Chapter 3, Problem 4P

(a)

To determine

To find: The quartic polynomial that best fits the given data.

Answer to Problem 4P

The quartic polynomial that best fits the given data is equal to $y=-55.9076x^4+1414.88x^3-12199.094x^2+42577.206x-25714.646$ .

Explanation of Solution

Given information:

Clothing sales tend to vary by season, with more clothes sold in spring and fall. The table gives sales figures for each month at a certain clothing center.

Month	Sales $\left(\$\right)$
January	$8000$
February	$18000$
March	$22000$
April	$31000$
May	$29000$
June	$21000$
July	$22000$
August	$26000$
September	$38000$
October	$40000$
November	$27000$
December	$15000$

Calculation:

Take the sales per thousand i.e. $y=1$ represents $1000$ dollars and the months represented by numbers.

For the quartic model of data follow the steps below:

First press the $\boxed{2\text{nd}}$ key and then $\boxed{\text{Y}=}$ and make sure that Plot $1$ is ON and choose the scatter plot icon in type.

Go to ${\text{Y}}_1$ and then press the key $\boxed{\text{STAT}}$ and then in edit enter the data in ${\text{L}}_1$ and ${\text{L}}_2$ .

Now enter the keystrokes $\boxed{\text{STAT}}\boxed{\to }\boxed{5}$ and press the $\boxed{\text{ENTER}}$ key for the regression.

  

Figure(1)

Therefore, the quartic polynomial that best fits the given data is equal to $y=-55.9076x^4+1414.88x^3-12199.094x^2+42577.206x-25714.646$ .

(b)

To determine

To graph: The scatter plot of the given data and the quadratic polynomial obtained from part(a) in the same viewing window.

Explanation of Solution

Given information:

Clothing sales tend to vary by season, with more clothes sold in spring and fall. The table gives sales figures for each month at a certain clothing center.

Month	Sales $\left(\$\right)$
January	$8000$
February	$18000$
March	$22000$
April	$31000$
May	$29000$
June	$21000$
July	$22000$
August	$26000$
September	$38000$
October	$40000$
November	$27000$
December	$15000$

Graph:

To graph the points on scatter plot, follow the steps using graphing calculator.

First press the $\boxed{2\text{nd}}$ key and then $\boxed{\text{Y}=}$ and make sure that Plot $1$ is ON and choose the scatter plot icon in type.

Go to ${\text{Y}}_1$ and then press the key $\boxed{\text{STAT}}$ and then in edit enter the data in ${\text{L}}_1$ and ${\text{L}}_2$ .

Now press the key $\boxed{\text{Y}=}$ and go to ${\text{Y}}_2$ and enter the right hand side of the function $y=-55.9076x^4+1414.88x^3-12199.094x^2+42577.206x-25714.646$ . Enter the keystrokes $\boxed{(-)}\boxed{55.9076}\boxed{\text{X}}\boxed{^}\boxed{4}\boxed{+}\boxed{1414.88}\boxed{\text{X}}\boxed{^}\boxed{3}\boxed{-}\boxed{12199.094}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{+}\boxed{42577.206}\boxed{\text{X}}\boxed{-}\boxed{25714.646}$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph in standard window as shown in Figure (2).

  

Figure (2)

(c)

To determine

To check: Whether the quartic polynomial is a good model for these data.

Answer to Problem 4P

Yes, the quartic model is good for these set of data as all the points are scattered in the graph of scatter plot and all the points lie near the line of model.

Explanation of Solution

Given information:

Clothing sales tend to vary by season, with more clothes sold in spring and fall. The table gives sales figures for each month at a certain clothing center.

Month	Sales $\left(\$\right)$
January	$8000$
February	$18000$
March	$22000$
April	$31000$
May	$29000$
June	$21000$
July	$22000$
August	$26000$
September	$38000$
October	$40000$
November	$27000$
December	$15000$

Calculation:

As observed from the graph in part(b), the points on the graph of scatter plot are scattered and all the points lies near the graph of the model.

Therefore, the quartic model is good for these set of data as all the points are scattered in the graph of scatter plot and all the points lie near the line of model.