Before we take up the discussion of linear regression and correlation, we need to examine a way to display the relation between two variables [latex]x[/latex] and [latex]y[/latex]. The most common and easiest way is a scatter plot. The following example illustrates a scatter plot.

Example

In Europe and Asia, m-commerce is popular. M-commerce users have special mobile phones that work like electronic wallets as well as provide phone and Internet services. Users can do everything from paying for parking to buying a TV set or soda from a machine to banking to checking sports scores on the Internet. For the years 2000 through 2004, was there a relationship between the year and the number of m-commerce users? Construct a scatter plot. Let [latex]x = \text{the year}[/latex], and let [latex]y = \text{the number of m-commerce users, in millions}[/latex].

The table below shows the data:

Table 1: Number of m-commerce users (in millions) (y-values) by Year (x-values)

[latex]x[/latex] (year)	[latex]y[/latex] (# of users)
2000	0.5
2002	20.0
2003	33.0
2004	47.0

Here is the scatter plot showing the number of m-commerce users (in millions) by year:

Example

The following data is real. The percent of declared ethnic minority students at De Anza College for selected years from 1970–1995 was:

Table 2: Year and Student Minority Percentage

Year (x)	Student Ethnic Minority Percentage (y)
1970	14.13
1973	12.27
1976	14.08
1979	18.16
1982	27.64
1983	28.72
1986	31.86
1989	33.14
1992	45.37
1995	53.1

The independent variable is "Year," while the independent variable is "Student Ethnic Minority Percent."

Here is the scatter plot for the data:

Your Turn!

Amelia plays basketball for her high school. She wants to improve to play at the college level. She notices that the number of points she scores in a game goes up in response to the number of hours she practices her jump shot each week. She records the following data:

Table 3: Points Scored versus Hours Practicing

X (hours practicing jump shot)	Y (points scored in a game)
5	15
7	22
9	28
10	31
11	33
12	36

Construct a scatter plot and state if what Amelia thinks appears to be true.

Solution

 

Yes, Amelia’s assumption appears to be correct. The number of points Amelia scores per game goes up when she practices her jump shot more.

A scatter plot shows the direction of a relationship between the variables. A clear direction happens when there is either:

You can determine the strength of the relationship by looking at the scatter plot and seeing how close the points are to a line, a power function, an exponential function, or some other type of function. For a linear relationship there is an exception. Consider a scatter plot where all the points fall on a horizontal line providing a "perfect fit." The horizontal line would in fact show no relationship.

When you look at a scatter plot, you want to notice the overall pattern and any deviations from the pattern. The following scatter plot examples illustrate these concepts.

Figure 4 - The first graph (a) is a scatter plot with 6 points plotted. The points form a pattern that moves upward to the right, almost in a straight line. The second graph (b) is a scatter plot with the same 6 points as the first graph. A 7th point is plotted in the top left corner of the quadrant. It falls outside the general pattern set by the other 6 points.

Figure 5 - The first graph (a) is a scatter plot with 6 points plotted. The points form a pattern that moves downward to the right, almost in a straight line. The second graph (b) is a scatter plot of 8 points. These points form a general downward pattern, but the points do not align in a tight pattern.

Figure 6 - The first graph (a) is a scatter plot of 7 points in an exponential pattern. The pattern of the points begins along the x-axis and curves steeply upward to the right side of the quadrant. The second graph (b) shows a scatter plot with many points scattered everywhere, exhibiting no pattern.

In this chapter, we are interested in scatter plots that show a linear pattern. Linear patterns are quite common. The linear relationship is strong if the points are close to a straight line, except in the case of a horizontal line where there is no relationship. If we think that the points show a linear relationship, we would like to draw a line on the scatter plot. This line can be calculated through a process called linear regression. However, we only calculate a regression line if one of the variables helps to explain or predict the other variable. If [latex]x[/latex] is the independent variable and [latex]y[/latex] the dependent variable, then we can use a regression line to predict [latex]y[/latex] for a given value of [latex]x[/latex].

Section 10.2 Practice

Does the scatter plot appear linear? Strong or weak? Positive or negative?

Does the scatter plot appear linear? Strong or weak? Positive or negative?

Does the scatter plot appear linear? Strong or weak? Positive or negative?

The Gross Domestic Product Purchasing Power Parity is an indication of a country’s currency value compared to another country. The table below shows the GDP PPP of Cuba as compared to US dollars. Construct a scatter plot of the data.

Table 4: Cuba's PPP versus Year

Year	Cuba’s PPP	Year	Cuba’s PPP
1999	1,700	2006	4,000
2000	1,700	2007	11,000
2002	2,300	2008	9,500
2003	2,900	2009	9,700
2004	3,000	2010	9,900
2005	3,500

The following table shows the poverty rates and cell phone usage in the United States. Construct a scatter plot of the data

Table 5: Year, Poverty Rate, and Cellular Usage per Capita

Year	Poverty Rate	Cellular Usage per Capita
2003	12.7	54.67
2005	12.6	74.19
2007	12	84.86
2009	12	90.82

Does the higher cost of tuition translate into higher-paying jobs?

The table lists the top ten colleges based on mid-career salary and the associated yearly tuition costs. Construct a scatter plot of the data.

Table 6: School, Midcareer Salary, and Yearly Tuition

School	Midcareer Salary (in thousands)	Yearly Tuition
Princeton	137	28,540
Harvey Mudd	135	40,133
CalTech	127	39,900
US Naval Academy	122	0
West Point	120	0
MIT	118	42,050
Lehigh University	118	43,220
NYU-Poly	117	39,565
Babson College	117	40,400
Stanford	114	54,506