Scatter Plots and Correlation Notes

Scatter Plots and Correlation

Objective 1: Drawing Scatter Plots

• A scatter plot is a graph of ordered pairs used to determine if a relationship exists between two variables.
• Types of linear relationships:
  • Positive linear relationship: As one variable increases, the other variable also tends to increase.
  • Negative linear relationship: As one variable increases, the other variable tends to decrease.
  • No linear relationship: There is no apparent pattern between the two variables.
• No relationship: There isn't a discernible connection between the variables.

Example: Age and Systolic Blood Pressure

• Data:
  • Age (x): 43, 48, 56, 61, 67, 70
  • Blood Pressure (y): 128, 120, 135, 143, 141, 152
• Task: Draw a scatter plot for the data and determine if there appears to be a linear relationship.
• Assessment: Determine if the relationship is positive or negative.

Objective 2: Computing the Correlation Coefficient

• Population Correlation Coefficient:
  • Denoted by the Greek letter ρ (rho).
  • Computed using all possible pairs of data values from a population: $(x, y)$
• Sample Correlation Coefficient:
  • Symbolized by $r$. Measures the strength and direction of a linear relationship between two quantitative variables.
• Formula for Sample Correlation Coefficient (r):
  • $r = \frac{n \sum xy - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}$
  • Where $n$ is the number of data pairs.

Example: Compute the Correlation Coefficient for Age and Systolic Blood Pressure

• Data:
  • Age (x): 43, 48, 56, 61, 67, 70
  • Blood Pressure (y): 128, 120, 135, 143, 141, 152
  • $x^2$: 1849, 2304, 3136, 3721, 4489, 4900
  • $y^2$: 16384, 14400, 18225, 20449, 19881, 23104
  • $xy$: 5504, 5760, 7560, 8723, 9447, 10640

Properties of the Linear Correlation Coefficient

• The value of $r$ will always be between $-1$ and $1$ inclusively: $-1 \le r \le 1$
• The closer $r$ is to $-1$ or $1$, the stronger the linear association between the variables.
• The closer $r$ is to $0$, the weaker the linear association between the variables.
• If the values of $x$ and $y$ are interchanged, the value of $r$ will not change.

Objective 3: Hypothesis Testing for Correlation

• Null Hypothesis ($H_0$):
  • $H_0: \rho = 0$ (There is no linear correlation between the two variables.)
• Alternative Hypothesis ($H_1$):
  • $H_1: \rho \neq 0$ (There is a linear correlation between the two variables.)
• Test Value (t-statistic):
  • $t = \frac{r}{\sqrt{\frac{1 - r^2}{n - 2}}}$
• P-value:
  • Degrees of freedom: $df = n - 2$

Example: Hypothesis Test for Age and Systolic Blood Pressure

• Using a significance level of 5% ($\alpha = 0.05$), test if there is a linear correlation between age and systolic blood pressure.
• Steps:
  1. State the null and alternative hypotheses.
  2. Calculate the test statistic $(t)$.
  3. Determine the degrees of freedom $(df)$.
  4. Find the p-value.
  5. Make a decision based on the p-value and significance level.
  6. Draw a conclusion.

Example: Driver Age and Accidents

• Data represents ages of drivers and the number of accidents reported for each age group in Pennsylvania for a selected year.
  • Age (x): 16, 17, 18, 19, 20, 21
  • Number of accidents (y): 6605, 8932, 8506, 7349, 6458, 5974
• Tasks:
  • Draw a scatter plot.
  • Compute the correlation coefficient (round to three decimal places).
  • Test the claim that there is a linear correlation between ages of drivers and the number of accidents reported using $\alpha = 10\%$.

Example: Driver Age and Accidents (Excluding 16-Year-Olds)

• Calculate the correlation coefficient without the ordered pair $(16, 6605)$.
  • Age (x): 17, 18, 19, 20, 21
  • Number of accidents (y): 8932, 8506, 7349, 6458, 5974
  • $x^2$: 289, 324, 361, 400, 441
  • $y^2$: 79780624, 72352036, 54007801, 41705764, 35688676
  • $xy$: 151844, 153108, 139631, 129160, 125454
• Determine if there is a linear correlation between the ages of drivers and the number of accidents reported when 16-year-olds are left out using $\alpha = 10\%$.
  • Follow the same hypothesis testing steps as before.