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 II - Constructed Response

1. 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 the points \((3,4)\), \((2,4)\), \((7,0)\), \((1,6)\), \((2,6)\), \((5,2)\), \((1,7)\), \((4,2)\), \((2,5)\), \((3,3)\) on the coordinate plane.

1. b. Line of Best Fit

• Drawing the Line: Visually estimate a line that passes through the middle of the cluster of 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.

1. c. Equation of the Line of Best Fit

To find the equation of the line of best fit (\( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept):

• Step 1: Calculate the mean of \( x \) and \( y \)

The formula for the mean of a set of values \( x_1,x_2,\cdots,x_n \) is \( \bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n} \) and for \( y \) is \( \bar{y}=\frac{\sum_{i=1}^{n}y_i}{n} \)
\( \sum_{i = 1}^{10}x_i=3 + 2+7 + 1+2 + 5+1 + 4+2 + 3=30 \), so \( \bar{x}=\frac{30}{10} = 3 \)
\( \sum_{i=1}^{10}y_i=4 + 4+0 + 6+6 + 2+7 + 2+5 + 3=39 \), so \( \bar{y}=\frac{39}{10}=3.9 \)

• Step 2: Calculate the slope \( m \)

The formula for the slope \( m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2} \)
First, calculate \( (x_i-\bar{x})(y_i - \bar{y}) \) and \( (x_i-\bar{x})^2 \) for each data point:

\( x_i \)	\( y_i \)	\( x_i-\bar{x} \)	\( y_i-\bar{y} \)	\( (x_i - \bar{x})(y_i-\bar{y}) \)	\( (x_i-\bar{x})^2 \)
2	4	- 1	0.1	- 0.1	1
7	0	4	- 3.9	- 15.6	16
1	6	- 2	2.1	- 4.2	4
2	6	- 1	2.1	- 2.1	1
5	2	2	- 1.9	- 3.8	4
1	7	- 2	3.1	- 6.2	4
4	2	1	- 1.9	- 1.9	1
2	5	- 1	1.1	- 1.1	1
3	3	0	- 0.9	0	0

\( \sum_{i = 1}^{10}(x_i-\bar{x})(y_i - \bar{y})=0-0.1 - 15.6-4.2-2.1-3.8-6.2-1.9-1.1 + 0=-34 \)
\( \sum_{i=1}^{10}(x_i-\bar{x})^2=0 + 1+16 + 4+1+4+4+1+1+0 = 32 \)
\( m=\frac{-34}{32}\approx - 1.0625 \)

• Step 3: Calculate the y - intercept \( b \)

Using the formula \( \bar{y}=m\bar{x}+b \)
\( 3.9=-1.0625\times3 + b \)
\( 3.9=-3.1875 + b \)
\( b=3.9 + 3.1875=7.0875 \)
So the equation of the line of best fit is approximately \( y=-1.06x + 7.09 \) (rounded to two decimal places)

1. d. Approximate value of \( y \) when \( x = 6 \)

Substitute \( x = 6 \) into the equation of the line of best fit \( y=-1.06x + 7.09 \)
\( y=-1.06\times6+7.09=-6.36 + 7.09 = 0.73\approx1 \)

1. e. Association in the Scatter Plot

• Direction: As the age of the chicken (\( x \)) increases, the number of eggs laid (\( y \)) generally decreases. So there is a negative association.
• Form: The points seem to follow a somewhat linear pattern (the line of best fit is a straight line), so the association is approximately linear.
• Strength: The points are somewhat clustered around the line of best fit, but not extremely tightly. So it is a moderate negative linear association.

1. f. Correlation or Causation

• This relationship is a correlation.
• Explanation: While there is a relationship between the age of the chicken and the number of eggs laid (as age increases, egg - laying tends to decrease), we cannot say for certain that age "causes" the change in egg - laying. There could be other factors such as the health of the chicken, diet, breed, etc., that also affect the number of eggs laid. Correlation just means that there is a relationship between two variables, while causation implies that one variable directly causes a change in the other.