A data set is found to have a linear correlation coefficient of r = − 0.92 r = -0.92 r = − 0.92 . Which of the following graphs most likely represents the relationship between these variables?

So now that we've learned a lot about the linear correlation coefficient, let's take a look at this practice problem here. So we're told that a data set is found to have a linear correlation coefficient. Remember, the letter that we use for that is little r and that little r is negative 0.92. We're given four graphs over here, and we're asked to find which of the following graphs most likely represents the relationship between these variables. Now, if you'll notice here, we have no idea what these variables even represent. We have no idea what y or x are. We just know what the linear correlation coefficient is. That's all we really need to know what the graph is going to look like. So let's go ahead and get started here. How do we use this number, negative 0.92, to figure out which graph it's going to be? Well, let's just remember a couple of things over here. I basically like to think of this number or this, this r value as kind of like a spectrum or a number line. Remember, it goes from negative one to positive one. That's all of the values that it could possibly be. We've got zero over here, and let me just sort of draw a couple tick marks. Right? We said here that the r values, as they get closer towards zero, they're actually weaker. And weaker r values basically means that the, the dots are sort of more loose or spread apart. Right? So this is what happens when you get r values that are closer to zero. Now on the other hand, the correlation gets stronger, so we have stronger correlation and higher r values whenever the dots that we see are sort of more tight together along that trend line. And this is gonna be where you have numbers that are closer towards either negative one or positive one. Now Now on the notes, the difference between negative and positive is just where the slope is gonna be. Right? So in other words, we have negative slopes over here, gonna point downwards, and then we have positive slopes over here. Alright? So we got our r value, which in this case is r equals negative 0.92. So that's a number that is close to negative one, and it's also negative. Right? So it's gonna be basically, we're gonna be looking for is we're gonna be looking for a very strong negative correlation. So we should expect to find a bunch of dots packed together in a line like this, in which that line is sort of very tight and also negative. So which one of these graphs could it possibly represent? So if we look at a over here, a doesn't look like it has really any specific correlation or any obvious relationship that's going on. I can't really draw a clear line that kind of cuts through a lot of these points. This would probably not be the answer because that correlation is very weak. So it's definitely not answer choice a. Now let's take a look at b over here. Right? So it looks like b, there's actually some kind of a correlation or some type of a relationship that's going on. Here, we see the values kind of go up like this. Then they get the you know, so they go down like this. So this is probably not going to be a strong negative correlation. It's probably just going to be a nonlinear correlation. We've seen those before as well. It's not answer choice B. So it's basically just between these two over here, which we have negative slopes in which these things are obviously pointing downwards. So which are the ones could it possibly be? Is it C or D? Now, you might be thinking that for C, because this trend line over here has a higher or steeper slope, that the r value is going to be higher. But actually, remember that the slope has nothing to do with the r value. Even though these dots are sort of steeper in the trend line, the dots in d actually form a tighter sort of trend or tighter relationship between y and x. Therefore, it's definitely not going to be answer choice c because, again, remember the slope has nothing to do with r value, and instead, your answer is going to be d. We have a very strong negative correlation, which means that the dots are very tightly packed together along this trend line. They don't deviate that much. And obviously, we have a negative slope. Alright? So this is gonna be your answer. Let me know if you have any questions, and I'll see you in the next one.

11. Correlation

Correlation Coefficient

Statistics for Business

11. Correlation

Correlation Coefficient

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Multiple Choice

A data set is found to have a linear correlation coefficient of $r = -0.92$ . Which of the following graphs most likely represents the relationship between these variables?

Scatter plot with orange dots showing a strong negative linear correlation between x and y variables.

Scatter plot with orange dots showing a strong negative linear correlation, sloping downward from left to right.

Scatter plot with orange dots showing a strong negative linear correlation between x and y axes.

Scatter plot showing a strong negative linear relationship between variables, with points descending from left to right.

Verified step by step guidance

Step 1: Understand the linear correlation coefficient (r). The value of r ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship. In this case, r = -0.92 indicates a strong negative linear relationship.

Step 2: Analyze the characteristics of a graph with a strong negative linear relationship. In such a graph, as the value of x increases, the value of y decreases consistently, forming a downward-sloping pattern.

Step 3: Examine the provided graphs. The first graph shows no clear pattern, indicating no correlation. The second graph shows a curved pattern, which suggests a non-linear relationship. The third graph shows a scattered pattern with no clear direction, indicating weak or no correlation. The fourth graph shows a clear downward-sloping pattern, consistent with a strong negative linear relationship.

Step 4: Match the graph to the correlation coefficient. Since r = -0.92 represents a strong negative linear relationship, the graph that best matches this description is the fourth graph.

Step 5: Conclude that the fourth graph most likely represents the relationship between the variables, as it visually aligns with the characteristics of a strong negative linear correlation.