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This week, we are looking at the relationship between quantitative variables. We will concentrate on two variables for the purpose of this Discussion. We will be looking at the direction—positive or negative—and the strength—weak, moderate, or strong—of the correlation. There are no hard and fast rules on what numbers mean weak, moderate, or strong correlations; however, here is a rule of thumb:
0 < 0.4 weak 0.4 < 0.6 moderate 0.6 – 1 strong, where the numbers can be positive or negative. Remember that you can only “prove” that one variable “causes” another by using an experiment. Be skeptical of observational studies that claim causation. To prepare for this Discussion: Review the following resources: Discussion Resources (click to expand/reduce) Discussion Videos Finding Correlation Using Statdisk (4m) Statdisk to Find r (2m) StatCrunch: Creating Scatterplots (5m) Discussion Documents Scatterplots (PDF) Correlation (PDF) Look at the data set that you used in Discussion 1 and/or 2. In order to do this week’s Discussion, you will need to have two quantitative variables that you think might be related. In other words, changes in one variable result in changes in the other variable.  If you do not have two variables like this in your data set, add another variable.  Example: (click to expand/reduce) Review the rubric that will be used for grading.   With these thoughts in mind: By Day 1 By the end of Day 1, you can post your scenario and data for your Instructor to review.  If you don’t submit this by the end of Day 1, you will need to proceed to submit the entire Discussion on Day 3. By Day 3 Post a 1- to 2-paragraph write-up that includes the following: Scatterplots Does smoking cause lung cancer? Does low unemployment lead to inflation? Does human use of fossil fuel cause global warming? A major goal of statistical studies is to determine if there is a relationship between different variables. Once we know there is a relationship between the variables, we can try to determine if one variable causes the other. One of the first steps in this process is to make a scatterplot. A scatterplot is a diagram that represents the relationship between two quantitative variable. It is a plot of paired values (x, y) with the horizontal axis representing the first, x, variable and the vertical axis representing the second or y variable. In choosing the x and y variables, x is the explanatory variable and y is the response variable. The choice of the x and y variables is important. Ask yourself which variable depends on the result of the other and that will be the response or y variable. The pattern of the dots in a scatterplot are important in determining whether there is a correlation or relationship between the two variables. Interpreting scatterplots involves examining the overall pattern for deviations from the pattern, or outliers. The overall pattern can be explained using the direction, form, and strength of the relationship. The direction would state whether the two variables have a positive or negative relationship. In the case of a positive relationship, both variables move in the same direction. As one variable increases, so does the other; and as one variable decreases, so does the other. A negative direction would see the variables move in opposite directions. As one variable increases, the other decreases. For a positive direction, the points would move in an upward direction while a negative direction would see the points move downward. The form of the scatterplot would be whether the points seem to cluster in the form of a straight line, a parabola, or a cubic function. We will only study linear or straight-line relationships. The strength of the relationship is seen in how closely the points are clustered together around a line. The measure of strength is the correlation coefficient, which we will discuss next. There are a number of ways to create a scatterplot. Using technology is preferred. Stat Disk Example: We want to determine if the weight of a car is related to its city miles per gallon. Solution: Open Stat Disk and choose Data Sets, 12th edition of the textbook, and the file Car Measurements. The data will populate the spreadsheet. Refer to page 4 in the Stat Disk User’s Manual for directions on how to open a data file. Once you have the dataset displayed, click on Data, Scatterplot, and choose column 3, weight as the x or explanatory variable, and column 8, city MPG, as the y or response variable. Then click Evaluate. Refer to page 13 of the Stat Disk User’s Manual for Correlation A correlation exists between two quantitative variables when a change in one variable is associated with a change in the second variable. It measures the direction and strength of the relationship. Types of Correlation Positive correlation: both variables tend to increase or decrease together. On the scatterplot, the points tend to move in an upward direction. Negative correlation: the two variables tend to change in opposite directions with one increasing and the other decreasing. On the scatterplot, the points tend to move in a downward direction. No correlation: There is no apparent relationship between the two variables. On the scatterplot, the points tend to have no visible pattern and are scattered. Nonlinear correlation: the two variables are related but the relationship does not appear to follow a straight-line pattern. On the scatterplot, the points tend to show a parabolic or cubic pattern. We will study only linear correlation. For most purposes, it is sufficient to state whether the correlation is strong, weak, or nonexistent by looking at the scatterplot. However, sometimes it is useful to be more precise and associate a numerical value to measure the strength using the correlation coefficient which is denoted by the letter r. Properties of the Correlation Coefficient, r • The correlation coefficient is a unitless number that measures the strength of the correlation and has a value between -1 and +1. • If there is no correlation between the two variables meaning the points on the scatterplot are random, the correlation coefficient will be close to 0. • If there is a positive correlation, the correlation coefficient will be positive, greater than 0 to 1. A correlation coefficient closer to one will be clustered together in an upward sloping straight-line pattern. • If there is a negative correlation, the correlation coefficient is negative, less than 0 to -1. A correlation coefficient closer to -1 will be clustered together in a downward sloping straight-line pattern. • Like the mean and standard deviation, the correlation coefficient is not resistant to outliers. • Changing the units of measurement on the two variables does not change the correlation coefficient because correlation does not have units of measure. The figure below shows the strength of the correlation and how it relates to the scatterplot. Notice that a perfect correlation of +1 in the first graphic shows the points along a straight line. The last graphic show a Correlation perfect negative correlation where the points are along a negative straight line. In the center, notice the correlation coefficient of 0, no correlation shows a random pattern. The other graphics represent different correlation coefficients with varying linear patterns. There are no hard and fast rules on what numbers mean weak, moderate or strong correlations; however, here is a

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