## 5.2 Inferential Statistics: Can the Results Be Generalized to Population?

5.2.3 Describing the Relationship between Two (Ratio Scaled) Variables

### Types of Relationships

Unless we perform a designed experiment, we can only claim an association between the predictor and response variables, not a causation

### Linear Correlation

Two linearly related variables are **positively associated** if an increase in one causes an increase in the other.

Two linearly related variables are **negatively associated **if an increase in one causes a decrease in the other.

### Linear Correlation Coefficient

(Pearson‘s) Linear Correlation Coefficient is a measure of the strength of the linear relationship between two variables.

**Properties:**

- The linear correlation coefficient is always between -1 and 1.
- If r = +1, there is a perfect positive linear relation between the two variables.
- If r = -1, there is a perfect negative linear relation between the two variables.
- The closer r is to +1, the stronger is the evidence of positive association between the two variables.
- The closer r is to -1, the stronger is the evidence of negative association between the two variables.
- If r is close to 0, there is little or no evidence of a linear relation between the two variables – this does not mean there is no relation, only that there is no linear relation.

**Strength of relationship between variables**

Value |
Interpretation |

0 to 0.3 | very weak |

0.3 to 0.5 | weak |

0.5 to 0.7 | medium |

0.7 to 0.9 | high |

0.9 to 1 | very high |

### Ordinary Least-Squares (OLS) Regression

Regression Analysis is a powerful and flexible procedure for analyzing **associative relationships **between a metric dependent and one or more independent variables.

**Allows to:**

- Determine whether the relationship exists
- Quantify the strength of the relationship
- Derive the mathematical model / equation of the relationship
- Predict the values of the dependent variable
- Control for other independent variables when evaluating the contributions of a specific variable or set of variables

**Examples:**

- Can variation in sales be explained in terms of variation in advertising expenditure?
- Can the variation in market share be accounted for by the size of the sales force?
- Are consumer’s perceptions of quality determined by their perceptions of price?

What amount of goods will we sale if we spend €85,000 on advertising?

Advertising expenditure€1,000 | Sales €1,000 |

40 | 377 |

60 | 507 |

70 | 555 |

110 | 779 |

150 | 869 |

160 | 818 |

190 | 862 |

200 | 817 |

Relationship between sales and advertisement

- Advertising expenditure explains 83.6% of the variance in sales.
- Each extra Euro spent on advertising gains €2.82 on additional sales.
- €85T ad spendings convert into

2.824∙85,000 + 325.07 = 240,383.57 sales. -
## To the beginning of the section