1. **Problem Statement:** Test if $x$ is necessary in predicting $y$ using the given data. 2. **Hypothesis:** - Null hypothesis $H_0$: $x$ is not necessary (slope $\beta_1 = 0$). - Alternative hypothesis $H_a$: $x$ is necessary (slope $\beta_1 \neq 0$). 3. **Key Formulas:** - Regression line: $y = a + bx$ - Total Sum of Squares (SST): $\sum (y_i - \bar{y})^2$ - Regression Sum of Squares (SSR): $\sum (\hat{y}_i - \bar{y})^2$ - Error Sum of Squares (SSE): $\sum (y_i - \hat{y}_i)^2$ - Coefficient of determination: $R^2 = \frac{SSR}{SST}$ - F-ratio: $F = \frac{MSR}{MSE} = \frac{SSR/1}{SSE/(n-2)}$ 4. **Simplified Steps:** - Calculate means $\bar{x}$ and $\bar{y}$. - Compute slope $b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$. - Compute intercept $a = \bar{y} - b\bar{x}$. - Calculate predicted values $\hat{y}_i = a + bx_i$. - Calculate SSR, SSE, SST using formulas above. - Calculate $R^2$. - Calculate F-ratio. 5. **Decision Rule:** - Compare calculated $F$ with critical $F$ at $\alpha=0.05$ and degrees of freedom 1 and $n-2$. - If $F$ is greater, reject $H_0$; otherwise, do not reject. 6. **Summary:** This simplified approach focuses on key calculations to test the significance of $x$ in predicting $y$.