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The advertisement cost per months and the corresponding Volume of sales of Chiemaka Limited are shown below: x Advert cost #000 y Sales Plot the scattered diagram for the data Determine the simple correlation coefficient (r) and find the coefficient for determination (r2) for the data.

Mathematics

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Answers

2 parts
ii

0.9637\text{0.9637}

iii

0.9287\text{0.9287}

Step-by-step solution

6 steps

Here are the solutions to your questions.

Part i

Plot the scattered diagram for the data

To plot a scattered diagram, you will represent each pair of data points (x,y)(x, y) as a single point on a graph. Let the Advert cost (xx) be on the horizontal axis and Sales (yy) be on the vertical axis.

The data points are:

  • (6, 21)
  • (10, 28)
  • (12, 34)
  • (11, 31)
  • (13, 40)

Plot these five points on a coordinate plane. The resulting graph will show the relationship between advert cost and sales.

Part ii

0.9637\text{0.9637}

Determine the sample correlation coefficient (rr)

  1. Calculate the necessary sums.

    Let xx be Advert cost and yy be Sales.

    xyxyx2y262112636441102828010078412344081441156113134112196113405201691600∑x=52∑y=154∑xy=1675∑x2=570∑y2=4942\begin{array}{|c|c|c|c|c|} \hline x & y & xy & x^2 & y^2 \\ \hline 6 & 21 & 126 & 36 & 441 \\ 10 & 28 & 280 & 100 & 784 \\ 12 & 34 & 408 & 144 & 1156 \\ 11 & 31 & 341 & 121 & 961 \\ 13 & 40 & 520 & 169 & 1600 \\ \hline \sum x = 52 & \sum y = 154 & \sum xy = 1675 & \sum x^2 = 570 & \sum y^2 = 4942 \\ \hline \end{array}

    The number of data points is n=5n=5.

  2. Apply the formula for the sample correlation coefficient.

    The formula for rr is:

    r=n∑xy−(∑x)(∑y)[n∑x2−(∑x)2][n∑y2−(∑y)2]r = \frac{n \sum xy - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}

  3. Substitute the calculated sums into the formula.

    r=5(1675)−(52)(154)[5(570)−(52)2][5(4942)−(154)2]r = \frac{5(1675) - (52)(154)}{\sqrt{[5(570) - (52)^2][5(4942) - (154)^2]}}

  4. Perform the calculations.

    Numerator:

    5(1675)−(52)(154)=8375−8008=3675(1675) - (52)(154) = 8375 - 8008 = 367

    Denominator, first part:

    5(570)−(52)2=2850−2704=1465(570) - (52)^2 = 2850 - 2704 = 146

    Denominator, second part:

    5(4942)−(154)2=24710−23716=9945(4942) - (154)^2 = 24710 - 23716 = 994

    Denominator:

    (146)(994)=145024≈380.82016\sqrt{(146)(994)} = \sqrt{145024} \approx 380.82016

    Calculate rr:

    r=367380.82016≈0.9637r = \frac{367}{380.82016} \approx 0.9637

    The sample correlation coefficient is 0.9637\boxed{\text{0.9637}}.

Part iii

0.9287\text{0.9287}

Find the coefficient of determination (r2r^2)

  1. Square the value of the correlation coefficient (rr) found in part (ii).

    r2=(0.9637)2r^2 = (0.9637)^2

  2. Calculate the value.

    r2≈0.9287r^2 \approx 0.9287

    The coefficient of determination is 0.9287\boxed{\text{0.9287}}.

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