Practice Problems: Regression & Correlation
Question 1
A civil engineer is analyzing the relationship between concrete compressive strength (psi) and curing time (days) for a construction project. The following data was collected from six test specimens:
Curing time (days): 7, 14, 21, 28, 35, 42
Compressive strength (psi): 3200, 4100, 4800, 5400, 5900, 6300
What is the correlation coefficient (r) for this dataset?
(a) 0.876
(b) 0.923
(c) 0.991
(d) 0.856
Question 2
A mechanical engineer is studying the relationship between operating temperature (°F) and efficiency (%) of a heat exchanger. The linear regression equation was determined to be: Efficiency = 95.2 - 0.15(Temperature). If the coefficient of determination (R²) is 0.81, what percentage of the variation in efficiency is NOT explained by temperature?
(a) 19%
(b) 81%
(c) 90%
(d) 9%
Question 3
An environmental engineer is analyzing the relationship between pollutant concentration (mg/L) and distance from an industrial source (km). The regression analysis yielded the following statistics: n = 15, Σx = 120, Σy = 450, Σxy = 2800, Σx² = 1200, Σy² = 15500. What is the slope (b) of the regression line for predicting pollutant concentration from distance?
(a) -1.75
(b) -2.25
(c) -3.00
(d) -1.50
Question 4
A structural engineer is evaluating the relationship between beam deflection (mm) and applied load (kN). From 8 test specimens, the following regression equation was obtained: Deflection = 2.5 + 1.8(Load). The standard error of estimate is 0.6 mm and the mean deflection is 12.1 mm. What is the coefficient of determination (R²)?
Given: Total sum of squares (SST) = 96.0 mm²
Sum of squared errors (SSE) = 2.88 mm²
(a) 0.85
(b) 0.92
(c) 0.97
(d) 0.88
Question 5
A transportation engineer collected traffic volume data (vehicles/hour) versus time of day (hour, 24-hr format) at an intersection. The regression analysis shows: ȳ = 850 vehicles/hour, regression sum of squares (SSR) = 125,000, and total sum of squares (SST) = 156,250. What is the standard error of the estimate if n = 10 observations?
(a) 58.9 vehicles/hour
(b) 62.4 vehicles/hour
(c) 68.2 vehicles/hour
(d) 54.7 vehicles/hour
Question 6
A chemical engineer is studying the relationship between reactor pressure (bar) and yield (%). Analysis of 12 data points gives: r = 0.88, x̄ = 15 bar, ȳ = 72%, Sx = 4.2 bar, Sy = 8.5%. What is the y-intercept (a) of the regression equation?
(a) 44.8%
(b) 38.2%
(c) 52.6%
(d) 41.4%
Question 7
An industrial engineer measured production time (minutes) versus number of units produced in a manufacturing process. The data from 7 shifts yielded: Σx = 210 units, Σy = 385 minutes, Σ(x - x̄)² = 280, Σ(y - ȳ)² = 462, Σ(x - x̄)(y - ȳ) = -336. What is the correlation coefficient?
(a) -0.93
(b) -0.87
(c) -0.78
(d) -0.96
Question 8
A geotechnical engineer analyzed soil moisture content (%) versus depth (m) from 10 boring samples. The regression equation is: Moisture = 28 - 3.2(Depth). If a sample at 4.5 m depth has actual moisture content of 12%, what is the residual for this observation?
(a) -2.4%
(b) +2.4%
(c) -1.6%
(d) +1.6%
Question 9
An electrical engineer is analyzing power consumption (kW) versus ambient temperature (°C) for a cooling system. From 15 measurements: n = 15, Σx = 375°C, Σy = 450 kW, Σx² = 10125, Σy² = 14250, Σxy = 11925. What is the regression equation for predicting power from temperature?
(a) Power = 8.0 + 0.8(Temp)
(b) Power = 6.5 + 0.9(Temp)
(c) Power = 10.0 + 0.7(Temp)
(d) Power = 5.0 + 1.0(Temp)
Question 10
A water resources engineer is correlating rainfall (mm) with runoff (m³/s) for a watershed. Given 9 storm events with correlation coefficient r = 0.76, what percentage of runoff variation is explained by rainfall?
(a) 76%
(b) 58%
(c) 87%
(d) 24%
Question 11
A materials engineer tested 8 concrete samples correlating water-cement ratio (x) with 28-day compressive strength (MPa). The analysis shows: x̄ = 0.50, ȳ = 35 MPa, b = -40 MPa per unit ratio, and standard error Se = 2.8 MPa. Predict the compressive strength for a water-cement ratio of 0.45.
(a) 38 MPa
(b) 42 MPa
(c) 45 MPa
(d) 40 MPa
Question 12
A petroleum engineer analyzed well production rate (barrels/day) versus days of operation. From 12 wells: SST = 8400, SSE = 1260. Testing the significance of regression at α = 0.05, the critical F-value is 4.96. What is the calculated F-statistic?
(a) 56.7
(b) 48.2
(c) 62.5
(d) 44.8
Question 13
An HVAC engineer correlates air velocity (ft/min) with heat transfer coefficient (BTU/hr·ft²·°F). From 6 experimental runs: Σx = 3600, Σy = 48, Σxy = 30240, Σx² = 2,340,000, Σy² = 424. What is the predicted heat transfer coefficient at 720 ft/min velocity?
(a) 9.2 BTU/hr·ft²·°F
(b) 8.8 BTU/hr·ft²·°F
(c) 10.4 BTU/hr·ft²·°F
(d) 9.6 BTU/hr·ft²·°F
Question 14
A traffic engineer studied vehicle speed (mph) and stopping distance (ft) on wet pavement. The regression equation is: Distance = -50 + 5.2(Speed). If the correlation coefficient is 0.94 and there were 20 observations, what is the standard deviation of stopping distances given that the standard deviation of speeds is 12 mph?
(a) 62.4 ft
(b) 58.6 ft
(c) 66.3 ft
(d) 54.8 ft
Question 15
A manufacturing engineer analyzed machine age (years) versus maintenance cost ($1000/year) for 10 machines. Results: r = 0.82, x̄ = 8 years, ȳ = 12.5, Sx = 3.2 years, Sy = 4.8. The engineer needs to predict maintenance cost for a 12-year-old machine. What is the prediction?
(a) $17,900
(b) $18,600
(c) $16,200
(d) $19,400
Question 16
A coastal engineer measured wave height (m) and erosion rate (m³/day) at a beach over 14 storms. The analysis yielded: SST = 156, SSR = 124.8, and n = 14. What is the correlation coefficient for this relationship?
(a) 0.80
(b) 0.89
(c) 0.85
(d) 0.93
Question 17
A biomedical engineer studied oxygen consumption rate (mL/min) versus heart rate (bpm) in 11 patients. The regression shows: ŷ = 45 + 2.8x with Se = 18 mL/min. For a patient with heart rate of 85 bpm, what is the 95% prediction interval width if t(0.025, 9) = 2.262?
Given: Σ(x - x̄)² = 1800, x̄ = 75 bpm
(a) 82.4 mL/min
(b) 89.6 mL/min
(c) 95.2 mL/min
(d) 78.8 mL/min
Question 18
A mining engineer correlated ore grade (% metal) with depth (m) from 16 drill cores. Statistical analysis shows: r² = 0.72, ȳ = 2.4%, and the regression equation has slope b = -0.015%/m. At what depth does the regression predict 1.8% ore grade?
(a) 38 m
(b) 42 m
(c) 46 m
(d) 40 m
Question 19
A reliability engineer analyzed component failure time (hours) versus operating temperature (°C) from 13 tests. Given: Σx = 650°C, Σy = 3900 hours, Σxy = 182,000, Σx² = 34,500, Σy² = 1,250,000. The engineer wants to test if the correlation is significant at α = 0.05 (critical value r(0.05,11) = 0.553). What is the calculated correlation coefficient?
(a) -0.68
(b) -0.72
(c) -0.64
(d) -0.76
Question 20
A process engineer developed a regression model for chemical reactor yield (%) based on catalyst amount (kg). From 18 batches: the regression equation is Yield = 62 + 8.5(Catalyst), with SSE = 225 and SST = 1800. A new batch uses 4.2 kg catalyst and achieves 96% yield. Is this observation an outlier based on standardized residual (using threshold of ±2)?
(a) Yes, standardized residual = -2.4
(b) No, standardized residual = -1.6
(c) Yes, standardized residual = +2.2
(d) No, standardized residual = +1.8
