To determine which explanatory variable ($x$ or $w$) has a stronger relationship with the response variable ($y$), we will compute the Pearson correlation coefficient for both datasets.

The Pearson correlation coefficient, $r$, is calculated as:

$r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2} \sqrt{\sum (Y_i - \bar{Y})^2}}$

Let's compute $r$ for both pairs: $(x, y)$ and $(w, y)$. numpy as np

Given data

x = np.array([71.9, 66.9, 81, 52.1, 59.6, 82.9, 75.9, 66.8]) y1 = np.array([74, 81, 79.5, 57.8, 82.3, 74.7, 104.3, 64.3])

w = np.array([69.2, 70, 75, 73.9, 79.8, 78, 66.1, 69]) y2 = np.array([80.3, 66.8, 60, 61.7, 42, 41.2, 77.1, 76.1])

Compute Pearson correlation coefficients

r_xy = np.corrcoef(x, y1)[0, 1] r_wy = np.corrcoef(w, y2)[0, 1]

r_xy, r_wyThe computed Pearson correlation coefficients are:

• Correlation between $x$ and $y$: $r = 0.463$ (rounded to three decimals)
• Correlation between $w$ and $y$: $r = -0.950$ (rounded to three decimals)

Since the absolute value of $r$ indicates the strength of the relationship, the second variable $w$ has a stronger relationship with the response variable $y$ because $| -0.950 | > | 0.463 |$.

Final Answer:

• The second variable ($w$) has a stronger relationship with $y$.

Would you like a more detailed explanation or any visualization of the correlation?

Related Questions:

1. What does a negative correlation indicate in this context?
2. How does the strength of correlation affect predictions?
3. Can we conclude causation from correlation?
4. What are some potential reasons for the difference in correlation strength?
5. How can we visualize the relationships between these variables?

Tip:

A correlation close to -1 or 1 indicates a strong relationship, while a correlation near 0 suggests a weak or no relationship.