inferential statistics

examination of a relationship between 2 variables

scatterplot

statistics provides us

with the ability to quantify the relationship between measurement variables

If there really was nothing going on in the population, what us the chance that this relationship would have been observed?

• if the chance was small, we declare the relationship to be statistically significant and not to be just a fluke.

• To believe an observed relationship, it must be shown to be statistically significant

a relationship is deemed to be statistically significant if the chance of observing the relationship, when there is actually nothing going on in the population, is less than

5%

statistical significance may be misinterpreted - 1

A minor or weak relationship will achieve statistical significance when the sample is very large - a relationship which is declared to be statistically significant is not necessarily a strong relationship

statistical significance may be misinterpreted - 2

A very strong relationship won't achieve statistical significance if the sample is very small - a small sample means not enough observations have been taken to rule out chance as an explanation of the relationship

correlation

• Pearson product-moment correlation

• Correlation coefficient

• r

correlation measures

the strength of linear relationships only

correlation between 2 measurement variables

a measure of how closely their values fall to a straight line

a positive correlation indicates

the variables increase together

a negative correlation indicates

as one variable increases the other variable decreases

correlation is unaffected by

units of measurement

correlation of r = +1

there is a perfect linear relationship between the 2 variables

r = -1

that there is a perfect linear relationship between the 2 variables

r = 0

there is no relationship between the two variables

2 main problems

1. Outliers can substantially inflate or deflate correlations

2. Combining groups within the population inappropriately can mask relationships

masking relationships

The variables may be correlated within a group, but when the data is merged these relationships will be masked

Correlation ⇏ causation

 if two variables are legitimately correlated do not be fooled into thinking that there is a causal connection between them

regression

we have one explanatory variable and one response variable

sample linear regression

fitted to these data to provide a quantitative summary of the relationship between sales and advertising budget

least squares line

Focus on the response variable and try to fit the line so that the observed response values are as close as possible to the line. This means that the vertical deviation between each response (or y) data point and the line is as small as possible

the vertical deviations

squared and then added up for all points in the scatter plot. The line which minimizes this sum of squared distance is the line which fits the data best. This is called the least squares line

R2

the amount of the total variation in the response variable that is explained by the explanatory variable

R2 = 1

all of the observed responses lie exactly on a straight line. In other words, our explanatory variable explains all of the variation in our response variable

R2 = 0

our explanatory variable explains none of the variation in our response variable - the linear regression model is not a good  model for the data

multiple linear regression

an extension of simple linear regression with more that one explanatory variable

significant relationships

To test whether a relationship is statistically significant or not we need to perform a hypothesis test

hypothesis testing allows

to use a sample of data to decide between 2 statements about a population characteristic

population characteristic

things like the mean of the population or the proportion of the population who have a particular property

the null hypothesis

hypothesis which we initially assume to be true

the alternative hypothesis

the reason the data was collected in the first place - suspect null false

significance levels

the probability of making a type I error when the null hypothesis is true

p-values

the probability of observing a value of the characteristic of interest the same, or more extreme than what was actually observed in the data

type I error

false positive

type II error

false negative

interpreting p-values

1. If p-value ≤ α fail to accept H0 at the α% significance level

2. If p-value ≥ α do not reject H0 at the α% significance level