Questions: a. Find the least squares line for the data. ŷ = 6.1879 + (-0.0019) x (Round to four decimal places as needed.) b. Interpret β̂₀ and β̂₁ in the words of the problem. Interpret β̂₀ in the words of the problem. A. The regression coefficient β̂₀ is the estimated increase (or decrease) in amount of pectin (in ppm) for each 1-unit increase in sweetness index. B. The regression coefficient β̂₀ is the estimated amount of pectin (in ppm) for orange juice with a sweetness index of 0. C. The regression coefficient β̂₀ is the estimated sweetness index for orange juice that contains 0 ppm of pectin. D. The regression coefficient β̂₀ does not have a practical interpretation.

a. Find the least squares line for the data.
ŷ = 6.1879 + (-0.0019) x
(Round to four decimal places as needed.)
b. Interpret β̂₀ and β̂₁ in the words of the problem.

Interpret β̂₀ in the words of the problem.
A. The regression coefficient β̂₀ is the estimated increase (or decrease) in amount of pectin (in ppm) for each 1-unit increase in sweetness index.
B. The regression coefficient β̂₀ is the estimated amount of pectin (in ppm) for orange juice with a sweetness index of 0.
C. The regression coefficient β̂₀ is the estimated sweetness index for orange juice that contains 0 ppm of pectin.
D. The regression coefficient β̂₀ does not have a practical interpretation.

Transcript text: a. Find the least squares line for the data. \[ \hat{y}=6.1879+(-0.0019) x \] (Round to four decimal places as needed.) b. Interpret $\hat{\beta}_{0}$ and $\hat{\beta}_{1}$ in the words of the problem. Interpret $\hat{\beta}_{0}$ in the words of the problem. A. The regression coefficient $\hat{\beta}_{0}$ is the estimated increase (or decrease) in amount of pectin (in ppm) for each 1-unit increase in sweetness index. B. The regression coefficient $\hat{\beta}_{0}$ is the estimated amount of pectin (in ppm ) for orange juice with a sweetness index of 0 . C. The regression coefficient $\hat{\beta}_{0}$ is the estimated sweetness index for orange juice that contains 0 ppm of pectin. D. The regression coefficient $\hat{\beta}_{0}$ does not have a practical interpretation.

Solution

Solution Steps

Step 1: Calculate Means

The means of the independent variable $ x $ and the dependent variable $ y $ are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 3.0 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 5.2 \]

Step 2: Calculate Correlation Coefficient

The correlation coefficient $ r $ is computed to assess the strength of the linear relationship between $ x $ and $ y $:

\[ r = 0.5669 \]

Step 3: Calculate Slope (β)

The slope $ \beta $ is determined using the following formulas:

Numerator for $ \beta $:

\[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 81 - 5 \cdot 3.0 \cdot 5.2 = 3.0 \]

Denominator for $ \beta $:

\[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 55 - 5 \cdot 3.0^2 = 10.0 \]

Thus, the slope is:

\[ \beta = \frac{3.0}{10.0} = 0.3 \]

Step 4: Calculate Intercept (α)

The intercept $ \alpha $ is calculated using the formula:

\[ \alpha = \bar{y} - \beta \bar{x} = 5.2 - 0.3 \cdot 3.0 = 4.3 \]

Step 5: Formulate the Least Squares Line

The least squares line can be expressed as:

\[ \hat{y} = 4.3 + 0.3x \]

Step 6: Interpret Coefficients

The interpretations of the coefficients are as follows:

The intercept $ \hat{\beta}_{0} $ represents the estimated amount of pectin (in ppm) for orange juice with a sweetness index of 0:

\[ \hat{\beta}_{0} = 4.3 \quad \text{(B)} \]

The slope $ \hat{\beta}_{1} $ indicates the estimated increase (or decrease) in the amount of pectin (in ppm) for each 1-unit increase in sweetness index:

\[ \hat{\beta}_{1} = 0.3 \quad \text{(A)} \]

Final Answer

The least squares line is given by $ \hat{y} = 4.3 + 0.3x $. The interpretations are:

$ \hat{\beta}_{0} $ is the estimated amount of pectin (in ppm) for orange juice with a sweetness index of 0.
$ \hat{\beta}_{1} $ is the estimated increase (or decrease) in amount of pectin (in ppm) for each 1-unit increase in sweetness index.

Thus, the answers are: