Question

The Consumer Price Index​ (CPI) is a measure of the change in the cost of goods over time. If 1982 is used as the base year of comparison in some country, (CPI =100 in​ 1982), then the CPI of 191.2 in 2006 would indicate that an item that cost $ 1.00 in 1982 would cost $ 1.91 in 2006 in this country. It is known that the CPI in this country has been increasing at an approximately linear rate for the past 30 years.
a. Use this information to determine a linear function for this​ data, letting x be the years since 1982.

b. Based on your​ function, what was the CPI in​ 2000? Compare this estimate to the actual CPI of 167.5 for this country.

c. How is the annual CPI​ changing?

Answer 1

Answer:

a) $y = 3.8 x +100$

b) $Abs. change= |168.4-167.5|=0.9$

So the calculated value is 0.9 points above the actual value.

$Relative. Change =\frac{|168.4 -167.5|}{167.5}x100 =0.537$%

And the calculated value it's 0.537% higher than the actual value.

c) For this case we can use the slope obtained from the linear model to answer this question, and we can conclude that the CPI is increasing at approximate 3.8 units per year.

Step-by-step explanation:

Data given

1982 , CPI=100

1986, CPI = 191.2

Notation

Let CPI the dependent variable y. And the time th independent variable x.

For this case we want to adjust a linear model givn by the following expression:

$y=mx+b$

Solution to the problem

Part a

For this case we can find the slope with the following formula:

$m =\frac{CPI_{2006}-CPI_{1982}}{2006-1982}$

And if we replace we got:

$m =\frac{191.2-100}{2006-1982}=3.8$

Let X represent the number of years after. Then for 1982 t = 0, and if we replace we can find b:

$100 = 3.8(0)+b$

And then $b=100$

So then our linear model is given by:

$y = 3.8 x +100$

Part b

For this case we need to find the years since 1982 and we got x = 2000-1982=18, and if we rpelace this into our linear model we got:

$y = 3.8(18) +100=168.4$

And the actual value is 167.5 we can compare the result using absolute change or relative change like this:

$Abs. change= |168.4-167.5|=0.9$

So the calculated value is 0.9 points above the actual value.

And we can find also the relative change like this:

$Relative. Change =\frac{|Calculated -Real|}{Real}x100$

And if we replace we got:

$Relative. Change =\frac{|168.4 -167.5|}{167.5}x100 =0.537$%

And the calculated value it's 0.537% higher than the actual value.

Part c

For this case we can use the slope obtained from the linear model to answer this question, and we can conclude that the CPI is increasing at approximate 3.8 units per year.