1. **Problem Statement:** We have data for 7 patients with ages $T$ and blood pressures $P$. We want to analyze the relationship between $P$ and $T$ using linear regression. 2. **Why linear regression?** Linear regression is appropriate if the relationship between the independent variable $T$ (age) and dependent variable $P$ (blood pressure) appears approximately linear. The scatter plot shows a trend where blood pressure tends to increase with age, suggesting a linear model is suitable. 3. **Formula for regression line:** The regression line is given by $$P = b_0 + b_1 T$$ where $b_1$ is the slope and $b_0$ is the intercept. 4. **Calculate means:** $$\bar{T} = \frac{42+74+48+35+56+26+60}{7} = 48.71$$ $$\bar{P} = \frac{98+130+120+88+182+80+135}{7} = 121.86$$ 5. **Calculate slope $b_1$:** $$b_1 = \frac{\sum (T_i - \bar{T})(P_i - \bar{P})}{\sum (T_i - \bar{T})^2}$$ Calculate numerator: $$(42-48.71)(98-121.86) + (74-48.71)(130-121.86) + \ldots + (60-48.71)(135-121.86) = 3246.86$$ Calculate denominator: $$(42-48.71)^2 + (74-48.71)^2 + \ldots + (60-48.71)^2 = 544.86$$ So, $$b_1 = \frac{3246.86}{544.86} = 5.96$$ 6. **Calculate intercept $b_0$:** $$b_0 = \bar{P} - b_1 \bar{T} = 121.86 - 5.96 \times 48.71 = -164.15$$ 7. **Regression line equation:** $$P = -164.15 + 5.96 T$$ 8. **Interpret slope:** The slope $5.96$ means that for each additional year of age, blood pressure increases on average by about 5.96 mmHg. 9. **Estimate blood pressure for $T=40$:** $$P = -164.15 + 5.96 \times 40 = 75.25$$ 10. **Confidence intervals:** To estimate confidence intervals for slope and prediction, we would calculate standard errors and use $t$-distribution. This requires more detailed calculations beyond this scope but generally provides a range around the slope and predicted values indicating reliability.