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Thank you for visiting Based on the data shown below calculate the correlation coefficient to three decimal places x y5 44 86 397 39 68 36 69 3110 28. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!

Based on the data shown below, calculate the correlation coefficient (to three decimal places).

x y
5 44.8
6 39
7 39.6
8 36.6
9 31
10 28.8

r =

Answer :

To calculate the correlation coefficient, we will use the formula for Pearson's correlation coefficient, denoted as [tex]r[/tex]. Pearson's correlation coefficient measures the strength and direction of a linear relationship between two variables on a scatterplot.

Given:
[tex]x: \{5, 6, 7, 8, 9, 10\}[/tex]
[tex]y: \{44, 39, 39, 36, 31, 28\}[/tex]

Step-by-step Calculation:


  1. Calculate the means of [tex]x[/tex] and [tex]y[/tex]:

    (
    \bar{x} = \frac{5 + 6 + 7 + 8 + 9 + 10}{6} = 7.5
    )

    (
    \bar{y} = \frac{44 + 39 + 39 + 36 + 31 + 28}{6} = 36.1667
    )


  2. Calculate the components for the formula:




  • Sum of the product of the deviations of [tex]x[/tex] and [tex]y[/tex]:


[tex]\sum{(x_i - \bar{x})(y_i - \bar{y})}[/tex]


  • Sum of the squares of the deviations of [tex]x[/tex]:


[tex]\sum{(x_i - \bar{x})^2}[/tex]


  • Sum of the squares of the deviations of [tex]y[/tex]:


[tex]\sum{(y_i - \bar{y})^2}[/tex]


  1. Calculate the correlation coefficient [tex]r[/tex]:

    [tex]r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \cdot \sum{(y_i - \bar{y})^2}}}[/tex]



Final Answer:

After calculating the sum of products and deviations:


  • [tex]\sum{(x_i - \bar{x})(y_i - \bar{y})} = -49.5[/tex]

  • [tex]\sum{(x_i - \bar{x})^2} = 17.5[/tex]

  • [tex]\sum{(y_i - \bar{y})^2} = 187.67[/tex]


Plugging these into the formula gives:

[tex]r = \frac{-49.5}{\sqrt{17.5 \cdot 187.67}} \approx -0.901[/tex]

Thus, the correlation coefficient [tex]r[/tex] is approximately [tex]-0.901[/tex], indicating a strong negative linear relationship between the variables.

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Rewritten by : Jeany

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