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**Question 3** involves properties of the sample correlation coefficient \( r \) for data pairs \((x_i, y_i)\). Let's go through each part of the question and provide the proofs or arguments:

### (a) Show that \( -1 \leq r \leq 1 \)

The correlation coefficient \( r \) is defined as:
\[
r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{(n-1)s_x s_y}
\]

where \( s_x \) and \( s_y \) are the sample standard deviations of \( x \) and \( y \) respectively. The numerator is the covariance between \( x \) and \( y \).

By the Cauchy-Schwarz inequality, which states that for any real-valued vectors \( a \) and \( b \):
\[
\left( \sum_{i=1}^{n} a_i b_i \right)^2 \leq \left( \sum_{i=1}^{n} a_i^2 \right) \left( \sum_{i=1}^{n} b_i^2 \right),
\]
we can set \( a_i = x_i - \bar{x} \) and \( b_i = y_i - \bar{y} \). This gives:
\[
\left( \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) \right)^2 \leq \left( \sum_{i=1}^{n} (x_i - \bar{x})^2 \right) \left( \sum_{i=1}^{n} (y_i - \bar{y})^2 \right).
\]

Dividing by \( (n-1)^2 s_x^2 s_y^2 \) gives:
\[
r^2 \leq 1.
\]
Since \( r^2 \) is always non-negative, \( -1 \leq r \leq 1 \).

### (b) If for constants \( a \) and \( b \), with \( b > 0 \), \( y_i = a + b x_i \) for \( i \in \{1, ..., n\} \), then \( r = 1 \)

If \( y_i = a + b x_i \), every change in \( x_i \) leads to a proportional change in \( y_i \). This indicates a perfect linear relationship.

Substitute \( y_i \) into the correlation formula:
\[
\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) = b \sum_{i=1}^{n} (x_i - \bar{x})^2.
\]

\( s_y = |b| s_x \) and since \( b > 0 \), \( s_y = b s_x \).

Therefore,
\[
r = \frac{b \sum_{i=1}^{n} (x_i - \bar{x})^2}{(n-1) s_x (b s_x)} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{(n-1) s_x^2} = 1.
\]

### (c) If for constants \( a \) and \( b \), with \( b \leq 0 \), \( y_i = a + b x_i \) for \( i \in \{1, ..., n\} \), then \( r = -1 \)

Similarly, if \( b < 0 \), there is an inverse relationship:
\[
s_y = |b| s_x = -b s_x.
\]

Since \( b \leq 0 \),
\[
r = \frac{b \sum_{i=1}^{n} (x_i - \bar{x})^2}{(n-1) s_x (-b s_x)} = -1.
\]

### (d) If \( r \) is the sample correlation coefficient for data pairs \((x_i, y_i)\), then it is also the correlation coefficient for the data pairs \((a + b x_i, c + d y_i)\), provided that \( b \) and \( d \) are both positive or both negative.

Given linear transformations:
\[
x'_i = a + b x_i, \quad y'_i = c + d y_i,
\]

The correlation coefficient is invariant under linear transformations if they maintain the direction of relationship:
\[
r' = \frac{\sum_{i=1}^{n} (x'_i - \bar{x}')(y'_i - \bar{y}')}{(n-1) s_{x'} s_{y'}} = r.
\]

Since scaling by positive or negative constants doesn't change the direction of \( x_i \) and \( y_i \), this maintains the original correlation.

Would you like further details or have any specific questions?

### 5 Relative Questions to Expand:
1. How is the correlation coefficient used to determine the strength and direction of a linear relationship between two variables?
2. Can you explain the implications if the correlation coefficient \( r \) is close to 0?
3. What are some real-world examples where correlation does not imply causation?
4. How does the correlation coefficient differ from the covariance?
5. What are some limitations of using the correlation coefficient in statistical analysis?

### Tip:
Remember that the correlation coefficient is sensitive to outliers. A single outlier can significantly affect the value of \( r \), potentially leading to misleading interpretations.

Show/dispute the following properties of the sample correlation coefficient: (a) -1 ≤ r ≤ 1, (b) If for constants a and b, with b > 0, yi = a + bxi, then r = 1, (c) If for constants a and b, with b ≤ 0, yi = a + bxi, then r = -1, (d) If r is the sample correlation coefficient for data pairs (xi, yi), then it is also the correlation coefficient for the data pairs (a + bxi, c + dyi), provided that b and d are both positive or both negative.

Explore the properties of the sample correlation coefficient with detailed proofs for bounds, perfect linear relationships, and invariance under transformations. This guide covers important statistical concepts like covariance and the Cauchy-Schwarz inequality.

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