To solve the parts of this question systematically, we will go step by step:

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Part 1a: Obtain the estimated regression equation

The general form of the regression equation is:

$\hat{y} = b_0 + b_1x_1 + b_2x_2$

We use multiple regression formulas to find $b_0$, $b_1$, and $b_2$. The calculations involve finding the coefficients using the least squares method:

$b_1 = \frac{SS_{x_1y}}{SS_{x_1x_1}}, \quad b_2 = \frac{SS_{x_2y}}{SS_{x_2x_2}}, \quad b_0 = \bar{y} - b_1\bar{x}_1 - b_2\bar{x}_2$

Given the provided estimated regression equation:

$\hat{y} = 10.8 - 0.1x_1 + 0.7x_2$

We already have the coefficients:

• $b_0 = 10.8$,
• $b_1 = -0.1$,
• $b_2 = 0.7$.

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Part 1b: Develop the correlation matrix

To develop the correlation matrix, we calculate pairwise correlations between the variables ($y$, $x_1$, $x_2$) using the formula:

$r_{xy} = \frac{\text{Cov}(x, y)}{\sqrt{\text{Var}(x) \cdot \text{Var}(y)}}$

The correlation matrix is:

	$y$	$x_1$	$x_2$
$y$	1	0.156	0.916
$x_1$	0.156	1	0.257
$x_2$	0.916	0.257	1

The independent variable with the smallest correlation magnitude with $y$ is $x_1$ ($r = 0.156$).

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Part 1c: Hypothesis testing for $x_1$

We test whether the correlation coefficient between $y$ and $x_1$ ($\rho$) is significant.

• Hypotheses:
  • $H_0$: $\rho = 0$ (no correlation)
  • $H_A$: $\rho \neq 0$ (correlation exists)

This corresponds to Option C.

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Part 1d: Calculate the test statistic

The test statistic for correlation is calculated as:

$t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}}$

Where:

• $r = 0.156$ (correlation between $x_1$ and $y$),
• $n = 10$ (sample size).

Substitute values into the formula:

$t = \frac{0.156 \cdot \sqrt{10 - 2}}{\sqrt{1 - 0.156^2}}$

$t = \frac{0.156 \cdot \sqrt{8}}{\sqrt{1 - 0.0243}}$

$t = \frac{0.156 \cdot 2.828}{\sqrt{0.9757}}$

$t = \frac{0.441}{0.9878} \approx 0.447$

The test statistic is $t = 0.447$.

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Conclusion:

1. Based on the $t$-test value and degrees of freedom ($df = n - 2 = 8$), compare the $p$-value with $\alpha = 0.05$. If $p > \alpha$, fail to reject $H_0$; otherwise, reject $H_0$.

Do you need further clarification on any step?

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Follow-up Questions:

1. Would you like to compute the $p$-value and decision for the hypothesis test?
2. Should I explain how the correlation matrix is calculated in detail?
3. Would you like to understand how the regression coefficients are computed from scratch?
4. Do you want a step-by-step analysis of the test statistic formula?
5. Should I provide a graph to visualize the correlations?

Tip:

When dealing with multiple regression, always examine the correlation matrix to assess multicollinearity before interpreting regression coefficients.