Solved Problems

A concrete testing laboratory recorded the curing time ($x$, in days) and the resulting compressive strength ($y$, in MPa) for five samples: (7, 21), (14, 28), (21, 35), (28, 41), and (35, 45). Determine the simple linear regression equation, $\hat{y} = b_0 + b_1 x$.

Using the data from Problem 1 (curing time $x$ and compressive strength $y$), calculate the Pearson correlation coefficient ($r$). The additional sum required is $\sum y^2 = 21^2 + 28^2 + 35^2 + 41^2 + 45^2 = 6156$.

For the concrete testing model derived in Problem 1 ($\hat{y} = 15.7 + 0.871 x$), what is the predicted compressive strength for a curing time of 21 days? Calculate the residual for the actual data point (21, 35).

Given the correlation coefficient $r = 0.9948$ from Problem 2, calculate the coefficient of determination ($R^2$) and interpret its meaning in the context of the concrete curing data.

A transportation engineer analyzes the relationship between speed limit ($x$) and average vehicle speed ($y$). The regression model yields $SST = 450$ and $SSE = 75$. Calculate the Sum of Squares due to Regression ($SSR$) and the Coefficient of Determination ($R^2$).

Using the data from Problem 5 ($SSE = 75$) and assuming a sample size of $n = 12$ roads, calculate the standard error of the estimate ($s_e$).

For the speed limit vs. average speed data, the estimated slope is $b_1 = 1.05$. The standard error of the slope ($s_{b1}$) is calculated to be $0.15$. Test the hypothesis that there is a significant linear relationship between the variables at $\alpha = 0.05$ (sample size $n=12$).

A geotechnical engineer measures the settlement of a foundation over time and finds it follows an exponential decay model: $y = a e^{bx}$, where $y$ is settlement and $x$ is time. Show how to transform this into a linear model to find parameters $a$ and $b$ using linear regression.

A multiple regression model predicts the heating load ($Y$, in kW) of a building based on wall area ($X_1$, in $m^2$) and roof area ($X_2$, in $m^2$). The estimated model is $\hat{Y} = 12.5 + 0.15 X_1 + 0.22 X_2$. Calculate the predicted heating load for a building with $200 \, \text{m}^2$ of wall area and $150 \, \text{m}^2$ of roof area.

For a multiple regression model predicting river flow rate based on rainfall and snowmelt, the sample size is $n = 30$, the number of predictors is $k = 2$, $SST = 1200$, and $SSR = 950$. Conduct an ANOVA F-test for the overall significance of the model at $\alpha = 0.05$.

Following Problem 10, the estimated coefficient for rainfall ($X_1$) is $b_1 = 2.4$ with standard error $s_{b1} = 0.8$. The coefficient for snowmelt ($X_2$) is $b_2 = 1.1$ with standard error $s_{b2} = 0.9$. Test the significance of each individual predictor at $\alpha = 0.05$ ($df = 27$).

A study shows a very high positive correlation ($r = 0.92$) between the amount of asphalt used in a city during the summer and the number of heatstroke incidents. Can the city's health department conclude that laying asphalt directly causes heatstroke?

A structural engineer models the yielding stress of an alloy ($y$) based on temperature ($x$) for temperatures ranging from $20^\circ \text{C}$ to $100^\circ \text{C}$, finding a strong linear relationship. Why might it be dangerous to use this regression model to predict the yielding stress at $300^\circ \text{C}$?

After fitting a simple linear regression model to predict pipe flow rate based on pressure, an engineer plots the residuals ($e_i$) on the y-axis against the predicted values ($\hat{y}_i$) on the x-axis. The plot shows a distinct "fan" or "funnel" shape, where the spread of the residuals increases as the predicted value increases. What does this indicate?

An environmental engineer adds a fifth predictor variable to a model predicting air pollution levels. The standard $R^2$ value increases slightly, but the Adjusted $R^2$ value decreases. Explain why this happens and what it means.