Question

Is the magnitude of an earthquake related to the depth below the surface at which the quake occurs? Let *x* be the magnitude of an earthquake (on the Richter scale), and let *y* be the depth (in kilometers) of the quake below the surface at the epicenter.

x |
2.8 | 3.9 | 3.3 | 4.5 | 2.6 | 3.2 | 3.4 |

y |
4.7 | 9.6 | 11.2 | 10.0 | 7.9 | 3.9 | 5.5 |

(a) Make a scatter diagram of the data.

Then visualize the line you think best fits the data.

(b) Use a calculator to verify that Σ*x* = 23.7, Σ*x*^{2} = 82.75, Σ*y* = 52.8, Σ*y*^{2} = 447.56 and Σ*xy* = 184.28.

Compute*r*. (Round to 3 decimal places.)

As*x* increases, does the value of *r* imply that *y* should tend to increase or decrease? Explain your answer.

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Then visualize the line you think best fits the data.

(b) Use a calculator to verify that Σ

Compute

As

Given our value of *r*, we can not draw any conclusions for the behavior of *y* as *x* increases.

Given our value of *r*, *y* should tend to remain constant as *x* increases.

Given our value of *r*, *y* should tend to decrease as *x* increases.

Given our value of *r*, *y* should tend to increase as *x* increases.

Step 1

**Part (a):**

The scatter diagram, along with the best fitted line (least-squares regression line) for the data is as follows:

Step 2

**Part (b):**

Here, ∑*x*, ∑*x*^{2}, ∑*y*, ∑*y*^{2}, and ∑*xy* are the sums of *X*, *X*^{2} *Y*, *Y*^{2}, and *XY*, respectively.

Using calculator, it is verified that:

∑*x* = (2.8 + 3.9 + 3.3 + 4.5 + 2.6 + 3.2 + 3.4) = 23.7.

∑*x*^{2} = (2.8^{2} + 3.9^{2} + 3.3^{2} + 4.5^{2} + 2.6^{2}+3.2^{2} +3.4^{2}) = 82.75.

∑*y* = (4.7 + 9.6 + 11.2 + 10.0 + 7.9 + 3.9+ 5.5) = 52.8.

∑*y*^{2} = (4.7^{2} + 9.6^{2} + 11.2^{2} + 10^{2} + 7.9^{2 }+3.9^{2 }+5.5^{2}) = 447.56.

∑*xy* = (2.8×4.7) + (3.9×9.6) + (3.3×11.2) + (4.5×10) + (2.6×7.9) + (3.2×3.9) + (3.4×5.5) = 184.28.

There are *n* = 7 pairs of observations.

Using the given notations, the correlation coefficient is calculated below:

Step 3

**Interpretation:**

The correlation coefficient is positive. This means that, as *x* increases from 2.8 to 3.4, **given our valu...**

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