Questions: Consider the following pairs of observations. Complete parts a through f below. y 4 2 5 3 3 x 1 4 5 3 3 a. Construct a scatterplot of the data. Choose the correct graph below. A. B. C. D. b. Use the method of least squares to fit a straight line to the six data points. ŷ = + × (Round to three decimal places as needed.) c. Plot the least squares line on the scatterplot of part a. Choose the correct graph below. A. B. C. D.

Solution Steps

The scatterplot should represent the given pairs of (x, y) values: (1, 4), (4, 2), (5, 5), (2, 2), (3, 3), and (4, 4). Plotting these points corresponds to graph B.

To find the least squares line, we need to calculate the slope (b1) and y-intercept (b0) using the following formulas:

• $b_1 = \frac{n(\sum{xy}) - (\sum{x})(\sum{y})}{n(\sum{x^2}) - (\sum{x})^2}$
• $b_0 = \bar{y} - b_1\bar{x}$

Here's the calculation:

1. $\sum{x} = 1+4+5+2+3+4 = 19$
2. $\sum{y} = 4+2+5+2+3+4 = 20$
3. $\sum{xy} = (1_4) + (4_2) + (5_5) + (2_2) + (3_3) + (4_4) = 4 + 8 + 25 + 4 + 9 + 16 = 66$
4. $\sum{x^2} = 1 + 16 + 25 + 4 + 9 + 16 = 71$
5. $n = 6$
6. $\bar{x} = \frac{\sum{x}}{n} = \frac{19}{6} \approx 3.167$
7. $\bar{y} = \frac{\sum{y}}{n} = \frac{20}{6} \approx 3.333$

Now, substitute these values into the formulas:

$b_1 = \frac{6(66) - (19)(20)}{6(71) - (19)^2} = \frac{396 - 380}{426 - 361} = \frac{16}{65} \approx 0.246$

$b_0 = 3.333 - (0.246)(3.167) \approx 3.333 - 0.779 = 2.554$

So, the least squares line is y = 2.554 + 0.246x.

Plotting the line y = 2.554 + 0.246x on the scatterplot from Step 1 (graph B) corresponds to graph A in part c.

Final Answer: