STATS 1123 - correlation and regression

single categorical variable charts

• pie chart

• bar graph

single numerical variable charts and statistics

• stemplot

• histogram

• boxplot

• sample mea/sample median

• standard deviation/IQR

what is the response variable

dependent

• y axis

what is the explanatory variable

independent

• x axis

scatterplot

• x and y axis ( independent and dependent)

• desc:

  • direction: positive, negative, no direction

  • form: linear, curved, cluster, no pattern

  • strength: 0-4 (weak), 4-7 (moderate), 7+ (strong)

  • outlier

r (correlation coefficient)

• between 2 numerical variables

• -1 < x < 1 (negative to positive)

• sensitive to outliers

• if r = 0, it means there is no linear relationship between the 2 variables

• no units

• 0-4 (weak), 4-7 (moderate), 7+ (strong)

• graph should be linear

hypothesis test

• Ho (null - original)

• Ha (alternative - what we want to test (linear relationship))

• conclusion:

  • reject Ho

  • fail to reject Ho

Steps to hypothesis

1. Write null and alternative hypothesis (x and y of all subjects do not have/have a linear correlation among all subjects in the population)

2. compare r with DP (if r is outside the box then we have sufficient statistical evidence - r > DP/ r < DP)

3. conclusion (we do not have/have sufficient statistical evidence to conclude that x and y have a linear relationship among all subjects within the population)

regression analysis

• Y (hat ) = a + bx (predicted value)

  • a = when x = 0, then a is this much (the predicted value of y is a when x = 0)

  • b = slope (for every 1 unit increase in x, the predicted value of y ↓/↑ by |b| units)

r²

• square of correlation

• the fraction of the variation in the values of y that is explained by the least squares regression of y on x

residuals

• difference between the predicted value (y hat) and the observed value

• (y-y hat) = residual

• sum of residual is always 0

• no pattern

RMSE

root mean square error

When using the least squares regression line with

explanatory variable x to predict y ,we expect about 95% of the observed values of y to lie within 2s of their respective

least square predicted values of y