Let's go through the problem step-by-step based on the questions.

Part (i)

The regression model for predicting the yield ($y$) based on hormone strength ($x$) can be written as:

$y = \beta_0 + \beta_1 x + \epsilon$

where:

• $\beta_0$ is the intercept (from the R output, $-3.72335$),
• $\beta_1$ is the slope or coefficient for $x$ (from the R output, $0.42999$),
• $\epsilon \sim N(0, \sigma^2)$ represents the error term, which is assumed to follow a normal distribution with a mean of 0 and some variance $\sigma^2$.

Therefore, the fitted model based on the R output is:

$\hat{y} = -3.72335 + 0.42999x$

Part (ii)

To calculate the parameter estimates using matrix notation and the information provided, we need to apply the formula:

$\hat{\beta} = (X^T X)^{-1} X^T y$

Given the inverse matrix $(X^T X)^{-1}$ as: $(X^T X)^{-1} = \begin{pmatrix} 2.6485 & -0.1758 \\ -0.1758 & 0.0121 \end{pmatrix}$

and the sums provided: $\sum_{i=1}^{10} y_i = 26.76 \quad \text{and} \quad \sum_{i=1}^{10} x_i y_i = 424.33$

Using this information, you can calculate $\hat{\beta}$ as shown in the matrix multiplication steps. This calculation helps verify the parameter estimates from R.

Part (iii)

To find the fitted value and residual for $x_3 = 11$ and $y_3 = 1.66$:

1. Substitute $x = 11$ into the regression equation to find the fitted value $\hat{y}_3$.

   $\hat{y}_3 = -3.72335 + (0.42999 \times 11)$

2. Compute the residual as the difference between the observed $y_3$ and the fitted $\hat{y}_3$:

   $\text{Residual} = y_3 - \hat{y}_3$

Part (iv)

The 95% confidence intervals for the intercept and slope can be calculated using the formula:

$\text{CI} = \hat{\beta} \pm t \cdot \text{SE}(\hat{\beta})$

where $t$ is the critical value from the $t$-distribution with 8 degrees of freedom (given as 2.306004), and $\text{SE}(\hat{\beta})$ is the standard error provided in the output (0.28072 for the intercept and 0.01899 for the slope).

Part (v)

To test the hypothesis that there is no relationship between hormone strength and yield (i.e., $H_0: \beta_1 = 0$), calculate the $t$-statistic:

$t = \frac{\hat{\beta}_1}{\text{SE}(\hat{\beta}_1)}$

Compare this $t$-value with the critical $t$-value (2.306004) to determine whether to reject $H_0$.

Part (vi)

The assumptions for the linear regression model include:

1. Linearity of the relationship.
2. Independence of errors.
3. Homoscedasticity (constant variance of errors).
4. Normality of errors.

A hypothesis test for normality, such as the Shapiro-Wilk test, could be used. Here, the null hypothesis is that the residuals are normally distributed.

Would you like further details on any part, or a more detailed calculation for specific sections?

Related Questions:

1. How is the matrix $(X^T X)^{-1}$ used in estimating regression parameters?
2. What does the standard error represent in a regression output?
3. How do you interpret the confidence interval for a regression coefficient?
4. What steps are involved in calculating residuals in regression?
5. How do hypothesis tests in regression determine if predictors are significant?

Tip:

When interpreting regression coefficients, the slope represents the change in the response variable for each unit change in the predictor variable, while the intercept is the expected value of the response variable when all predictors are zero.