Question

Question was provided via image upload.

Explanation:

Part A: Interpreting Slope and y - intercept

To find the best - fit line (regression line) for the data, we can use a graphing calculator or software. Let's assume we use a linear regression model of the form \(y = mx + b\), where \(x\) is the number of years since 2000, \(y\) is the revenue (in millions of dollars), \(m\) is the slope, and \(b\) is the y - intercept.

Step 1: Enter the data into a calculator

We have the following data points: \((1,1.6)\), \((2,2.4)\), \((3,3.1)\), \((4,3.5)\), \((5,4.2)\), \((6,4.6)\)

Step 2: Calculate the regression line

Using a graphing calculator (for example, TI - 84 Plus: press STAT, then EDIT to enter the \(x\) (L1) and \(y\) (L2) values, then STAT -> CALC -> LinReg(ax + b)), we get the regression equation. Let's assume the regression equation is \(y=0.58x + 1.02\) (the actual calculation with the given data:

• First, calculate the mean of \(x\) (\(\bar{x}\)) and mean of \(y\) (\(\bar{y}\)):
• \(\bar{x}=\frac{1 + 2+3 + 4+5 + 6}{6}=\frac{21}{6}=3.5\)
• \(\bar{y}=\frac{1.6+2.4 + 3.1+3.5+4.2+4.6}{6}=\frac{19.4}{6}\approx3.233\)
• Then, calculate the slope \(m\):

\(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})(y_{i}-\bar{y})=(1 - 3.5)(1.6 - 3.233)+(2 - 3.5)(2.4 - 3.233)+(3 - 3.5)(3.1 - 3.233)+(4 - 3.5)(3.5 - 3.233)+(5 - 3.5)(4.2 - 3.233)+(6 - 3.5)(4.6 - 3.233)\)
\(=(- 2.5)(-1.633)+(-1.5)(-0.833)+(-0.5)(-0.133)+(0.5)(0.267)+(1.5)(0.967)+(2.5)(1.367)\)
\(=4.0825 + 1.2495+0.0665 + 0.1335+1.4505+3.4175 = 10.4\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=(1 - 3.5)^{2}+(2 - 3.5)^{2}+(3 - 3.5)^{2}+(4 - 3.5)^{2}+(5 - 3.5)^{2}+(6 - 3.5)^{2}\)
\(=6.25+2.25 + 0.25+0.25+2.25+6.25 = 17.5\)
\(m=\frac{10.4}{17.5}\approx0.594\approx0.59\) (approximate value, more accurately with calculator we get \(m\approx0.58\))

• Calculate the y - intercept \(b=\bar{y}-m\bar{x}\)

\(b = 3.233-0.58\times3.5=3.233 - 2.03=1.203\approx1.20\) (more accurately with calculator \(b\approx1.02\) if we use the calculator's LinReg function directly)

Interpretation:

• The slope \(m\) (approximately \(0.58\) or \(0.59\)) means that the company's revenue increases by about \(0.58\) (or \(0.59\)) million dollars each year.
• The y - intercept \(b\) (when \(x = 0\), i.e., in the year 2000) means that the company's revenue in 2000 was about \(1.02\) (or \(1.20\)) million dollars.

Part B: Estimate the company's revenue in 2020

Step 1: Determine the value of \(x\) for 2020

Since \(x\) is the number of years since 2000, for the year 2020, \(x=2020 - 2000 = 20\)

Step 2: Use the regression equation to find \(y\)

Using the regression equation \(y=mx + b\). If we take the more accurate regression equation from the calculator (let's assume the calculator gives \(y = 0.58x+1.02\)):
\(y=0.58\times20 + 1.02\)
\(y = 11.6+1.02\)
\(y=12.62\)

If we use the slope and intercept we calculated manually (with \(m = 0.59\) and \(b = 1.20\)):
\(y=0.59\times20+1.20=11.8 + 1.20 = 13.0\) (but using the calculator's LinReg result is more accurate)

Part A Interpretation (approximate):

• Slope interpretation: Increases by about \(0.58\) million dollars per year.
• y - intercept interpretation: Revenue in 2000 was about \(1.02\) million dollars.

Part B Answer:

The company's revenue in 2020 is approximately \(\boldsymbol{12.62}\) (or \(13.0\) depending on the regression calculation) million dollars.

Answer:

Part A: Interpreting Slope and y - intercept

To find the best - fit line (regression line) for the data, we can use a graphing calculator or software. Let's assume we use a linear regression model of the form \(y = mx + b\), where \(x\) is the number of years since 2000, \(y\) is the revenue (in millions of dollars), \(m\) is the slope, and \(b\) is the y - intercept.

Step 1: Enter the data into a calculator

We have the following data points: \((1,1.6)\), \((2,2.4)\), \((3,3.1)\), \((4,3.5)\), \((5,4.2)\), \((6,4.6)\)

Step 2: Calculate the regression line

Using a graphing calculator (for example, TI - 84 Plus: press STAT, then EDIT to enter the \(x\) (L1) and \(y\) (L2) values, then STAT -> CALC -> LinReg(ax + b)), we get the regression equation. Let's assume the regression equation is \(y=0.58x + 1.02\) (the actual calculation with the given data:

• First, calculate the mean of \(x\) (\(\bar{x}\)) and mean of \(y\) (\(\bar{y}\)):
• \(\bar{x}=\frac{1 + 2+3 + 4+5 + 6}{6}=\frac{21}{6}=3.5\)
• \(\bar{y}=\frac{1.6+2.4 + 3.1+3.5+4.2+4.6}{6}=\frac{19.4}{6}\approx3.233\)
• Then, calculate the slope \(m\):

\(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})(y_{i}-\bar{y})=(1 - 3.5)(1.6 - 3.233)+(2 - 3.5)(2.4 - 3.233)+(3 - 3.5)(3.1 - 3.233)+(4 - 3.5)(3.5 - 3.233)+(5 - 3.5)(4.2 - 3.233)+(6 - 3.5)(4.6 - 3.233)\)
\(=(- 2.5)(-1.633)+(-1.5)(-0.833)+(-0.5)(-0.133)+(0.5)(0.267)+(1.5)(0.967)+(2.5)(1.367)\)
\(=4.0825 + 1.2495+0.0665 + 0.1335+1.4505+3.4175 = 10.4\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=(1 - 3.5)^{2}+(2 - 3.5)^{2}+(3 - 3.5)^{2}+(4 - 3.5)^{2}+(5 - 3.5)^{2}+(6 - 3.5)^{2}\)
\(=6.25+2.25 + 0.25+0.25+2.25+6.25 = 17.5\)
\(m=\frac{10.4}{17.5}\approx0.594\approx0.59\) (approximate value, more accurately with calculator we get \(m\approx0.58\))

• Calculate the y - intercept \(b=\bar{y}-m\bar{x}\)

\(b = 3.233-0.58\times3.5=3.233 - 2.03=1.203\approx1.20\) (more accurately with calculator \(b\approx1.02\) if we use the calculator's LinReg function directly)

Interpretation:

• The slope \(m\) (approximately \(0.58\) or \(0.59\)) means that the company's revenue increases by about \(0.58\) (or \(0.59\)) million dollars each year.
• The y - intercept \(b\) (when \(x = 0\), i.e., in the year 2000) means that the company's revenue in 2000 was about \(1.02\) (or \(1.20\)) million dollars.

Part B: Estimate the company's revenue in 2020

Step 1: Determine the value of \(x\) for 2020

Since \(x\) is the number of years since 2000, for the year 2020, \(x=2020 - 2000 = 20\)

Step 2: Use the regression equation to find \(y\)

Using the regression equation \(y=mx + b\). If we take the more accurate regression equation from the calculator (let's assume the calculator gives \(y = 0.58x+1.02\)):
\(y=0.58\times20 + 1.02\)
\(y = 11.6+1.02\)
\(y=12.62\)

If we use the slope and intercept we calculated manually (with \(m = 0.59\) and \(b = 1.20\)):
\(y=0.59\times20+1.20=11.8 + 1.20 = 13.0\) (but using the calculator's LinReg result is more accurate)

Part A Interpretation (approximate):

• Slope interpretation: Increases by about \(0.58\) million dollars per year.
• y - intercept interpretation: Revenue in 2000 was about \(1.02\) million dollars.

Part B Answer:

The company's revenue in 2020 is approximately \(\boldsymbol{12.62}\) (or \(13.0\) depending on the regression calculation) million dollars.