Topic 5 - Inferential Statistics

what is inferential statistics

a branch of stats that uses sample data to make generalizations, predictions or inferences abt a larger population

OR 

using the make-up of a sample to infer if it is likely or unlikely to have been drawn from a particular population 

what is probability theory

a branch of maths that deals with the analysis of random phenomena and the likelihood of various outcomes

‘the doctrine of chances’

what is the more dominant approach to probability

the frequentist view

how does the frequentist view define probability? 

as a long-run frequency 

data is treated as a repeatable random sample

• focus on hypothesis testing

• p values

• confidence intervals

what is the subjectivist view of probability?

the bayesian view

what is the most common way of thinking about subjective probability

to define the probability of an event as the DEGREE OF BELIEF that a rational agent assigns to that truth of that event

how do you operationalise a degree of belief 

‘Rational gambling’ 

what is the advantage of the bayesian approach?

allows you to assign probabilities to any event you want to

very broad

what is the disadvantage of the bayesian approach?

can’t be purely objective

what are the desirable characteristics of the frequentist definition?

1. objective - necessarily grounded in the world

2. unambiguous

what are the undesirable characteristics of a frequentist definition

infinite sequences don’t really exist in the real world has a narrow scope 

elementary event/simple event

a single outcome of a random experiment (pants)

sample space

the set of all possible events (wardrobe) 

law of total probability 

The probability of an event is the sum of its conditional probabilities across a set of mutually exclusive and exhaustive events, which form a partition of the sample space

The probabilities of the elementary events need to add up to 1 

probability distribution

a function that assigns a probability to each possible outcome of a random event, describing the likelihood of all possible values a random variable can take

non elementary events

a compound event in probability that includes more than one outcome from the sample space.

population of scores

the set of all scores for a complete group

memory test scores, mean scores, age scores, variance scores etc.

z score

indicates how far a particular raw score is ABOVe or BELOW the mean in standard deviation units

sample

a subset of the population (could be ppl OR scores) 

finite in size 

random sampling variability

whenever random samples of scores are taken from the same population of score, the samples will differ from one another purely by chance.

when the mean and SD appropriate for our data we can…

model our data with the normal distribution 

compute standardized scores 

quantify relationships between variables 

conduct more sophisticated inferential stats 

what is the equation for a z score (standard score) 

observed difference (deviation score) / expected difference (standard deviation) 

what can z scores do for us

1. communicates where a score is compared to others (how unusual the score is)

2. make meaningful comparisons across diff measures

   1. by standardizing variables to a common metric

3. determine probabilities in normal distributions

effect size 

magnitude of relationship between variables (how much? strength) 

what does a cohen’s d do?

measures the strength of a relationship between 2 levels of a categorical variable and a quantitative variable

• standardized diff between 2 group means

• expressed in SD units

correlation coefficient r

measures the strength and direction of relationship between two quantitative variables

• 1 indicates a perfect positive correlation.

• -1 indicates a perfect negative correlation.

• 0 indicates no correlation.

degree of freedom 

indicates how many values in a calculation can vary without violating any constraints imposed on the data 

why n-1

provides an unbiased estimator of the population variance

accounts for the loss of one degree of freedom due to the constraint of the sample mean

sampling error reflects…

the fact that stats of randomly drawn samples will deviate from the corresponding population parameters

random sampling variability reflects…

the fact that owing to chance two random samples drawn from the same population will have different stats

population distribution consists of…

all individuals in a population

• has SD

a distribution of sample means consists of…

all possible samples of a given size (N) in a population 

• has standard error of the mean 

every distribution is defined by…

• mean

• standard deviation

• shape