STA 2023 Module 1 Notes

Unit: Introduction to Statistics

• Unit: the smallest material from which we collect data

  • Examples: person, family, store, machine, animal, event, transaction, etc.

• Population: the collection of all outcomes, responses, measurements, or counts that are of interest

• Sample: a subset, or part of the population

• Variable: an observed property of every unit in the population or sample

  • Examples: height ($X$), gender ($G$), race ($R$), income ($Y$)

• Two Types of variable

  • (i) Qualitative (Attribute/Categorical) Variable

  • A property/characteristic obtained by classifying each object or unit as belonging to one of several categories

  • Examples: gender, race, method of payment, nationality, patients status at hospital discharge (dead/alive), outcome of a coin toss (H/T), etc.

  • (ii) Quantitative (Numerical) Variable

  • A numerical characteristic obtained by measuring or counting sample/population units

  • Measurement variables: height, weight, length, temperature, etc.

  • Count variables: number of children in families, daily number of accidents, number of credit cards you have, etc.

  • Distinctions: measurement variables are continuous; count variables are discrete

• Quick exercise: Is the response qualitative or quantitative?

  • A UCF instructor asks her students if they have scholarships. Is the response quantitative or qualitative? Qualitative

  • 1. Political party affiliation: Qualitative

  • 2. How many times you’ve checked Instagram today: Quantitative

  • 3. Temperature of the room: Quantitative

  • 4. Eye color: Qualitative

  • 5. Zip code: Qualitative

Discrete vs Continuous; Classification of variables

• Discrete: the random variable has a finite or countable number of possible outcomes that can be listed

• Continuous: the random variable has an uncountable number of possible outcomes, represented by an interval on a number line

• Exercise: Is the variable discrete or continuous?

  1. The weight, in pounds, of a food item bought at the grocery store: Continuous

  2. The number of customers in line at the grocery store: Discrete

  3. The length of time it takes a student to complete an exam: Continuous

  4. The number of seizures an epileptic patient has in a given week: Discrete

  5. The amount of money you get back from paying your Target total in cash: Continuous

• Displaying data of a sample or population

  • Data Set: a collection of values of all variables often displayed in the form of an array with one column for each variable and one row for each unit in the sample or population

  • Example: A hypothetical data set

  • A data set based on a sample of five students

  • We may be interested in:

  • average GPA ($\bar{X}$)

  • average GPA by gender ($\bar{X}_{gender}$ )

  • proportion of male students ($ext{p}_{m}$)

  • percentage of students with GPA > 3.5 ($ext{P}(GPA>3.5)$)

  • most typical number of courses taken by students

  • distribution of number of courses by gender

• More Key Terms

  • Statistic: a function of the sample (e.g., sample average $\bar{X}$, sample proportion $\, ext{p̂}$, range of data in a sample, etc.)

  • Parameter: a function of the population (e.g., population average $\mu$, population proportion $p$, range of data in the population, etc.)

• Types of statistical applications

  • Descriptive statistics: numerical summaries (e.g., average, variance, percentile, etc.), basic features of data often displayed in graphs, tables, etc.

  • Inferential statistics: estimation, prediction, and hypothesis testing about some unknown characteristics of a population based on information in a random sample drawn from the population.

• Example data (five students)

  • Student | Gender | GPA | # of Courses

  • 1 | M | 3.5 | 4

  • 2 | F | 3.7 | 5

  • 3 | F | 2.8 | 5

  • 4 | M | 2.9 | 6

  • 5 | M | 3.6 | 5

Research Question and Data Collection

• Research Question: What proportion of UCF students have scholarships?

• Data collection: We ask 30 UCF students if they have scholarships, and 60% said yes.

• Tasks:

  • Identify the population

  • Identify the sample

  • Descriptive statistics

  • Suppose that 50% of students at UF have scholarships. Give an example of inferential statistics: infer about the population of UF and/or compare populations

Statistical Studies: Observational vs Experiment

• Observational study: a researcher observes and measures characteristics of interest of part of a population but does not change existing conditions

• Experiment: a researcher deliberately applies a treatment before observing the responses

  • Treatment: applied to part of the population (treatment group)

  • Control group: no treatment is applied, subjects may be given a placebo

  • Experimental units: subjects in the treatment and control groups

• Example: Detecting Lung Cancer: CT vs. X-ray

  • The H. Lee Moffitt Cancer Center at the University of South Florida conducted a clinical trial of 50,000 smokers to compare the effectiveness of CT scans with X-rays for detecting lung cancer

  • Each participating smoker was randomly assigned to be screened by either CT or chest X-ray, and his or her progress was tracked over time

  • The age at which the scanning method first detects a tumor is the variable of interest

• Questions to consider for this example

  • Was this an experiment or observational study? Experiment (randomized clinical trial)

  • Identify the experimental units: the participating smokers

  • Was the variable measured qualitative or quantitative? Quantitative (time-to-detection, a continuous variable)

  • Identify the population and sample: Population = all smokers; Sample = the 50,000 smokers in the trial

  • What is the inference that will ultimately be drawn from the clinical trial? Comparison of effectiveness/delay in tumor detection between CT and X-ray; generalizable conclusions about screening method effectiveness (with caveats about external validity)

Sampling Methods

• Simple random sample: every possible member of the population has the same chance of being selected

• Stratified sample: members of the population are divided into two or more subsets (strata) by a characteristic. A sample is then randomly selected from each stratum, ensuring that all strata are sampled in proportion to their actual percentages of occurrence in the population

• Cluster sample: divide the population into groups (clusters) and select all of the members in one or more (but not all) of the clusters. All clusters should have similar characteristics. Selecting all of the members of a population is called a census

• Systematic sample: members of the population are selected at regular intervals from a randomly determined starting point

• Sampling examples (from the transcript):
  1) Starting with the third vehicle, every 5th student pulling into a parking garage is asked how many minutes they commuted to campus. — Systematic sample
  2) On a football game day, Memory Mall is divided into 20 equal plots. 3 plots are selected at random, and every person in the plot is asked about their GPA. — Cluster sampling
  3) From UCF, 20 students are randomly chosen for a scholarship. From UF, 17 students are randomly chosen for a scholarship. From USF, 9 students are randomly chosen from a scholarship… — Stratified sampling by university
  4) 400 UCF students are chosen at random and receive a free textbook. — Simple random sample (with treatment in an experimental context)
  5) 100 freshmen, 85 sophomores, 80 juniors, 75 seniors, and 50 graduate students are chosen at random from their classification and asked about their drinking habits. — Stratified sampling by class level
  6) A grocery manager selects every other egg carton to check for broken eggs. — Systematic sample
  7) In an effort to improve nursing conditions in Florida hospitals, the Governor randomly selects 10 Florida hospitals and interviews every nurse in the hospital. — Cluster sampling (randomly chosen clusters, census within clusters)
  8) A polling agency randomly selects 300 registered voters in a small city and asks if the voter favors a particular candidate for mayor. — Simple random sample