Question

part ii – constructed response

1. the table below shows the age of chickens in years (x) and the number of eggs (y) each chicken laid in one week.

table: age in years (x): 3, 2, 7, 1, 2, 5, 1, 4, 2, 3; eggs laid in one week (y): 4, 4, 0, 6, 6, 2, 7, 2, 5, 3

a. draw a scatter plot of the data. label the axes.

b. draw a line of best fit. explain why your line is a good fit.

c. find an equation for the line of best fit.

d. use your line of best fit to find the approximate value of y when x = 6.

e. describe the association in the scatter plot.

f. would this relationship be categorized as correlation or causation? explain.

Answer:

Part a: Scatter Plot

To draw the scatter plot:

• x - axis (Horizontal): Label it "Age in Years (x)". The values of \( x \) are \( 3, 2, 7, 1, 2, 5, 1, 4, 2, 3 \).
• y - axis (Vertical): Label it "Eggs Laid in One Week (y)". The values of \( y \) are \( 4, 4, 0, 6, 6, 2, 7, 2, 5, 3 \).
• Plot each point \((x,y)\) on the coordinate plane. For example, the point \((3,4)\) means at age 3 years, the chicken laid 4 eggs. Similarly, plot \((2,4)\), \((7,0)\), \((1,6)\), \((2,6)\), \((5,2)\), \((1,7)\), \((4,2)\), \((2,5)\), \((3,3)\)

Part b: Line of Best Fit

• Drawing the Line: Visually estimate a line that passes through the middle of the data points, balancing the number of points above and below the line.
• Explanation of Good Fit: A good line of best fit should have approximately the same number of data points above and below it. This line minimizes the overall distance between the data points and the line, representing the general trend of the data.

Part c: Equation of Line of Best Fit

First, we can use the method of finding the slope and y - intercept. Let's choose two points on the line of best fit. Let's assume we pick two points, say \((1, 6.5)\) and \((7, 0.5)\) (approximate points on the line of best fit).

Step 1: Calculate the slope (\(m\))

The formula for slope is \(m=\frac{y_2 - y_1}{x_2 - x_1}\)
Let \((x_1,y_1)=(1, 6.5)\) and \((x_2,y_2)=(7, 0.5)\)
\(m=\frac{0.5 - 6.5}{7 - 1}=\frac{- 6}{6}=- 1\)

Step 2: Calculate the y - intercept (\(b\))

Using the point - slope form \(y=mx + b\) and the point \((1,6.5)\)
\(6.5=-1\times1 + b\)
\(b=6.5 + 1=7.5\)

So the equation of the line of best fit is approximately \(y=-x + 7.5\) (Note: The equation may vary slightly depending on the points chosen for the line of best fit)

Part d: Approximate value of \(y\) when \(x = 6\)

We use the equation of the line of best fit \(y=-x + 7.5\)

Step 1: Substitute \(x = 6\) into the equation

\(y=-6 + 7.5\)

Step 2: Calculate the value of \(y\)

\(y = 1.5\approx2\) (We can also estimate from the line on the scatter plot. When \(x = 6\), we look at the line of best fit and find the corresponding \(y\) - value. The approximate number of eggs is around 1 or 2. Using our equation, we get \(y = 1.5\), which we can round to 2)

Part e: Association in the Scatter Plot

• Direction: The association is negative because as the age of the chicken (\(x\)) increases, the number of eggs laid (\(y\)) generally decreases.
• Form: The association appears to be linear because the points seem to follow a roughly straight - line pattern.
• Strength: The association is moderately strong. Most of the points are close to the line of best fit, but there is some scatter.

Part f: Correlation or Causation

• Category: Correlation
• Explanation: While there is a relationship (correlation) between the age of the chicken and the number of eggs laid (as age increases, egg production tends to decrease), we cannot say that age "causes" the change in egg production with absolute certainty from this data alone. There could be other factors such as the health of the chicken, diet, or living conditions that also affect egg production. A correlation just shows that two variables are related, while causation implies that one variable directly causes a change in the other.

Final Answers (for each part)

a. Scatter plot with points \((3,4)\), \((2,4)\), \((7,0)\), \((1,6)\), \((2,6)\), \((5,2)\), \((1,7)\), \((4,2)\), \((2,5)\), \((3,3)\) and axes labeled "Age in Years (x)" and "Eggs Laid in One Week (y)".
b. Line of best fit drawn through the middle of the data points, with approximately equal number of points above and below.
c. Approximately \(y=-x + 7.5\) (varies by line choice).
d. Approximately \(y = 2\) (or \(1.5\)).
e. Negative, linear, moderately strong association.
f. Correlation, because age and egg production are related but other factors may influence egg production.