206 test

weeks 5-9

Probability and Statistical inference

Probability matters in stats - probability provides a way of quantifying uncertainty and is the foundation of statistical inference

Why probability matters?

• Sample results vary because observations depend on which individuals are included

• Probability provides framework for understanding and modelling this variability

• Statistical inference uses probability to link sample data to population conclusions

Parameters, Statistical and uncertainty

All samples statistics involve uncertainty because of sampling variation

Sample distributions

• A sample distribution describes how a statistic varies across repeated random samples from the same population

• it is a distribution of statistics not raw data

• it is usually theoretical or stimulated

The sampling distribution of the sampling Mean: how is it formed?

• take a large number of random samples of the same size (n) from the same population

• calculate the mean for each sample

• plot all of those sample means

Key properties of sampling distribution

Centre: the mean of the sampling distribution equals the population mean

Spread: the variability of the sampling distribution is smaller than the variability of individual scores

Shape: as sample size increases, the sampling distribution becomes approximately normal

Standard error: measuring uncertainty in estimates (SEM)

• The standard error quantifies the variability of a statistic across repeated samples

• the standard error of the mean (SEM) desribes how much sample means typically differ from the population mean

what is SEM formula?

SD / square root of n

Key ideas

SD = describes variability in individual scores

SEM = describes variability in sample means - decreases as sample size increases

larger SEM = more certainty in the estimate

smaller SEM = more precise estimate

Sample size and precision

• larger samples reduce sampling variation

• larger samples lead to smaller standard errors

• larger samples produce more precise estimates of population parameters

why estimation requires probability

• model how samples behave when drawn at random from a population

• quantify uncertainty in our estimates

• make statements about plausible population values

what is correlational research

correlational research examines the relationship between two or more measured variables, focusing on how they co-vary

association claims and

most correlational studies make association claims, not causal claims

3 criteria for causation

1. covariation

2. temporal precedence

3. elimination of alternative explanations

correlation coefficients ( r )

the correlation coefficients describes the strength and direction of the relationship between two variables

Key properties

Range: r ranges from -1 to +1

Direction: positive = as one variable increases, the other increases, negative = as one variable increases, the other decreases

approximate benchmarks:

Small: r= .10

medium r= .30

large r= .50

what do scatterplots show?

• direction of the relationship

• strength of the relationship

• shape of the relationship

• presence of outliers

threats to statistical validity

• outliers

• restriction of range

• curvilinear relationship

confidence intervals for correlations

• confidence intervals (CIs) provides information

• a 95% confidence intervals for a correlation

• indicates a range of plausible population correlation values

• reflects uncertainty due to sampling variability

interpretation of CIs

narrow CI = more precise estimate

wide CI = less precise estimate

Association claims

association claims should be evaluated using mulitple forms of validity

construct validity

concerned with how well the variables are measured and defined

statistical validity

concerned with whether the statistical conclusions are accurate and reasonable

external validity

concerned with whether the findings generalise beyond the study

internal validity

concerned with whether a causal conclusion can be made

key features of simple linear regression

• describes the relationship using an equation

• predicts values of one variable from another

• identifies the line of best fit

what is intercept (bo) and slope (B1)

intercept: predicted value of y when X = 0

slope: expected change in y for a one-unit increase in X