The four data sets provided seem to differ in terms of their linear or nonlinear relationships between the variables. Based on the information provided, here’s how you can approach answering the questions:

(a) Perfect negative linear relationship:

A perfect negative linear relationship is indicated when one variable decreases in a perfectly linear fashion as the other increases. In such a case, the correlation coefficient (r) would be $-1$.

Looking at the data sets:

• The scatter plot for Figure 2 (u, v) shows a clear negative trend as $u$ increases, $v$ decreases. This suggests a strong negative correlation, likely perfect.

Answer: Data set $u, v$ (Figure 2)

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(b) Strongest positive linear relationship:

A strong positive linear relationship occurs when one variable increases in a linear fashion as the other increases, with a correlation coefficient close to 1.

Looking at the data sets:

• Figure 4 (m, n) shows a perfectly increasing relationship where $m$ increases, $n$ increases in the same manner. The data points lie perfectly on a straight line with a slope of 1.

Answer: Data set $m, n$ (Figure 4)

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(c) Strong nonlinear relationship:

A nonlinear relationship is indicated when the relationship between variables is not linear, meaning the data points do not follow a straight line.

Looking at the data sets:

• Figure 3 (w, t) shows a pattern where the relationship between $w$ and $t$ is not strictly linear. There are rises and falls in the data that suggest a non-linear trend.

Answer: Data set $w, t$ (Figure 3)

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(d) Apparent positive, but not perfect, linear relationship:

This occurs when the data shows a general positive trend, but the points do not lie perfectly on a line.

Looking at the data sets:

• Figure 1 (x, y) shows a general positive trend but with some variation in the data points. It's not perfect, but the relationship between $x$ and $y$ appears to be linear overall.

Answer: Data set $x, y$ (Figure 1)

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Summary of answers:

• (a) $u, v$ (Figure 2)
• (b) $m, n$ (Figure 4)
• (c) $w, t$ (Figure 3)
• (d) $x, y$ (Figure 1)

Would you like to go into more details on the interpretation of linear relationships?

Here are five related questions to explore:

1. What is the difference between a positive and negative correlation?
2. How do you calculate the correlation coefficient for a data set?
3. What are the characteristics of a strong linear relationship in data?
4. How can scatter plots help identify nonlinear relationships?
5. Why is a perfect linear relationship rare in real-world data?

Tip: For bivariate data, always check the scatter plot first. It gives a visual representation of the relationship before calculating statistical measures like correlation.