Question

this question has two parts. first, answer part a. then, answer part b.
part a
physical fitness the table shows the percentage of students in public school who have met all six of california’s physical fitness standards each year since the 2011–2012 school year.

year	percentage
2012–2013	22.1%
2013–2014	22.6%
2014–2015	21.2%

a. write the equation for the best - fit line for the data. round to the nearest hundredth, if necessary.
(y=square x+square)
part b
b. find and interpret the correlation coefficient.
(r=) (\boxed{\text{select choice}})

Explanation:

Part A:

Step 1: Define Variables

Let \( x \) be the number of years since 2011 - 2012 (so \( x = 0 \) for 2011 - 2012, \( x = 1 \) for 2012 - 2013, \( x = 2 \) for 2013 - 2014, \( x = 3 \) for 2014 - 2015). Let \( y \) be the percentage of students. The data points are \( (0, 20.5) \), \( (1, 22.1) \), \( (2, 22.6) \), \( (3, 21.2) \).

Step 2: Calculate Mean of \( x \) and \( y \)

Mean of \( x \) (\( \bar{x} \)): \( \bar{x}=\frac{0 + 1+2 + 3}{4}=\frac{6}{4}=1.5 \)
Mean of \( y \) (\( \bar{y} \)): \( \bar{y}=\frac{20.5 + 22.1+22.6 + 21.2}{4}=\frac{86.4}{4}=21.6 \)

Step 3: Calculate Slope (\( m \))

Slope formula: \( m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2} \)
Calculate \( (x_i-\bar{x})(y_i-\bar{y}) \) for each point:

• For \( (0, 20.5) \): \( (0 - 1.5)(20.5 - 21.6)=(-1.5)(-1.1)=1.65 \)
• For \( (1, 22.1) \): \( (1 - 1.5)(22.1 - 21.6)=(-0.5)(0.5)=-0.25 \)
• For \( (2, 22.6) \): \( (2 - 1.5)(22.6 - 21.6)=(0.5)(1)=0.5 \)
• For \( (3, 21.2) \): \( (3 - 1.5)(21.2 - 21.6)=(1.5)(-0.4)=-0.6 \)

Sum of these: \( 1.65-0.25 + 0.5-0.6 = 1.3 \)

Calculate \( (x_i-\bar{x})^2 \) for each point:

• For \( x = 0 \): \( (0 - 1.5)^2 = 2.25 \)
• For \( x = 1 \): \( (1 - 1.5)^2 = 0.25 \)
• For \( x = 2 \): \( (2 - 1.5)^2 = 0.25 \)
• For \( x = 3 \): \( (3 - 1.5)^2 = 2.25 \)

Sum of these: \( 2.25+0.25 + 0.25+2.25 = 5 \)

So, \( m=\frac{1.3}{5}=0.26 \)

Step 4: Calculate Y - Intercept (\( b \))

Using \( y = mx + b \) and \( \bar{y}=m\bar{x}+b \)
\( 21.6=0.26\times1.5 + b \)
\( 21.6 = 0.39 + b \)
\( b=21.6 - 0.39 = 21.21 \)

Step 5: Write the Equation

The equation of the best - fit line is \( y = 0.26x+21.21 \)

Part B:

Step 1: Recall the Formula for Correlation Coefficient (\( r \))

The formula for the correlation coefficient is \( r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i-\bar{y})^2}} \)

Step 2: Calculate \( \sum_{i = 1}^{n}(y_i-\bar{y})^2 \)

For each point:

• For \( (0, 20.5) \): \( (20.5 - 21.6)^2=(-1.1)^2 = 1.21 \)
• For \( (1, 22.1) \): \( (22.1 - 21.6)^2=(0.5)^2 = 0.25 \)
• For \( (2, 22.6) \): \( (22.6 - 21.6)^2=(1)^2 = 1 \)
• For \( (3, 21.2) \): \( (21.2 - 21.6)^2=(-0.4)^2 = 0.16 \)

Sum of these: \( 1.21+0.25 + 1+0.16 = 2.62 \)

Step 2: Calculate the Numerator and Denominator

We know that \( \sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y}) = 1.3 \), \( \sum_{i = 1}^{n}(x_i-\bar{x})^2 = 5 \), and \( \sum_{i = 1}^{n}(y_i-\bar{y})^2 = 2.62 \)

The denominator is \( \sqrt{5\times2.62}=\sqrt{13.1}\approx3.62 \)

The numerator is \( 1.3 \)

Step 3: Calculate \( r \)

\( r=\frac{1.3}{3.62}\approx0.36 \)

The positive value of \( r \) (close to 0.36) indicates a weak positive linear relationship between the number of years since 2011 - 2012 and the percentage of students who met the physical fitness standards.

Answer:

(Part A):
\( y=\boldsymbol{0.26}x+\boldsymbol{21.21} \)