Question

this question has two parts. first, answer part a. then, answer part b.
part a
farming some crops, such as barley, are very sensitive to how acidic the soil is. barley grows best in soil with a ph range of 6 to 7.5. to determine the ideal level of acidity, a farmer measures how many bushels of barley he harvests in different fields with varying acidity levels.

soil acidity (ph)	5.7	6.2	6.6	6.8	7.1

a. find the equation for the best - fit line for the data and the correlation coefficient. round to the nearest hundredth, if necessary.
y = \boxed{}x + \boxed{}

Explanation:

Step 1: Identify variables and data points

Let \( x \) be the soil acidity (pH) and \( y \) be the bushels harvested. The data points are: \((5.7, 3)\), \((6.2, 20)\), \((6.6, 48)\), \((6.8, 61)\), \((7.1, 73)\).

Step 2: Calculate necessary sums

First, calculate \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \):

• \( \sum x = 5.7 + 6.2 + 6.6 + 6.8 + 7.1 = 32.4 \)
• \( \sum y = 3 + 20 + 48 + 61 + 73 = 205 \)
• \( \sum xy = (5.7 \times 3) + (6.2 \times 20) + (6.6 \times 48) + (6.8 \times 61) + (7.1 \times 73) \)

\( = 17.1 + 124 + 316.8 + 414.8 + 518.3 = 1391 \)

• \( \sum x^2 = (5.7)^2 + (6.2)^2 + (6.6)^2 + (6.8)^2 + (7.1)^2 \)

\( = 32.49 + 38.44 + 43.56 + 46.24 + 50.41 = 211.14 \)

Step 3: Calculate slope (\( m \)) and y-intercept (\( b \))

The formula for the slope \( m \) of the best - fit line \( y = mx + b \) is:
\( m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}} \)
where \( n = 5 \) (number of data points).

Substitute the values:
\( m=\frac{5\times1391 - 32.4\times205}{5\times211.14-(32.4)^{2}} \)
First, calculate the numerator: \( 5\times1391=6955 \), \( 32.4\times205 = 32.4\times(200 + 5)=6480+162 = 6642 \)
Numerator: \( 6955 - 6642=313 \)
Denominator: \( 5\times211.14 = 1055.7 \), \( (32.4)^{2}=32.4\times32.4 = 1049.76 \)
Denominator: \( 1055.7-1049.76 = 5.94 \)
\( m=\frac{313}{5.94}\approx52.69 \)

The formula for the y - intercept \( b \) is:
\( b=\frac{\sum y - m\sum x}{n} \)
Substitute the values:
\( \sum y=205 \), \( m\approx52.69 \), \( \sum x = 32.4 \), \( n = 5 \)
\( b=\frac{205-52.69\times32.4}{5} \)
\( 52.69\times32.4 = 52.69\times(30 + 2.4)=1580.7+126.456 = 1707.156 \)
\( b=\frac{205 - 1707.156}{5}=\frac{-1502.156}{5}\approx - 300.43 \)

Step 4: Calculate correlation coefficient (\( r \))

The formula for the correlation coefficient \( r \) is:
\( r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}} \)
First, calculate \( \sum y^{2}=(3)^{2}+(20)^{2}+(48)^{2}+(61)^{2}+(73)^{2}=9 + 400+2304 + 3721+5329 = 11763 \)

Substitute the values:
Numerator: \( 5\times1391-32.4\times205 = 313 \) (same as before)
Denominator part 1: \( \sqrt{[n\sum x^{2}-(\sum x)^{2}]}=\sqrt{5.94}\approx2.44 \)
Denominator part 2: \( \sqrt{[n\sum y^{2}-(\sum y)^{2}]}=\sqrt{5\times11763-(205)^{2}} \)
\( 5\times11763 = 58815 \), \( (205)^{2}=42025 \)
\( 58815 - 42025 = 16790 \), \( \sqrt{16790}\approx129.58 \)
Denominator: \( 2.44\times129.58\approx316.18 \)
\( r=\frac{313}{316.18}\approx0.99 \)

Answer:

The equation of the best - fit line is \( y\approx52.69x - 300.43 \) and the correlation coefficient \( r\approx0.99 \)

For the equation of the line, filling in the blanks: \( y=\boxed{52.69}x+\boxed{- 300.43} \) (and the correlation coefficient is approximately \( 0.99 \))