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Problem Analysis and Solution

The worksheet focuses on understanding linear correlation, scatter plots, and linear regression. Below is a detailed solution for each part of the worksheet.

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#### 1. The Linear Correlation Coefficient

a) Give an example of two variables that you believe would be positively correlated. Explain your reasoning.

- Example: Height and weight of individuals.
- Reasoning: Generally, taller people tend to weigh more because height and body mass are related. As height increases, weight tends to increase as well, indicating a positive correlation.

b) Give an example of two variables that you believe would be negatively correlated. Explain your reasoning.

- Example: Number of hours spent studying and the number of mistakes made on a test.
- Reasoning: Typically, the more time a person spends studying, the fewer mistakes they make on a test. This inverse relationship indicates a negative correlation.

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#### 2. Interpretation of Correlation Coefficients

a) What does the linear correlation coefficient \( r = -0.90 \) indicate?

- Interpretation: A correlation coefficient of \( r = -0.90 \) indicates a strong negative linear relationship between the two variables. As one variable increases, the other tends to decrease significantly. The value being close to -1 suggests a very strong negative correlation.

b) What does the linear correlation coefficient \( r = 0.35 \) indicate?

- Interpretation: A correlation coefficient of \( r = 0.35 \) indicates a weak positive linear relationship between the two variables. As one variable increases, the other tends to increase slightly, but the relationship is not very strong. The value being close to 0 suggests a weak positive correlation.

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#### 3. Scatter Plot Analysis

Given scatter plots:

- a)
- i) State whether the variables appear to be positively correlated, negatively correlated, or have no correlation.
- The points in the scatter plot show a general upward trend. As the x-values increase, the y-values also tend to increase.
- Conclusion: Positively correlated.

- ii) Describe the strength of the correlation as weak, moderately weak, moderately strong, or strong.
- The points are relatively close to a straight line, indicating a strong linear relationship.
- Conclusion: Strong.

- iii) Estimate the value of the linear correlation coefficient.
- Given the strong positive correlation and how closely the points align with a straight line, the correlation coefficient is likely close to 1.
- Estimate: \( r \approx 0.85 \).

- b)
- i) State whether the variables appear to be positively correlated, negatively correlated, or have no correlation.
- The points in the scatter plot do not show a clear upward or downward trend. They appear randomly scattered.
- Conclusion: No correlation.

- ii) Describe the strength of the correlation as weak, moderately weak, moderately strong, or strong.
- Since there is no discernible pattern, the correlation is very weak.
- Conclusion: Very weak.

- iii) Estimate the value of the linear correlation coefficient.
- Given the lack of any apparent relationship, the correlation coefficient is close to 0.
- Estimate: \( r \approx 0.05 \).

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#### 4. Scatter Plot Analysis (Age vs. Test Score)

Given scatter plot:

- a) Draw an approximate line of best fit.
- Visually, draw a straight line that minimizes the distances from the points to the line. The line should slope downward since the test scores generally decrease with age.

- b) Determine the equation for the line you drew.
- Assume the line passes through points approximately at:
- Point 1: \( (30, 270) \)
- Point 2: \( (50, 220) \)
- Calculate the slope (\( m \)):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{220 - 270}{50 - 30} = \frac{-50}{20} = -2.5
\]
- Use the point-slope form of the line equation:
\[
y - y_1 = m(x - x_1)
\]
Using point \( (30, 270) \):
\[
y - 270 = -2.5(x - 30)
\]
Simplify:
\[
y - 270 = -2.5x + 75 \implies y = -2.5x + 345
\]
- Equation: \( y = -2.5x + 345 \)

- c) Interpret the slope and the y-intercept of your line.
- Slope (\( m = -2.5 \)): For every additional year of age, the test score decreases by approximately 2.5 points.
- Y-intercept (\( b = 345 \)): If a person were 0 years old, their predicted test score would be 345. However, this interpretation may not be meaningful in this context since age cannot be 0 in this scenario.

- d) Use your equation to predict the test score for a 60-year-old test-taker.
- Substitute \( x = 60 \) into the equation:
\[
y = -2.5(60) + 345 = -150 + 345 = 195
\]
- Prediction: The test score for a 60-year-old is 195.

- e) Estimate the value of the linear correlation coefficient for your line. Explain your reasoning.
- The scatter plot shows a clear downward trend, indicating a strong negative correlation. The points are relatively close to the line of best fit.
- Estimate: \( r \approx -0.8 \).

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#### 5. Scatter Plot Analysis (Age vs. Test Score)

Given scatter plot:

- a) Draw an approximate line of best fit.
- Visually, draw a straight line that minimizes the distances from the points to the line. The line should slope upward since the test scores generally increase with age.

- b) Determine the equation for the line you drew.
- Assume the line passes through points approximately at:
- Point 1: \( (25, 220) \)
- Point 2: \( (35, 280) \)
- Calculate the slope (\( m \)):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{280 - 220}{35 - 25} = \frac{60}{10} = 6
\]
- Use the point-slope form of the line equation:
\[
y - y_1 = m(x - x_1)
\]
Using point \( (25, 220) \):
\[
y - 220 = 6(x - 25)
\]
Simplify:
\[
y - 220 = 6x - 150 \implies y = 6x + 70
\]
- Equation: \( y = 6x + 70 \)

- c) Interpret the slope and the y-intercept of your line.
- Slope (\( m = 6 \)): For every additional year of age, the test score increases by approximately 6 points.
- Y-intercept (\( b = 70 \)): If a person were 0 years old, their predicted test score would be 70. However, this interpretation may not be meaningful in this context since age cannot be 0 in this scenario.

- d) Use your equation to predict the test score for a 50-year-old taking the test.
- Substitute \( x = 50 \) into the equation:
\[
y = 6(50) + 70 = 300 + 70 = 370
\]
- Prediction: The test score for a 50-year-old is 370.

- e) Estimate the value of the linear correlation coefficient for your line. Explain your reasoning.
- The scatter plot shows a clear upward trend, indicating a strong positive correlation. The points are relatively close to the line of best fit.
- Estimate: \( r \approx 0.8 \).

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Final Answers

1. a) Height and weight (positively correlated).
b) Study hours and test mistakes (negatively correlated).

2. a) Strong negative correlation.
b) Weak positive correlation.

3. a)
- i) Positively correlated.
- ii) Strong.
- iii) \( r \approx 0.85 \).
b)
- i) No correlation.
- ii) Very weak.
- iii) \( r \approx 0.05 \).

4. a) Line of best fit drawn.
b) \( y = -2.5x + 345 \).
c) Slope: -2.5 (decrease of 2.5 points per year), Y-intercept: 345.
d) 195.
e) \( r \approx -0.8 \).

5. a) Line of best fit drawn.
b) \( y = 6x + 70 \).
c) Slope: 6 (increase of 6 points per year), Y-intercept: 70.
d) 370.
e) \( r \approx 0.8 \).

\boxed{r \approx -0.8 \text{ and } r \approx 0.8}