Constructing Scatterplots and Drawing Linear Models

Explanatory Variable ( $x$ ): Also known as the independent variable, it is the factor used to explain or predict changes in another variable.
Response Variable ( $y$ ): Also known as the dependent variable, it is the outcome being measured or affected.
Examples of Variable Identification:
- Washing time ( $w$ ) vs. Percentage of stains removed ( $p$ ): $w$ is explanatory, $p$ is the response.
- Man's height ( $h$ ) vs. Percentage of people who find him attractive ( $p$ ): $h$ is explanatory, $p$ is the response.
- Traveling speed ( $s$ ) vs. Stopping distance ( $d$ ): $s$ is explanatory, $d$ is the response.

Ordered Pairs: Data points are represented as $(x, y)$ , where the explanatory variable is always listed first.
- Example: An individual with a height of $66\,\text{inches}$ and weight of $142\,\text{pounds}$ is represented by the ordered pair $(66, 142)$ .
- Example: A person ingesting $300\,\text{mg}$ of caffeine daily and sleeping $7\,\text{hours}$ is represented by $(300, 7)$ .
Time-Based Variables: When recording data over years, a reference year is often used. For a study since 2010 ( $t = 0$ ), the year 2017 is expressed as $t = 7$ .
Visual Organization:
- Tables: The explanatory variable is listed in the first column and the response variable in the second column.
- Coordinate Systems: The horizontal axis represents the explanatory variable, and the vertical axis represents the response variable.

Positive Association: Occurs when the response variable tends to increase as the explanatory variable increases.
Negative Association: Occurs when the response variable tends to decrease as the explanatory variable increases.

Negative Association Case Study: Atlantic City gambling revenue dropped from $5.0\,\text{billion}$ in 2005 to $2.4\,\text{billion}$ in 2017. Although the revenue trend was downward, time itself did not cause the decrease.
External Factors: Factors such as competition from online gambling sites owned by casinos outside Atlantic City can drive trends.
Trend Limitations: Historical associations do not guarantee future performance. For instance, monthly revenue increased after the Hard Rock Hotel & Casino opened in June 2018, reversing the previous multi-year decline.