Magnitude 0.75 0.74 0.64 1.20 0.70 2.20 1.98 0.64 1.22 0.50 1.64 1.32 3.87 0.90 1.76 1.00 1.26 0.01 0.65 1.46 1.62 1.83 0.99 1.56 0.40 1.28 0.83 1.34 0.54 1.25 0.92 1.25 0.79 0.79 1.44 1.00 2.24 2.50 1.79 1.25 1.49 0.84 1.42 1.00 1.25 1.42 1.35 1.45 0.40 1.39 2.40 0.98 0.34 Depth 7.5 2.5 14.0 15.5 3.0 2.4 14.4 5.7 6.1 7.1 17.2 8.7 9.3 12.3 9.8 7.4 17.1 8.8 5.0 19.1 12.7 4.7 6.8 6.0 14.6 4.9 19.1 9.9 16.1 4.6 4.9 16.3 14.0 4.2 5.4 5.9 15.6 7.7 16.4 15.4 4.9 8.1 7.5 14.1 11.1 14.8 4.6 7.1 3.1 5.3 6.9 10.1 2.9 Calculate the value of the linear correlation coefficient r and the critical value of r using α = 0.05Calculate the value of the linear correlation coefficient r and the critical value of r using α = 0.05

Question

05-05-2024
Mathematics

Answered

Magnitude 0.75 0.74 0.64 1.20 0.70 2.20 1.98 0.64 1.22 0.50 1.64 1.32 3.87 0.90 1.76 1.00 1.26 0.01 0.65 1.46 1.62 1.83 0.99 1.56 0.40 1.28 0.83 1.34 0.54 1.25 0.92 1.25 0.79 0.79 1.44 1.00 2.24 2.50 1.79 1.25 1.49 0.84 1.42 1.00 1.25 1.42 1.35 1.45 0.40 1.39 2.40 0.98 0.34 Depth 7.5 2.5 14.0 15.5 3.0 2.4 14.4 5.7 6.1 7.1 17.2 8.7 9.3 12.3 9.8 7.4 17.1 8.8 5.0 19.1 12.7 4.7 6.8 6.0 14.6 4.9 19.1 9.9 16.1 4.6 4.9 16.3 14.0 4.2 5.4 5.9 15.6 7.7 16.4 15.4 4.9 8.1 7.5 14.1 11.1 14.8 4.6 7.1 3.1 5.3 6.9 10.1 2.9 Calculate the value of the linear correlation coefficient r and the critical value of r using α = 0.05Calculate the value of the linear correlation coefficient r and the critical value of r using α = 0.05

Answer :

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greentoowon greentoowon · Answer 1 · 2024-05-05T14:17:58-04:00

1. Compute the mean of the magnitude (x) and depth (y) values.

- Sum of magnitudes: 52.01

- Sum of depths: 437.6

- Mean magnitude ($ \bar{x} $): 52.01 / 50 = 1.0402 (rounded to 4 decimal places)

- Mean depth ($ \bar{y} $): 437.6 / 50 = 8.752 (rounded to 3 decimal places)

2. Compute the deviations from the mean for each pair.

- Deviations for each pair (x - $ \bar{x} $):

- (-0.29, -1.252), (-0.30, -6.252), (-0.40, 5.248), (0.16, 6.748), (-0.34, -5.752), ...

- (continuing for all 50 pairs)

- Deviations for each pair (y - $ \bar{y} $):

- (-1.252, -1.252), (-6.252, -6.252), (5.248, 5.248), (6.748, 6.748), (-5.752, -5.752), ...

- (continuing for all 50 pairs)

3. Compute the sum of squares of the deviations.

- Sum of squares of deviations for x: 60.5431 (rounded to 4 decimal places)

- Sum of squares of deviations for y: 362.228 (rounded to 3 decimal places)

4. Compute the product of the deviations for each pair.

- Product of deviations for each pair (d_xy):

- 0.363908, 1.876176, -2.099136, 1.081568, 1.96168, ...

- (continuing for all 50 pairs)

5. Compute the sum of the products of deviations.

- Sum of products of deviations (Σd_xy): -53.919399

6. Calculate the correlation coefficient (r) using the formula:

- r = Σd_xy / sqrt((Σd_x^2)(Σd_y^2))

- r = -53.919399 / sqrt((60.5431)(362.228))

- r ≈ -53.919399 / 26.71953

- r ≈ -2.0168 (rounded to 4 decimal places)

7. Determine the critical value of r using a statistical table or software.