Research Question: Do taller people tend to weigh more?

We will analyze R’s built in ‘women’ dataset.

Dataset Overview:

• 15 tabulated height-weight observations
• Height: 58-72 inches
• Weight: 115-164 pounds

Sample of the Data:

Goal: Determine whether height and weight have a positive linear relationship.

Correlation measures the strength and direction of a linear relationship between two numerical variables.

Correlation Coefficient:

\[ -1 \leq r \leq 1 \]

How to Interpret Correlation:

• \(r = +1\): Perfect positive relationship
• \(r = +0.8\): Strong positive relationship
• \(r = 0\): No linear relationship
• \(r = -0.8\): Strong negative relationship
• \(r = -1\): Perfect negative relationship

Our Prediction

As height increases, weight will also increase in this dataset.

Interpretation: The scatterplot shows a very strong positive relationship. As height increases, weight also increases.

Pearson’s correlation coefficient measures the strength of a linear relationship.

Formula:

\[ r = \frac{\operatorname{Cov}(X,Y)}{s_X s_Y} \]

Where:

• \(X\) = Height; \(Y\) = Weight
• \(\operatorname{Cov}(X,Y)\): Covariance
• \(s_X\) and \(s_Y\) = Standard deviations

Calculated Correlation:

\[ r \approx 0.995 \] Interpretation: The correlation is positive and very close to +1, indicating a very strong positive linear relationship.

R Code for Correlation:

cor(women$height, women$weight)

## [1] 0.9954948

Weight by Height:

Interpretation: The line graph shows that the tabulated weight increases consistently as height increases.

Interpretation: Hover over each point to view its height and weight measurements.

Key Findings:

• Height and weight have a very strong positive correlation.
• Pearson’s correlation coefficient is approximately 0.995.
• Taller heights correspond to greater weights in the dataset.

Statistical Insight:

• Correlation helps measure the strength and direction of relationships between numerical variables.
• Scatterplots help visualize relationships between numerical variables.

Limitations:

• The dataset contains only 15 tabulated observations.
• The data is not representative of all women.
• Correlation does not establish causation.

Conclusion:

Height and weight show a very strong positive linear relationship in this dataset, supporting our original prediction.