Problem 14 The consumer price index (CPI) i... [FREE SOLUTION]

Chapter 7: Problem 14

The consumer price index (CPI) is shown in the following table. Fit a least squares line to the data. Then use the line to predict the CPI in the year $2020 \quad(x=7)$. $$ \begin{array}{ccc} \hline & \boldsymbol{x} & \mathbf{C P I} \\ 1990 & 1 & 130.7 \\ 1995 & 2 & 152.4 \\ 2000 & 3 & 172.2 \\ 2005 & 4 & 195.3 \\ 2010 & 5 & 218.1 \\ \hline \end{array} $$

Short Answer

Expert verified

The predicted CPI for 2020 is 351.1.

Step by step solution

Understanding the data

The table provides CPI (Consumer Price Index) values for different years. The year and corresponding CPI are given: 1990 (year 1) has 130.7, 1995 (year 2) has 152.4, 2000 (year 3) has 172.2, 2005 (year 4) has 195.3, and 2010 (year 5) has 218.1.

Set up the linear regression formula

To fit a least squares line, the equation of a line, $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept, needs to be determined.

Calculate the slope (m)

The formula for the slope of the least squares line is: $ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} $.Substitute: $ n = 5 $, $ \sum x = 1+2+3+4+5 = 15 $,$ \sum y = 130.7+152.4+172.2+195.3+218.1 = 868.7 $,$ \sum xy = 1(130.7)+2(152.4)+3(172.2)+4(195.3)+5(218.1) = 3049.5 $,$ \sum x^2 = 1^2+2^2+3^2+4^2+5^2 = 55 $.Plugging these into the formula:$ m = \frac{5(3049.5) - 15(868.7)}{5(55) - 15^2} = \frac{15247.5 - 13030.5}{275 - 225} = \frac{2217}{50} = 44.34 $.

Calculate the y-intercept (b)

The formula for the y-intercept is: $ b = \frac{\sum y - m(\sum x)}{n} $.Using $ m = 44.34 $ and previous calculations:$ b = \frac{868.7 - 44.34(15)}{5} = \frac{868.7 - 665.1}{5} = \frac{203.6}{5} = 40.72 $.

Construct the equation of the line

Using the calculated values of $ m $ and $ b $, the equation of the least squares line is:$ y = 44.34x + 40.72 $.

Predict the CPI for 2020 (x=7)

Use the line equation to predict the CPI in the year 2020 (x=7): $ y = 44.34(7) + 40.72 = 310.38 + 40.72 = 351.1 $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Consumer Price Index

The Consumer Price Index (CPI) plays a vital role in understanding the cost of living and inflation over time. It measures the average change over time in the prices paid by urban consumers for a market basket of consumer goods and services.
The CPI serves multiple purposes:

It's used as an economic indicator.
Employers may use it to adjust wages based on inflation.
Governments rely on it to make informed financial decisions.

For students, the CPI often appears in mathematical problems involving historical economic data and predictions. By grasping the concept of CPI, you can better understand how data from the past can inform present and future economic decisions.

Linear Regression

Linear regression is a statistical method used to establish the relationship between two variables. It's represented by the equation of a line, which is generally given by the formula: $ y = mx + b $ where:

$ y $ is the dependent variable (the variable we're trying to predict).
$ x $ is the independent variable (the variable we're using to make predictions).
$ m $ is the slope (the rate of change of $ y $ with respect to $ x $).
$ b $ is the y-intercept (the starting value of $ y $ when $ x $ is zero).

In the context of the CPI data, linear regression allows us to fit a line through historical data points to predict future CPIs, providing a tangible way to foresee trends and potential changes in economic conditions.

Slope Calculation

Calculating the slope in linear regression is crucial as it tells us how steep the line is and the direction of the relationship between the independent variable and the dependent variable. The slope $ m $ is determined by the formula: \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] Key points to remember during calculation:

$ n $ is the number of data points.
$ \sum xy $ is the sum of the product of the paired scores.
$ \sum x $ is the sum of the x-values, and $ \sum y $ is the sum of the y-values.
$ \sum x^2 $ is the sum of the squares of the x-values.

The outcome provides an insight into how much the dependent variable (CPI in this case) is expected to increase or decrease as the independent variable (year) increases by one unit.

Y-intercept Determination

The y-intercept $ b $ is another critical component in the linear regression equation. It represents the value of $ y $ when $ x $ is zero. This intercept is calculated using the formula: \[ b = \frac{\sum y - m(\sum x)}{n} \] When working it out:

Remember that $ m $ is the slope you previously calculated.
$ \sum y $ is the total of your dependent variable data.
$ \sum x $ is the sum of your independent variable data.
$ n $ is the count of your data points.

Finding $ b $ gives the starting point of the trend line in the context of your data, determining where the line crosses the y-axis. This value is crucial for accurately plotting the least squares regression line and predicting future values.

91影视

Short Answer

Step by step solution

Understanding the data

Set up the linear regression formula

Calculate the slope (m)

Calculate the y-intercept (b)

Construct the equation of the line

Predict the CPI for 2020 (x=7)

Key Concepts

Consumer Price Index

Linear Regression

Slope Calculation

Y-intercept Determination

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