
  • E.g. collect data by email only

• To avoid selection bias & improve representativeness, most sampling methods involve randomness

• Sampling frame, i.e. list of available objects for sampling
  • Ideally covers entire population
• Simple Random Sample (SRS): every object sampled randomly and with equal probability
  • Avoids bias when no other information is available
  • Average of all SRS averages equal to population average

• Distribution of statistics from different sampling frames (SRS, \(n=100\))

• Randomness alone is not enough for representativeness
  • Must pay attention to sampling details

Two common sources of bias are:

• Participation or Non-response bias: respondents are not representative of entire population

• Coverage bias: sampling frame does not align well with population

• Population often divided into groups, called strata

• Stratified Sampling combines SRS from every straturm to ensure representation
  • Samples proportional to strata sizes

## # A tibble: 3 x 3
##   MINIMUM_INSPECTIONS_PERYEAR `mean(N_INSPECTIONS)` `n()`
##                         <int>                 <dbl> <int>
## 1                           1                  1.66  3887
## 2                           2                  3.45  8775
## 3                           3                  5.21  3629

• Observational studies collect data by observing what happens (no intervention)
  • E.g. survey sampling, polls, etc.
• Experiments collect data after manipulating aspects of the process
  • E.g. drug testing
• Observational studies are used for descriptions (what is happening), whereas experiments are used for decisions (what should be done)