Scatter Plot and Lines of Best Fit Worksheet with questions on real-world data trends and line of best fit equations.

Here are the solutions to the problems on your worksheet.

1. MUSIC (Scatter Plot Prediction)

Problem: The scatter plot shows the number of CDs sold from 1999 to 2005. If the trend continued, about how many CDs were sold in 2006?

Step-by-step Solution:
1. Look at the pattern: Look at the dots on the graph. They are going down as you move to the right. This means sales are decreasing every year.
2. Find the last data point: The last dot is for the year 2005. Looking at the y-axis (vertical axis), the dot for 2005 is at 300 million.
3. Determine the rate of change: Let's look at the years before that to see how much it drops each year.
* In 2004, the dot is at roughly 350 million.
* From 2004 to 2005, it dropped by about 50 million ($350 - 300 = 50$).
* In 2003, the dot is at roughly 400 million. It also dropped by about 50 million to get to 2004.
* The pattern seems to be a steady drop of about 50 million per year.
4. Predict 2006: If the trend continues, we subtract another 50 million from the 2005 value.
* $300 \text{ million} - 50 \text{ million} = 250 \text{ million}$.

Answer: About 250 million CDs.

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2. FAMILY (Cost vs. Age)

Problem: Draw a scatter plot and describe what relationship exists within the data.

Data Table:
* Age: 1, Cost: 750
* Age: 2, Cost: 1,100
* Age: 3, Cost: 1,450
* Age: 4, Cost: 1,800
* Age: 5, Cost: 2,150
* Age: 6, Cost: 2,500
* Age: 7, Cost: 2,850
* Age: 8, Cost: 3,200

Step-by-step Solution:
1. Plotting: You would put "Age" on the bottom (x-axis) and "Cost" on the side (y-axis). As you plot the points $(1, 750), (2, 1100)$, etc., you will see they form a straight line going up.
2. Describing the Relationship:
* As the age of the child increases (goes from 1 to 8), the cost increases (goes from 750 to 3200).
* Because both numbers go up together, this is a positive correlation.
* Because the points line up almost perfectly straight, it is a strong linear relationship.

Answer: There is a positive linear relationship. As the child gets older, the predicted annual cost increases steadily.

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3. Line of Best Fit Equation

Problem: Make a scatter plot of the data in the table. Draw a line of best fit. What is the equation of the line of best fit?

Data Table:
* X: -2, Y: 2
* X: -1, Y: 0
* X: 0, Y: -2
* X: 1, Y: -1
* X: 2, Y: -3
* X: 3, Y: -5

Step-by-step Solution:
1. Check the slope (rise over run): Let's look at how $Y$ changes when $X$ increases by 1.
* From $X=-2$ to $X=-1$: $Y$ goes from $2$ to $0$ (Change of $-2$).
* From $X=-1$ to $X=0$: $Y$ goes from $0$ to $-2$ (Change of $-2$).
* From $X=0$ to $X=1$: $Y$ goes from $-2$ to $-1$ (Change of $+1$). *This point is slightly off the perfect line.*
* From $X=1$ to $X=2$: $Y$ goes from $-1$ to $-3$ (Change of $-2$).
* From $X=2$ to $X=3$: $Y$ goes from $-3$ to $-5$ (Change of $-2$).

Most of the points follow a pattern where $Y$ decreases by 2 for every step in $X$. So, the slope ($m$) is -2.

2. Find the y-intercept ($b$): The y-intercept is the value of $Y$ when $X$ is 0.
* Looking at the table, when $X = 0$, $Y = -2$.
* So, $b = -2$.

3. Write the Equation: Using $y = mx + b$:
* $m = -2$
* $b = -2$
* Equation: $y = -2x - 2$

*(Note: The point $(1, -1)$ is a little bit above this line, but the line $y = -2x - 2$ fits the majority of the data perfectly.)*

Answer: The equation is $y = -2x - 2$.

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4. EDUCATION (Study Hours vs. Grade)

Problem: Analyze the table showing study hours and test grades.

Data Table:
* Hours: 3, Grade: 84
* Hours: 2, Grade: 77
* Hours: 5, Grade: 92
* Hours: 1, Grade: 70
* Hours: 0, Grade: 60
* Hours: 4, Grade: 90
* Hours: 3, Grade: 79

a. Draw a scatter plot...
*(You would plot these points on the grid provided. The points generally go up from left to right.)*

b. What is the equation for the line of best fit?
Let's estimate the line.
1. Pick two points that seem to represent the middle and end of the trend.
* Point A: $(0, 60)$ — This is a good starting point (y-intercept).
* Point B: $(5, 92)$ — This is near the top right.
2. Calculate Slope ($m$):
* $\text{Slope} = \frac{\text{Change in } Y}{\text{Change in } X} = \frac{92 - 60}{5 - 0} = \frac{32}{5} = 6.4$
* Let's check another pair to be sure. Let's try $(1, 70)$ and $(4, 90)$.
* $\text{Slope} = \frac{90 - 70}{4 - 1} = \frac{20}{3} \approx 6.67$
* The slopes are close. Let's use an average or a simple integer if possible. A slope of roughly 6.5 seems reasonable.
* Let's try fitting $y = 6.5x + 60$.
* If $x=2$, $y = 6.5(2) + 60 = 13 + 60 = 73$ (Actual is 77). Close.
* If $x=3$, $y = 6.5(3) + 60 = 19.5 + 60 = 79.5$ (Actuals are 84 and 79). Very close.
* Another common method is just picking two clear points like $(0,60)$ and $(4,90)$.
* Slope = $\frac{90-60}{4-0} = \frac{30}{4} = 7.5$.
* Equation: $y = 7.5x + 60$.
* Let's test $x=5$: $7.5(5) + 60 = 37.5 + 60 = 97.5$. (Actual is 92). This is a bit high.
* Let's stick with the calculation from $(0,60)$ and $(5,92)$ which gave us 6.4. It fits the data well.
* Equation: $y = 6.4x + 60$

*(Note: In school settings, answers may vary slightly depending on exactly where you draw the line. $y = 6x + 62$ or $y = 6.5x + 60$ are also very good estimates.)*

c. Predict the grade for a student who studied for 6 hours.
Using our equation $y = 6.4x + 60$:
* Substitute $x = 6$.
* $y = 6.4(6) + 60$
* $y = 38.4 + 60$
* $y = 98.4$

Rounding to the nearest whole number, the grade would be 98.

d. Could this line go on forever? Why or why not?
No.
* Reason 1 (Grades): Test grades usually have a maximum limit (like 100%). If you plug in a huge number of hours, the equation will eventually predict a grade higher than 100, which is impossible.
* Reason 2 (Time): There are only 24 hours in a day. You cannot study for 100 hours in a single day. Eventually, the physical limit of time stops the line.

Final Answer:
1. MUSIC: About 250 million CDs.
2. FAMILY: Positive linear relationship (as age increases, cost increases).
3. LINE OF BEST FIT: $y = -2x - 2$
4. EDUCATION:
b. $y = 6.4x + 60$ (Approximate)
c. 98
d. No, because grades cannot exceed 100% and there is a limit to how many hours one can study in a day.