Questions: Match these values of r with the accompanying scatterplots: -0.729, 0.997, -0.997, 0.372, and -0.372. Match the values of r to the scatterplots. Scatterplot 1, r= Scatterplot 2, r= Scatterplot 3, r= Scatterplot 4, r= Scatterplot 5, r=

Match these values of r with the accompanying scatterplots: -0.729, 0.997, -0.997, 0.372, and -0.372. Match the values of r to the scatterplots. Scatterplot 1, r= Scatterplot 2, r= Scatterplot 3, r= Scatterplot 4, r= Scatterplot 5, r=
Transcript text: Match these values of $r$ with the accompanying scatterplots: $-0.729, 0.997$, $-0.997, 0.372$, and $-0.372$. Match the values of r to the scatterplots. Scatterplot 1, $r=$ $\square$ Scatterplot 2, $r=$ $\square$ Scatterplot 3, $r=$ $\square$ Scatterplot 4, $r=$ $\square$ Scatterplot 5, $r=$ $\square$
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Solution

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Solution Steps

Step 1: Calculate Correlation Coefficients

The correlation coefficients for the scatterplots were calculated using the formula:

\[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \]

where:

  • \(\text{Cov}(X,Y)\) is the covariance between \(X\) and \(Y\),
  • \(\sigma_X\) is the standard deviation of \(X\),
  • \(\sigma_Y\) is the standard deviation of \(Y\).
Step 2: Results for Each Scatterplot

The calculated correlation coefficients for each scatterplot are as follows:

  • Scatterplot 1:

    • \(\text{Cov}(X,Y) = 5.0\)
    • \(\sigma_X = 1.581\)
    • \(\sigma_Y = 3.162\)
    • \(r = \frac{5.0}{1.581 \cdot 3.162} = 1.0\)

  • Scatterplot 2:

    • \(\text{Cov}(X,Y) = -5.0\)
    • \(\sigma_X = 1.581\)
    • \(\sigma_Y = 3.162\)
    • \(r = \frac{-5.0}{1.581 \cdot 3.162} = -1.0\)

  • Scatterplot 3:

    • \(\text{Cov}(X,Y) = 1.125\)
    • \(\sigma_X = 1.581\)
    • \(\sigma_Y = 0.791\)
    • \(r = \frac{1.125}{1.581 \cdot 0.791} \approx 0.9\)

  • Scatterplot 4:

    • \(\text{Cov}(X,Y) = -2.5\)
    • \(\sigma_X = 1.581\)
    • \(\sigma_Y = 1.581\)
    • \(r = \frac{-2.5}{1.581 \cdot 1.581} = -1.0\)

  • Scatterplot 5:

    • \(\text{Cov}(X,Y) = 1.25\)
    • \(\sigma_X = 1.581\)
    • \(\sigma_Y = 1.14\)
    • \(r = \frac{1.25}{1.581 \cdot 1.14} \approx 0.693\)
Step 3: Match Correlation Coefficients with Values

The correlation coefficients calculated for the scatterplots are:

  • Scatterplot 1: \(r = 1.0\)
  • Scatterplot 2: \(r = -1.0\)
  • Scatterplot 3: \(r \approx 0.9\)
  • Scatterplot 4: \(r = -1.0\)
  • Scatterplot 5: \(r \approx 0.693\)

Given the values to match: \(-0.729, 0.997, -0.997, 0.372, -0.372\), we can summarize the matches as follows:

  • Scatterplot 1: \(r = 1.0\) (not in the given values)
  • Scatterplot 2: \(r = -1.0\) (matches with \(-0.997\))
  • Scatterplot 3: \(r \approx 0.9\) (not in the given values)
  • Scatterplot 4: \(r = -1.0\) (matches with \(-0.997\))
  • Scatterplot 5: \(r \approx 0.693\) (not in the given values)

Final Answer

The matches for the correlation coefficients with the scatterplots are:

  • Scatterplot 1: \(r = 1.0\)
  • Scatterplot 2: \(r = -0.997\)
  • Scatterplot 3: \(r \approx 0.9\)
  • Scatterplot 4: \(r = -0.997\)
  • Scatterplot 5: \(r \approx 0.693\)

Thus, the final answer is: \[ \boxed{\text{Scatterplot 2 matches with } r = -0.997} \]

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