Let's work through the solution step-by-step and double-check each calculation.

Step 1: Organize the Data

• Speeds (x): 30, 50, 40, 55, 30, 24, 61, 25, 50, 55
• Miles per Gallon (MPG, y): 28, 24, 24, 22, 31, 33, 20, 35, 25, 24

Step 2: Calculate the Mean of x and y

The mean of $x$ (speed) is: $\bar{x} = \frac{30 + 50 + 40 + 55 + 30 + 24 + 61 + 25 + 50 + 55}{10} = \frac{480}{10} = 48$

The mean of $y$ (miles per gallon) is: $\bar{y} = \frac{28 + 24 + 24 + 22 + 31 + 33 + 20 + 35 + 25 + 24}{10} = \frac{267}{10} = 26.7$

Step 3: Calculate Deviations from the Mean

• $x - \bar{x}$: $-18, 2, -8, 7, -18, -24, 13, -23, 2, 7$
• $y - \bar{y}$: $1.3, -2.7, -2.7, -4.7, 4.3, 6.3, -6.7, 8.3, -1.7, -2.7$

Step 4: Multiply the Deviations for Each Pair of x and y

Now, let's compute the product of the deviations for each pair:

$(x_i - \bar{x})(y_i - \bar{y}) = (-18)(1.3) + (2)(-2.7) + (-8)(-2.7) + (7)(-4.7) + (-18)(4.3) + (-24)(6.3) + (13)(-6.7) + (-23)(8.3) + (2)(-1.7) + (7)(-2.7)$

Calculating each term:

$-18 \times 1.3 = -23.4, \quad 2 \times -2.7 = -5.4, \quad -8 \times -2.7 = 21.6, \quad 7 \times -4.7 = -32.9$ $-18 \times 4.3 = -77.4, \quad -24 \times 6.3 = -151.2, \quad 13 \times -6.7 = -87.1, \quad -23 \times 8.3 = -190.9$ $2 \times -1.7 = -3.4, \quad 7 \times -2.7 = -18.9$

Now, sum these results: $-23.4 + (-5.4) + 21.6 + (-32.9) + (-77.4) + (-151.2) + (-87.1) + (-190.9) + (-3.4) + (-18.9) = -569.2$

Step 5: Compute the Sum of the Squared Deviations for $x$ and $y$

Squared Deviations of $x$:

$(x_i - \bar{x})^2 = (-18)^2 + (2)^2 + (-8)^2 + (7)^2 + (-18)^2 + (-24)^2 + (13)^2 + (-23)^2 + (2)^2 + (7)^2$ $= 324 + 4 + 64 + 49 + 324 + 576 + 169 + 529 + 4 + 49 = 2144$

Squared Deviations of $y$:

$(y_i - \bar{y})^2 = (1.3)^2 + (-2.7)^2 + (-2.7)^2 + (-4.7)^2 + (4.3)^2 + (6.3)^2 + (-6.7)^2 + (8.3)^2 + (-1.7)^2 + (-2.7)^2$ $= 1.69 + 7.29 + 7.29 + 22.09 + 18.49 + 39.69 + 44.89 + 68.89 + 2.89 + 7.29 = 220.39$

Step 6: Calculate the Correlation Coefficient $r$

The formula for the sample correlation coefficient is: $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$ Substituting the values: $r = \frac{-569.2}{\sqrt{2144 \times 220.39}} = \frac{-569.2}{\sqrt{472792.16}} = \frac{-569.2}{686.9} \approx -0.829$

Step 7: Conclusion

• Sample Correlation Coefficient: $r \approx -0.83$, which is a strong negative linear relationship.
• Based on the value of $r$, the correct answer is:
  • There is a strong negative linear relationship between speed and miles per gallon.

Interpretation of the Sample Correlation Coefficient:

Since the correlation is negative and significant, the correct interpretation is:

• For driving speeds between 24 and 61 miles per hour, higher speeds are associated with lower miles per gallon.

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Additional Practice Questions:

1. What would happen to the correlation coefficient if all the values of speed and miles per gallon were increased by 10?
2. How can you visualize the relationship between speed and miles per gallon using a scatter plot?
3. What does it mean if the correlation coefficient is close to 0?
4. How would you interpret a correlation coefficient of +0.85?
5. What other statistical measures might help in understanding the relationship between speed and miles per gallon?

Tip: Always check for linearity before calculating correlation. If the relationship isn't linear, the correlation coefficient may not be meaningful!