Answer to Which correlation coefficient represents the strongest relationship?a. - b. +c. +d. + Correlation coefficient: Indicates the direction, positively or negatively of the relationship, and how strongly the 2 variables are related. Which of the following sequences of correlation coefficients correctly arranges the relationships between three pairs of two variables in order of increasing.
In fact, we would say that there is no relationship between the variables. If the coefficient is near 1. The correlation coefficient is 0. This is a weak positive relationship. This scatterplot depicts the relationship between the Number of Sports Events and the Number of Museums for the same cities: This is a moderate positive relationship.
This scatterplot depicts the relationship between the Number of Authors and the Number of Musicians for these cities: This is a strong positive relationship; the correlation coefficient is 0. The Direction of the Relationship The sign of the correlation coefficient indicates the direction of the relationship.
A positive relationship means that larger scores on one variable are associated with larger scores the other variable. A negative, or inverse, relationship means that larger scores on one variable are associated with smaller scores on the other variable.
Interpreting the Correlation Coefficient There is no rule for determining what size of correlation is considered strong, moderate or weak. The interpretation of the coefficient depends, in part, on the topic of study. When we are studying things that are difficult to measure, such as the contents of someone's mental life, we should expect the correlation coefficients to be lower.
In these kinds of studies, we rarely see correlations above 0. For this kind of data, we generally consider correlations above 0. When we are studying things that are more easily countable, we expect higher correlations. For example, with demographic data, we we generally consider correlations above 0. One useful way to interpret the correlation coefficient is based on explained variation. The square of the correlation coefficient is equal to the proportion of variation in the dependent variable that is accounted for, or explained, by variation in the independent variable.
A correlation of 0. The Significance Test If we collect data from a random sample, and calculate the correlation coefficient for two variables, we need to know how reliable the result is.
This calls for a statistical test. Let's say we have collected data on U. We want to know if there is a relationship between the number of artists in a community and the amount of arts funding it received. Start, as always, with the hypotheses. The null hypothesis states that there is no linear relationship between the independent variable and the dependent variable. In our example, the null hypothesis is that there is no relationship between the number of artists in a community and the amount of grant funding it received.
Next, we calculate the correlation coefficient for the sample. As always, if the significance, p, is less than or equal to 0. We can reject the null hypothesis and interpret the correlation coefficient.
The number of artists in a community is positively related to the amount of grant funding it received. Communities with more artists tended to receive more grant funding. Partial Correlations Based on the data from our sample, we concluded that there is a positive relationship between number of artists and amount of grant funding.
The main idea is that correlation coefficients are trying to measure how well a linear model can describe the relationship between two variables. For example, let me do some coordinate axes here. Let's say that's one variable.
Say that's my y variable and let's say that is my x variable. Let's say when x is low, y is low.Testing the Significance of the Correlation Coefficient
When x is a little higher, y is a little higher. When x is a little bit higher, y is higher. When x is really high, y is even higher. A linear model would describe it very, very well.
It's quite easy to draw a line that essentially goes through those points. So something like this would have an r of 1, r is equal to one. A linear model perfectly describes it and it's a positive correlation.
Example: Correlation coefficient intuition (video) | Khan Academy
When one increases, when one variable gets larger, then the other variable is larger. When one variable is smaller then other variable is smaller and vice versa. Now what would an r of negative one look like? Well, that would once again be a situation where a linear model works really well but when one variable moves up, the other one moves down and vice versa. Let me draw my coordinates, my coordinate axes again. I'm gonna try to draw a dataset where the r would be negative one.
Maybe when y is high, x is very low. When y becomes lower, x become higher. When y becomes a good bit lower, x becomes a good bit higher. Once again, when y decreases, x increases or as x increases, y decreases.
They're moving in opposite directions but you can fit a line very easily to this. The line would look something like this.
This would have an r of negative one, and r of zero, r is equal to zero, would be a dataset which a line doesn't really fit very well at all. I'll do that one really small, since I don't have much space here. An r of zero might look something like this.
Maybe I'll have a data point here, maybe have a data point here, maybe I have one there. And it wouldn't necessarily be this well organized but this gives you a sense of things. How would you actually try to fit a line here? You could equally justify a line that looks like that or a line that looks like that, or a line that looks like that. A linear model really does not describe the relationship between the two variables that well, right over here. So with that, is a primer.
Let's see if we can tackle these scatterplots.
The way I'm gonna do it is I'm just gonna try to eyeball what a linear model might look like. There's different methods of trying to fit a linear model to a dataset, an imperfect dataset.
I drew very perfect ones, at least for the r equals negative one and r equals one but these are what the real world actually looks like. Very few times will things perfectly sit on a line. For scatterplot A, if I were to try to fit a line, it would look something like that. If I were to try to minimize distances from the points to the line, I do see a general trend if we look at these data points over here, when y is high, x is low.
When x is larger, y is smaller. Looks like r is going to be less than zero, and a reasonable bit less than zero.
It's going to approach this thing here.
Which of these correlation numbers shows the strongest relationship?
If we look at our choices, it wouldn't be r equals 0. These are positive so I wouldn't use that one or that one.
And this one is almost no correlation.
R equals negative 0. I feel good with r is equal to negative 0.