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Problem Analysis and Solution

The worksheet involves several tasks related to linear regression, correlation, and scatter plots. Let's solve each part step by step.

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#### Part 1: Match the r-values to the graphs

The task is to match the given r-values to the corresponding scatter plots. Recall that:
- r ≈ -0.990: Strong negative correlation.
- r ≈ -0.419: Moderate negative correlation.
- r ≈ -0.019: Very weak or no correlation.
- r ≈ 0.857: Strong positive correlation.
- r ≈ 0.669: Moderate positive correlation.

Graph Analysis:
1. Graph 1: The points are scattered with no clear trend. This indicates a very weak or no correlation.
→ Matches A] r ≈ -0.019.
2. Graph 2: The points show a strong downward trend, indicating a strong negative correlation.
→ Matches B] r ≈ -0.990.
3. Graph 3: The points show a moderate downward trend, indicating a moderate negative correlation.
→ Matches C] r ≈ -0.419.
4. Graph 4: The points show a strong upward trend, indicating a strong positive correlation.
→ Matches D] r ≈ 0.857.
5. Graph 5: The points show a moderate upward trend, indicating a moderate positive correlation.
→ Matches E] r ≈ 0.669.

Answer:
1. A
2. B
3. C
4. D
5. E

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#### Part 2: Match the graphs to the description of the relationship

The task is to match the given graphs (A, B, C) to the descriptions of relationships.

Graph Analysis:
- Graph A: Shows a strong positive correlation.
- Graph B: Shows a moderate positive correlation.
- Graph C: Shows a strong positive correlation.

Descriptions:
6. Number of college classes taken and hours of free time: As the number of classes increases, free time decreases. This is a negative correlation.
→ Does not match any graph.
7. Number of college classes taken and the cost of tuition: As the number of classes increases, tuition cost increases. This is a positive correlation.
→ Matches Graph A (strong positive correlation).
8. Number of college classes taken and number of roommates: As the number of classes increases, the number of roommates might increase slightly. This is a moderate positive correlation.
→ Matches Graph B (moderate positive correlation).

Answer:
6. None (negative correlation not shown)
7. Graph A
8. Graph B

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#### Part 3: Describe the trend in the scatter plot

The task is to complete statements based on the trends in the scatter plots.

Scatter Plot Analysis:
1. Graph for SAT Score vs. College GPA: As SAT scores increase, college GPA tends to increase.
→ Statement: "As SAT Score increases, college GPA increases."
2. Graph for Cups of Coffee vs. Hours of Sleep: As the number of cups of coffee increases, the number of hours of sleep decreases.
→ Statement: "The more cups of coffee you drink, the fewer hours of sleep you get."
3. Graph for Distance from Baltimore vs. Airfare: As the distance from Baltimore increases, the cost of airfare increases.
→ Statement: "The cost of airfare increases as the distance from Baltimore increases."

Answer:
9. increases, increases
10. more, fewer
11. increases, increases

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#### Part 4: Use the line of best fit for predictions

The task is to use the data to find the line of best fit and make predictions.

Data Table:
| Years since 1999 | 0 | 2 | 3 | 4 |
|-------------------|---|---|---|---|
| Number of teams | 163 | 149 | 143 | 119 |

Step 1: Calculate the line of best fit.

The general form of the line of best fit is \( y = mx + b \), where:
- \( m \) is the slope.
- \( b \) is the y-intercept.

Calculate the slope \( m \):
\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]

Data Calculation:
- \( n = 4 \)
- \( \sum x = 0 + 2 + 3 + 4 = 9 \)
- \( \sum y = 163 + 149 + 143 + 119 = 574 \)
- \( \sum xy = (0 \cdot 163) + (2 \cdot 149) + (3 \cdot 143) + (4 \cdot 119) = 0 + 298 + 429 + 476 = 1203 \)
- \( \sum x^2 = 0^2 + 2^2 + 3^2 + 4^2 = 0 + 4 + 9 + 16 = 29 \)

\[ m = \frac{4(1203) - (9)(574)}{4(29) - (9)^2} = \frac{4812 - 5166}{116 - 81} = \frac{-354}{35} \approx -10.11 \]

Calculate the y-intercept \( b \):
\[ b = \frac{\sum y - m(\sum x)}{n} = \frac{574 - (-10.11)(9)}{4} = \frac{574 + 90.99}{4} = \frac{664.99}{4} \approx 166.25 \]

Thus, the line of best fit is:
\[ y = -10.11x + 166.25 \]

Step 2: Make predictions.

C] How many softball teams there were in 2000:
- Year 2000 corresponds to \( x = 1 \).
\[ y = -10.11(1) + 166.25 = -10.11 + 166.25 = 156.14 \]
Rounded to two decimal places: \( 156.14 \).

D] Approximate the number of softball teams in 2020:
- Year 2020 corresponds to \( x = 21 \).
\[ y = -10.11(21) + 166.25 = -212.31 + 166.25 = -46.06 \]
Rounded to two decimal places: \( -46.06 \).

Reasonableness of Predictions:
- 2000 Prediction (156.14): Reasonable, as it is close to the observed values.
- 2020 Prediction (-46.06): Unreasonable, as the number of teams cannot be negative. This suggests the linear trend may not hold over such a long period.

Answer:
A] Line of best fit: \( y = -10.11x + 166.25 \)
B] Correlation: Negative correlation (teams decrease over time).
C] 156.14 teams in 2000. Reasonable.
D] -46.06 teams in 2020. Unreasonable.

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Final Answer:

\[
\boxed{
\begin{array}{l}
\text{Part 1: } 1. \text{A}, 2. \text{B}, 3. \text{C}, 4. \text{D}, 5. \text{E} \\
\text{Part 2: } 6. \text{None}, 7. \text{Graph A}, 8. \text{Graph B} \\
\text{Part 3: } 9. \text{increases, increases}, 10. \text{more, fewer}, 11. \text{increases, increases} \\
\text{Part 4: } \\
\text{A] Line of best fit: } y = -10.11x + 166.25 \\
\text{B] Correlation: Negative correlation} \\
\text{C] 156.14 teams in 2000. Reasonable} \\
\text{D] -46.06 teams in 2020. Unreasonable}
\end{array}
}
\]