Week 1 discussion

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Elementary Statistics

Thirteenth Edition

Chapter 2

Summarizing and Graphing Data

2-1 Frequency Distributions for Organizing and Summarizing Data

2-2 Histograms

2-3 Graphs that Enlighten and Graphs that Deceive

2-4 Scatterplots, Correlation, and Regression

Key Concept

Introduce the analysis of paired sample data.

Discuss correlation and the role of a graph called a scatterplot, and provide an introduction to the use of the linear correlation coefficient.

Provide a very brief discussion of linear regression, which involves the equation and graph of the straight line that best fits the sample paired data.

Scatterplot and Correlation (1 of 2)

Correlation

A correlation exists between two variables when the values of one variable are somehow associated with the values of the other variable.

Linear Correlation

A linear correlation exists between two variables when there is a correlation and the plotted points of paired data result in a pattern that can be approximated by a straight line.

Scatterplot and Correlation (2 of 2)

Scatterplot (or Scatter Diagram)

A scatterplot (or scatter diagram) is a plot of paired (x, y) quantitative data with a horizontal x-axis and a vertical y-axis. The horizontal axis is used for the first variable (x), and the vertical axis is used for the second variable (y).

Example: Waist and Arm Correlation (1 of 2)

Correlation: The distinct pattern of the plotted points suggests that there is a correlation between waist circumferences and arm circumferences.

Example: Waist and Arm Correlation (2 of 2)

No Correlation: The plotted points do not show a distinct pattern, so it appears that there is no correlation between weights and pulse rates.

Linear Correlation Coefficient r

The linear correlation coefficient is denoted by r, and it measures the strength of the linear association between two variables.

Using r for Determining Correlation

The computed value of the linear correlation coefficient, r, is always between −1 and 1.

If r is close to −1 or close to 1, there appears to be a correlation.

If r is close to 0, there does not appear to be a linear correlation.

Example: Correlation between Shoe Print Lengths and Heights? (1 of 2)

Shoe Print Length (cm)	29.7	29.7	31.4	31.8	27.6
Height (cm)	175.3	177.8	185.4	175.3	172.7

Example: Correlation between Shoe Print Lengths and Heights? (2 of 2)

It isn’t very clear whether there is a linear correlation.

P-Value

If there really is no linear correlation between two variables, the P-value is the probability of getting paired sample data with a linear correlation coefficient r that is at least as extreme as the one obtained from the paired sample data.

Interpreting a P-Value from the Previous Example

The P-value of 0.294 is high. It shows there is a high chance of getting a linear correlation coefficient of r = 0.591 (or more extreme) by chance when there is no linear correlation between the two variables.

Interpreting a P-Value from the Example Where n = 5

Because the likelihood of getting r = 0.591 or a more extreme value is so high (29.4% chance), we conclude there is not sufficient evidence to conclude there is a linear correlation between shoe print lengths and heights.

Interpreting a P-Value

Only a small P-value, such as 0.05 or less (or a 5% chance or less), suggests that the sample results are not likely to occur by chance when there is no linear correlation, so a small P-value supports a conclusion that there is a linear correlation between the two variables.

Example: Correlation between Shoe Print Lengths and Heights (n = 40)

Example: Correlation between Shoe Print Lengths and Heights

The scatterplot shows a distinct pattern. The value of the linear correlation coefficient is r = 0.813, and the P-value is 0.000. Because the P-value of 0.000 is small, we have sufficient evidence to conclude there is a linear correlation between shoe print lengths and heights.

Regression

Given a collection of paired sample data, the regression line (or line of best fit, or least-squares line) is the straight line that “best” fits the scatterplot of the data.

Example: Regression Line (1 of 2)

Example: Regression Line (2 of 2)

The general form of the regression equation has a y-intercept of b0 = 80.9 and slope b1 = 3.22.

Using variable names, the equation is:

Height = 80.9 + 3.22 (Shoe Print Length)