Let's work through each part:

Part 1: Regression Line Equation

The regression line is given by: $\hat{y} = 34.0438 + 0.8707x$ This equation represents the line of best fit with an intercept of 34.0438 and a slope of 0.8707.

Part 2: Hypotheses and Significance Test

To determine if there is a linear relationship between the dependent ($y$) and independent ($x$) variables, we test if the slope $\beta_1$ is significantly different from zero.

Hypotheses:

Since we are looking for a relationship between $x$ and $y$, the appropriate hypotheses would be:

• Null hypothesis ($H_0$): $\beta_1 = 0$ (no linear relationship).
• Alternative hypothesis ($H_A$): $\beta_1 \neq 0$ (there is a linear relationship).

So, the correct choice is: B. $H_0: \beta_1 = 0 \quad \text{and} \quad H_A: \beta_1 \neq 0$

Part 3: Test Statistic

To calculate the test statistic $t$ for the slope $\beta_1$, we use the formula: $t = \frac{\hat{\beta_1}}{SE(\hat{\beta_1})}$ where:

• $\hat{\beta_1} = 0.8707$ (the estimated slope).
• $SE(\hat{\beta_1}) = \sqrt{\frac{\sum (y_i - \hat{y})^2}{(n - 2) \cdot \sum (x_i - \bar{x})^2}}$

Using the values given:

• $\sum (y_i - \hat{y})^2 = 39.532$
• $\sum (x_i - \bar{x})^2 = 176.4$
• $n = 15$

Let's calculate $SE(\hat{\beta_1})$ and then $t$:

1. Calculate $SE(\hat{\beta_1})$: $SE(\hat{\beta_1}) = \sqrt{\frac{39.532}{(15 - 2) \cdot 176.4}}$

2. Substitute and solve: $t = \frac{0.8707}{SE(\hat{\beta_1})}$

I'll calculate these values next.The test statistic $t$ for the slope $\beta_1$ is approximately $6.632$ (rounded to three decimal places).

Summary:

• Regression Line: $\hat{y} = 34.0438 + 0.8707x$
• Hypotheses: $H_0: \beta_1 = 0$, $H_A: \beta_1 \neq 0$
• Test Statistic $t$: $6.632$

Would you like further details on any part of the process?

1. What is the interpretation of the regression line?
2. How can we interpret the $t$-value obtained?
3. What are the next steps if we were to perform the hypothesis test fully?
4. How does the significance level influence our interpretation?
5. What if we had a different confidence level—would it impact the conclusions?

Tip: When interpreting regression analysis results, always examine both the slope and $t$-value to understand the strength and significance of relationships between variables.