Introduction to Hypothesis Testing, Probability, and Sampling Distributions

This week's lecture, delivered by Josh Sabio, focuses on introducing hypothesis testing, probability, and sampling distributions. It builds upon previous lectures and introduces a cornerstone concept for inferential statistics.

Acknowledgement of Country

The University of Queensland acknowledges the traditional owners and their custodianship of the lands on which we meet, paying respects to their ancestors and descendants who maintain cultural and spiritual connections to country, and recognizing their valuable contributions to Australian and global society.

Recalling the Normal Distribution

We begin by recalling the normal distribution, a fundamental concept from previous lectures. It is useful because many variables of interest to psychologists, such as IQ, short-term memory capacity, personality traits, and even facial attractiveness, tend to be normally distributed. If a distribution is normal, we can use z-tables to determine the probability of certain values occurring within it. For example:

The probability that an individual's score is one standard deviation below the mean.
The probability that values fall between specific intervals (e.g., within $ ext{one standard deviation} $ of the mean).

However, the utility of z-tables is limited to variables that are explicitly normally distributed. If a variable of interest is not normal, direct application of z-tables or finding the area under the curve is not possible. This week introduces a distribution that is always normal (under certain conditions), forming the foundation of inferential statistics.

Core Concepts of Inferential Statistics

This lecture lays the groundwork for understanding inferential statistics, covering:

Characteristics of populations versus samples.
Factors affecting sampling, such as sampling variability and sampling error.
The supremely important sampling distribution of the mean, including its characteristics, the standard error of the mean (SEM), and its connection to the Central Limit Theorem.
Practical applications in exam-style questions.

Populations vs. Samples

Population: Refers to the entire group of individuals or observations that share a particular characteristic and to which researchers wish to generalize their conclusions. Its size and characteristics depend on how it's defined (e.g., all Australian citizens, all marmosets). Researchers typically aim to make conclusions about the population at large.
Sample: A subset of the population, chosen for research due to feasibility and affordability limitations (e.g., it's impractical to study all $30$ million Australian residents). In rare cases (e.g., studying a rare disease or a specific, small cohort like all students in a course), access to a full population might be possible.

Notation: Parameters vs. Statistics

Parameters: Characteristics of a Population, described using Parameters and Greek alphabet.
- Population mean: $ ext{mu} ext{ } ( ext{ } oldsymbol{oldsymbol{ ext{ extmu}}} ext{ }) $.
- Population standard deviation: $ ext{sigma} ext{ } ( ext{ } oldsymbol{oldsymbol{ ext{ extsigma}}} ext{ }) $.
Statistics: Characteristics of a Sample, described using Statistics and Roman alphabet.
- Sample mean: $ M $.
- Sample standard deviation: $ S $.

Sampling Variability and Sampling Error

Random Sampling: The principle that individuals are selected from a population such that each has an equal and independent chance of being chosen. This aligns with the concept of independent draws.
Sampling Error: The inherent difference between a randomly drawn sample's statistics and the corresponding population parameters. A sample's mean will inevitably deviate from the population's mean (e.g., a handful of balls from a basket will have a mean different from the whole basket).
Sampling Variability: The fact that, due to chance, two random samples drawn from the same population will have different statistics. Iteratively drawing samples will demonstrate this fluctuation (e.g., multiple petri dish samples from the same pool of bacteria will show varying bacterial counts).

Literary Digest Example (1936 Presidential Election)

This historical example illustrates the importance of unbiased random sampling. Literary Digest predicted a Landon victory based on $2$ million responses from $10$ million questionnaires. However, Roosevelt won. The survey was biased because it used car registries and telephone numbers during the Great Depression, inherently sampling wealthier individuals who could afford such luxuries, thus not representing the general population.

Probability - First Principles

Understanding probability is crucial for inferential statistics.

Basic Rule: For any event, the $ ext{probability that it will occur} (P( ext{event})) + ext{the probability that it will not occur} (P( ext{not event})) = 1 $.
Exact Probabilities: These can be derived from frequency distributions through recursive computation.

Birthday Problem Example

The probability that $2$ of $3$ people share a birthday is $ P = 0.008 $. This is computed by considering the probability that each person does not share a birthday with the previous ones ($365/365 imes 364/365 imes 363/365$, etc.) and subtracting from $1$.
When plotted, the probability of a shared birthday rapidly increases with the number of people in a room. In a classroom of $30$ people, the probability of at least one shared birthday is approximately $70.63 ext{\textperthousand} $.

Jar of Balls Example

Jar 1: $10$ green, $10$ red balls ($20$ total). $P( ext{red ball}) = 10/20 = 0.5$ ($50 ext{\textperthousand} $).
Jar 2: $19$ green, $1$ red ball ($20$ total). $P( ext{red ball}) = 1/20 = 0.05$ ($5 ext{\textperthousand} $).
Conclusion: If we know the composition of a population, we can make statements about the probability of events. This connection is fundamental to inferential statistics.

Certainty and Statistical Convention

$ P=1.0 $: Absolutely certain (e.g., death and taxes).
$ P=0.5 $: $50/50$ chance (e.g., pulling a red card from a deck).
$ P=0.25 $: Pulling a diamond from a deck ($13 ext{ diamonds } / 52 ext{ cards } = 0.25$).
$ P=0.038 $: Pulling a red two from a $52$-card deck ($2 ext{ red } 2s / 52 ext{ cards } ext{ } oldsymbol{oldsymbol{ ext{ extapprox }}} 0.038$).
Convention: In statistics, $ oldsymbol{oldsymbol{ ext{P } < 0.05}} $ is arbitrarily adopted as the threshold for a rare event or a significant effect. This means an event occurring only $5 ext{\textperthousand} $ of the time is considered unusual or truly different.

The Gambler's Fallacy and Independent Draws

Independent Draws: The outcome of one trial does not influence the distribution of outcomes in the next (e.g., one coin flip doesn't affect the next). The