A data set is found to have a linear correlation coefficient of 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.

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1. Intro to Stats and Collecting Data1h 14m

Intro to Stats
24m

Levels of Measurement
18m

Intro to Collecting Data
8m

Sampling Methods
23m

2. Describing Data with Tables and Graphs1h 55m

Visualizing Qualitative vs. Quantitative Data
4m

Frequency Distributions
35m

Histograms
14m

Bar Graphs and Pareto Charts
11m

Pie Charts
8m

Frequency Polygons
10m

Dot Plots
6m

Stemplots (Stem-and-Leaf Plots)
13m

Time-Series Graph
9m

3. Describing Data Numerically2h 5m

Mean
9m

Median
17m

Mode
7m

Standard Deviation
16m

Interpreting Standard Deviation
20m

Percentiles & Quartiles
14m

Describing Data Numerically Using a Graphing Calculator
10m

Boxplots
8m

Descriptive Statistics-Excel
11m

Boxplots-Excel
8m

4. Probability2h 16m

Basic Concepts of Probability
7m

Complements
6m

Addition Rule
17m

Multiplication Rule: Independent Events
11m

Introduction to Contingency Tables
17m

Multiplication Rule: Dependent Events
15m

Bayes' Theorem
13m

Fundamental Counting Principle
8m

Counting
37m

5. Binomial Distribution & Discrete Random Variables3h 6m

Discrete Random Variables
31m

Binomial Distribution
1h 7m

Finding Binomial Probabilities-Excel
17m

Poisson Distribution
40m

Finding Poisson Probabilities-Excel
15m

Hypergeometric Distribution
14m

6. Normal Distribution and Continuous Random Variables2h 11m

Uniform Distribution
18m

Standard Normal Distribution
39m

Probabilities & Z-Scores w/ Graphing Calculator
19m

Non-Standard Normal Distribution
21m

Finding Probabilities, Z Values, and X Values with the Normal Distribution-Excel
32m

7. Sampling Distributions & Confidence Intervals: Mean3h 23m

Sampling Distribution of the Sample Mean and Central Limit Theorem
19m

Distribution of Sample Mean - Excel
23m

Introduction to Confidence Intervals
15m

Confidence Intervals for Population Mean
1h 18m

Determining the Minimum Sample Size Required
12m

Finding Probabilities and T Critical Values - Excel
28m

Confidence Intervals for Population Means - Excel
25m

8. Sampling Distributions & Confidence Intervals: Proportion2h 10m

Sampling Distribution of Sample Proportion
29m

Confidence Intervals for Population Proportion
42m

Confidence Intervals for Population Proportion - Excel
12m

Chi Square Distribution
20m

Confidence Intervals for Population Variance
24m

9. Hypothesis Testing for One Sample5h 9m

Steps in Hypothesis Testing
1h 6m

Performing Hypothesis Tests: Means
1h 4m

Hypothesis Testing: Means - Excel
42m

Performing Hypothesis Tests: Proportions
37m

Hypothesis Testing: Proportions - Excel
27m

Performing Hypothesis Tests: Variance
12m

Critical Values and Rejection Regions
28m

Link Between Confidence Intervals and Hypothesis Testing
12m

Type I & Type II Errors
17m

10. Hypothesis Testing for Two Samples5h 37m

Two Proportions
1h 13m

Two Proportions Hypothesis Test - Excel
28m

Two Means - Unknown, Unequal Variance
1h 3m

Two Means - Unknown Variances Hypothesis Test - Excel
12m

Two Means - Unknown, Equal Variance
15m

Two Means - Unknown, Equal Variances Hypothesis Test - Excel
9m

Two Means - Known Variance
12m

Two Means - Sigma Known Hypothesis Test - Excel
21m

Two Means - Matched Pairs (Dependent Samples)
42m

Matched Pairs Hypothesis Test - Excel
12m

Two Variances and F Distribution
29m

Two Variances - Graphing Calculator
16m

11. Correlation1h 24m

Scatterplots & Intro to Correlation
26m

Correlation Coefficient
21m

Creating Scatterplots and FInding Correlation Coefficient - Excel
6m

Hypothesis Tests for Correlation Coefficient Using TI-84
17m

Inferences for the Correlation Coefficient - Excel
11m

12. Regression3h 33m

Linear Regression & Least Squares Method
26m

Residuals
12m

Coefficient of Determination
12m

Regression Line Equation and Coefficient of Determination - Excel
8m

Finding Residuals and Creating Residual Plots - Excel
11m

Inferences for Slope
31m

Enabling Data Analysis Toolpak
1m

Regression Readout of the Data Analysis Toolpak - Excel
21m

Prediction Intervals
13m

Prediction Intervals - Excel
19m

Multiple Regression - Excel
29m

Quadratic Regression
15m

Quadratic Regression - Excel
10m

13. Chi-Square Tests & Goodness of Fit2h 21m

Goodness of Fit Test
41m

Goodness of FIt Test Using TI-84
17m

Goodness of Fit Test - Excel
10m

Contingency Tables
12m

Independence Tests
14m

Homogeneity Tests
11m

Using Matrices on a TI-84
6m

Independence Test Using TI-84
12m

Independence Tests - Excel
13m

14. ANOVA2h 28m

Introduction to ANOVA
30m

One-Way ANOVA - Excel
12m

Multiple Comparisons: Tukey Test
14m

Multiple Comparisons: Tukey-Kramer Test
15m

Multiple Comparisons: Bonferoni Test
24m

Two-Way ANOVA
32m

Two-Way ANOVA - Excel
18m

11. Correlation

Correlation Coefficient

Statistics

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.

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Verified step by step guidance

Step 1: Understand the linear correlation coefficient (r). A value of r = -0.92 indicates a strong negative linear relationship between the two variables. This means that as one variable increases, the other variable tends to decrease in a linear fashion.

Step 2: Analyze the graphs provided. Look for a graph where the data points form a clear downward trend, indicating a negative linear relationship. The points should be relatively close to a straight line, reflecting the strong correlation (r close to -1).

Step 3: Eliminate graphs that do not show a linear relationship. For example, graphs with scattered points (no clear trend) or a curved pattern (non-linear relationship) are not consistent with a strong negative linear correlation.

Step 4: Identify the graph that best matches the description of a strong negative linear relationship. This graph should show data points forming a downward-sloping line with minimal scatter.

Step 5: Based on the analysis, the graph that most likely represents the relationship between the variables is the fourth graph, as it shows a clear downward trend with points closely aligned to a straight line.