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X value 56789 y value 22 29 28 31 37 find the correlation coefficient of the data around your r value to two decimal places is if necessary 0.94 6.40 0.90 0.93

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To find the correlation coefficient \( r \) for the given data points, we will calculate \( r \) using the formula for Pearson's correlation coefficient, which is given as:

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

Step 1: Organize the data.

Given that \( x = 56789 \) and \( y \) values are \( 22, 29, 28, 31, 37 \).

Let's denote the \( y \) values as:

  • \( y_1 = 22 \)
  • \( y_2 = 29 \)
  • \( y_3 = 28 \)
  • \( y_4 = 31 \)
  • \( y_5 = 37 \)

Step 2: Calculate necessary sums.

We only have one \( x \) value repeated for each of the \( y \) values, essentially treating it as a constant.

  1. Number of data points \( n = 5 \).
  2. Calculate sums:
    • \( \sum y = 22 + 29 + 28 + 31 + 37 = 147 \)
    • \( \sum y^2 = 22^2 + 29^2 + 28^2 + 31^2 + 37^2 = 484 + 841 + 784 + 961 + 1369 = 3439 \)
  3. The repeated \( x \) values lead to:
    • \( \sum x = 5 \times 56789 = 283945 \)
    • \( \sum xy = 56789(22 + 29 + 28 + 31 + 37) = 56789 \times 147 = 8357023 \)
    • \( \sum x^2 = 5 \times (56789^2) = 5 \times 3232106521 = 16160532605 \)

Step 3: Plug values into the \( r \) formula.

Now we can calculate:

\[ r = \frac{5(8357023) - (283945)(147)}{\sqrt{[5(16160532605) - (283945)^2][5(3439) - (147)^2]}} \]

Calculating individually:

  1. \( n \sum xy = 5 \times 8357023 = 41785115 \)
  2. \( \sum x \sum y = 283945 \times 147 = 41785515 \)

So:

\[ 5 \sum xy - \sum x \sum y = 41785115 - 41785515 = -400 \]

Calculating squares:

  1. \( n \sum x^2 = 5(16160532605) = 80802663025 \)
  2. \( \sum x^2 = (283945)^2 = 80503679225 \)
  3. \( n \sum y^2 = 5(3439) = 17195 \)
  4. \( \sum y^2 = (147)^2 = 21609 \)

Putting these into the square root terms:

\[ [5 \sum x^2 - (\sum x)^2] = 80802663025 - 80503679225 = 298183800 \] \[ [5 \sum y^2 - (\sum y)^2] = 17195 - 21609 = -4400 \quad \text{(which is not feasible)} \]

We notice this indicates a problem with our x values being constant influences the calculations of correlation drastically.

Conclusion

In this simple case of all x being constant, the correlation is undefined (or may be considered negligible). In essence, correlation calculations generally require variation in both x and y variables. Hence, we can't find valid results leading to those options of \( r \) at 0.94, 6.40, 0.90, or 0.93 given your data structure.

If looking for valid correlations, consider datasets where x shows significant variation relative to y for practical r evaluations.