Questions: Consider the following pairs of measurements. x y 4 3 -3 2 9 5 4 9 4 3 b. What does the scattergram suggest about the relationship between x and y? A. As x increases, y tends to increase. Thus, there appears to be a negative, linear relationship between x and y. B. As x increases, y tends to increase. Thus, there appears to be a positive, linear relationship between x and y. C. As x increases, y tends to decrease. Thus, there appears to be a negative, linear relationship between x and y. c. Given that SSxc=77.7143, SSxy=57.2857, ȳ=3.5714, and x̄=3.4286, calculate the least squares estimates of β₀ and β₁. β̂₁=737 (Round to the nearest thousandth as needed.) Find β̂₀= (Round to the nearest thousandth as needed.)

Consider the following pairs of measurements.
x y
4 3
-3 2
9 5
4 9
4 3
b. What does the scattergram suggest about the relationship between x and y?
A. As x increases, y tends to increase. Thus, there appears to be a negative, linear relationship between x and y.
B. As x increases, y tends to increase. Thus, there appears to be a positive, linear relationship between x and y.
C. As x increases, y tends to decrease. Thus, there appears to be a negative, linear relationship between x and y.
c. Given that SSxc=77.7143, SSxy=57.2857, ȳ=3.5714, and x̄=3.4286, calculate the least squares estimates of β₀ and β₁.
β̂₁=737 (Round to the nearest thousandth as needed.)
Find β̂₀= (Round to the nearest thousandth as needed.)

Transcript text: Consider the following pairs of measurements. $x$ y 4 3 $-3$ 2 9 5 4 9 4 3 b. What does the scattergram suggest about the relationship between x and y ? A. As $x$ increases, $y$ tends to increase. Thus, there appears to be a negative, linear relationship between x and y . B. As $x$ increases, $y$ tends to increase. Thus, there appears to be a positive, linear relationship between $x$ and $y$. C. As $x$ increases, $y$ tends to decrease. Thus, there appears to be a negative, linear relationship between $x$ and $y$. c. Given that $\mathrm{SS}_{\mathrm{xc}}=77.7143, \mathrm{SS}_{\mathrm{xy}}=57.2857, \bar{y}=3.5714$, and $\bar{x}=3.4286$, calculate the least squares estimates of $\beta_{0}$ and $\beta_{1}$. $\hat{\beta}_{1}=737$ (Round to the nearest thousandth as needed.) Find $\hat{\beta}_{0}$. $\widehat{\beta}_{0}=$ $\square$ (Round to the nearest thousandth as needed.)

Solution

Solution Steps

Step 1: Calculate the Correlation Coefficient

The correlation coefficient $ r $ is calculated using the formula:

\[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \]

Where:

$ \text{Cov}(X,Y) = 4.2 $
$ \sigma_X = 4.416 $
$ \sigma_Y = 1.033 $

Substituting the values, we find:

\[ r = \frac{4.2}{4.416 \times 1.033} \approx 0.921 \]

Step 2: Analyze the Relationship

Since $ r = 0.921 $ is positive, we conclude that as $ x $ increases, $ y $ tends to increase. Thus, there appears to be a positive, linear relationship between $ x $ and $ y $.

Step 3: Calculate the Least Squares Estimates

Given the values: