Question: Year 2011 2012 2013 2014 2015 2016
Profit (dollars) 100,000 85,000 91,000 82,000 79,500 74,000
y =
r =
Select the correct answer from the drop-down menu.
The linear regression equation appropriate for the data set.
Graph/table shape: a two-row table in the top-left showing Year versus Profit (dollars); position_hint = top-left.
1. **State the problem:** We have yearly profit data from 2011 to 2016 and want to find the linear regression equation $y = mx + b$ that models profit as a function of year, and calculate the correlation coefficient $r$.
2. **Assign variables:** Let $x$ represent the year, transformed to $x = \text{Year} - 2010$ for simplicity, so $x = 1, 2, 3, 4, 5, 6$.
3. **Data points:**
$$
\begin{array}{c|c}
x & y \\\hline
1 & 100000 \\
2 & 85000 \\
3 & 91000 \\
4 & 82000 \\
5 & 79500 \\
6 & 74000
\end{array}
$$
4. **Calculate sums:**
$$
\sum x = 1+2+3+4+5+6 = 21
$$
$$
\sum y = 100000 + 85000 + 91000 + 82000 + 79500 + 74000 = 511500
$$
$$
\sum xy = 1\times100000 + 2\times85000 + 3\times91000 + 4\times82000 + 5\times79500 + 6\times74000 = 1,735,500
$$
$$
\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 91
$$
$$
\sum y^2 = 100000^2 + 85000^2 + 91000^2 + 82000^2 + 79500^2 + 74000^2 = 44,089,250,000
$$
5. **Number of points:** $n=6$
6. **Calculate slope $m$:**
$$
m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{6 \times 1,735,500 - 21 \times 511,500}{6 \times 91 - 21^2} = \frac{10,413,000 - 10,741,500}{546 - 441} = \frac{-328,500}{105} = -3138.10
$$
7. **Calculate intercept $b$:**
$$
b = \frac{\sum y - m \sum x}{n} = \frac{511,500 - (-3138.10) \times 21}{6} = \frac{511,500 + 65,900.1}{6} = \frac{577,400.1}{6} = 96,233.35
$$
8. **Linear regression equation:**
$$
y = -3138.10 x + 96,233.35
$$
9. **Calculate correlation coefficient $r$:**
$$
r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}} = \frac{10,413,000 - 10,741,500}{\sqrt{(546 - 441)(6 \times 44,089,250,000 - 511,500^2)}}
$$
Calculate denominator:
$$
546 - 441 = 105
$$
$$
6 \times 44,089,250,000 = 264,535,500,000
$$
$$
511,500^2 = 261,635,225,000
$$
$$
264,535,500,000 - 261,635,225,000 = 2,900,275,000
$$
$$
r = \frac{-328,500}{\sqrt{105 \times 2,900,275,000}} = \frac{-328,500}{\sqrt{304,529,875,000}} = \frac{-328,500}{551,844.5} = -0.595
$$
10. **Interpretation:** The negative slope indicates profit decreases over years. The correlation coefficient $r \approx -0.595$ shows a moderate negative linear relationship.
**Final answers:**
$$
y = -3138.10 x + 96,233.35
$$
$$
r = -0.595
$$