Chapter 1.5, Problem 50E

(a)

To determine

To find: The natural logarithm regression equation and a scatter plot of the given data.

Answer to Problem 50E

The natural logarithm regression equation is $y=0.43+0.83\ln \left(x\right)$ and a scatter plot of the given data is shown in Figure (1).

Explanation of Solution

Given information: The table given below shows Natural gas production of a country in different years:

China’s Natural Gas Production
Year	Cubic Feet(trillions)
2002	1.15
2003	1.21
2004	1.44
2005	1.76
2006	2.07

Calculation:

Assume that $x=0$ represents year 2000, $x=1$ represents year 2001, $x=2$ represents year 2002 and so on.

To find the natural logarithm regression equation of the given data, use graphing calculator.

Step 1: Press $\boxed{\text{STAT}}\boxed{1}$ keys to enter the given data.

Step 2: List the input values 2, 3, 4, 5 and 6 in the L1 column.

Step 3: List the input values 1.15, 1.21, 1.44, 1.76 and 2.07 in the L2 column.

Step 4: Press the keystrokes $\boxed{\text{STAT}}\boxed{\Rightarrow }\boxed{9}\boxed{\text{ENTER}}$ to find line of regression.

  $y=0.43+0.83\ln \left(x\right)$

Now, to make the scatter plot, follow the steps using graphing calculator.

Step 1: Press $\boxed{2\text{nd}}\boxed{\text{Y}=}$ and make sure that Plot 1 is ON. Choose the scatter plot icon in type.

Step 2: Press $\boxed{\text{GRAPH}}$ key. Set the window at $\left[-1,13\right]$ by $\left[-1,5\right]$ .

Step 3: Press the keystrokes $\boxed{\text{STAT}}\boxed{\Rightarrow }\boxed{9}\boxed{\text{ENTER}}$ and $\boxed{\text{VARS}}\boxed{\Rightarrow }\boxed{1}\boxed{1}\boxed{\text{ENTER}}\boxed{\text{ENTER}}$ . Now, press the $\boxed{\text{GRAPH}}$ key to plot the graph as shown below.

  

Figure (1)

Therefore, the natural logarithm regression equation is $y=0.43+0.83\ln \left(x\right)$ and a scatter plot of the given data is shown in Figure (1).

(b)

To determine

To find: The estimated number of cubic feet of natural gas produced in 2008.

Answer to Problem 50E

The estimated natural gas produced by China in 2008 is 2.16 trillion cubic feet.

Explanation of Solution

Given information: The table given below shows Natural gas production of a country in different years:

China’s Natural Gas Production
Year	Cubic Feet(trillions)
2002	1.15
2003	1.21
2004	1.44
2005	1.76
2006	2.07

Calculation:

To find the natural gas produced in 2008, find the value of y at $x=8$ .

Press $\boxed{2\text{nd}}\boxed{\text{TRACE}}\boxed{1}\boxed{8}\boxed{\text{ENTER}}$ to find the value at $x=8$ as shown in the graph below.

  

Figure (2)

From the graph it can be seen that the value of rom the graph it can be seen that the value of y is 2.1594397.

Therefore, the estimated natural gas produced by China in 2008 is 2.16 trillion cubic feet.

(c)

To determine

To find: The year in which natural gas production reaches to 2.25 trillion cubic feet.

Answer to Problem 50E

In the year 2009 the natural gas production reaches to 2.25 trillion cubic feet.

Explanation of Solution

Given information: The table given below shows Natural gas production of a country in different years:

China’s Natural Gas Production
Year	Cubic Feet(trillions)
2002	1.15
2003	1.21
2004	1.44
2005	1.76
2006	2.07

Calculation:

From the graph shown in part (b), estimated natural gas produced in 2008 is 2.16 trillion cubic feet.

So, it can be predicted that the production of natural gas by China reaches to 2.25 trillion cubic feet in 2009.

Therefore, in year 2009 the natural gas production reaches to 2.25 trillion cubic feet.