Tree canopy and surface temperature

Use the given information to respond to parts A, B, C, D, E, and F. Label any subparts (e.g., i and ii) that may be present.

1. A.

   Describe the association between tree canopy and average midday surface temperature for the 18 blocks.

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   Score part A

   • 1 point Describes the association as strong, negative, and linear, in context: blocks with a higher percent of tree canopy tend to have lower average midday surface temperatures. Direction, form, and strength must all be included.Earns it: "There is a strong, negative, linear association: blocks with more tree canopy tend to have lower midday surface temperatures."Does not: "Negative correlation." (no form, no strength, and no context); "As canopy goes up, temperature goes down." (gives direction only)

   Model response

   There is a strong, negative, linear association between tree canopy and surface temperature. Blocks with a higher percent of tree canopy tend to have lower average midday surface temperatures, and there are no obvious unusual points.

2. B.

   i. Interpret the slope of the least-squares regression line in context.

   ii. Interpret the coefficient of determination, $r^2=0.835$, in context.

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   Score part B

   • 1 point Interprets the slope as the predicted change in the response for a one-unit increase in the explanatory variable, in context: for each additional percentage point of tree canopy, the predicted average midday surface temperature decreases by 0.367 degrees Fahrenheit. The word "predicted" (or "on average") is required.Earns it: "For each additional 1 percent of canopy, the predicted midday surface temperature decreases by 0.367 degrees F."Does not: "For each 1 percent of canopy, the temperature decreases by 0.367 degrees." (states a certain change, not a predicted one); "For every 0.367 percent of canopy, the temperature drops 1 degree." (the variables are reversed)
   • 1 point Interprets $r^2$ in context: about 83.5% of the variation in average midday surface temperature among these blocks is explained by the linear relationship with percent tree canopy.Earns it: "83.5% of the variation in midday surface temperature is explained by the linear relationship with tree canopy."Does not: "83.5% of the temperatures are explained by canopy." (the statement must be about variation in temperature); "The line makes correct predictions 83.5% of the time." (r-squared describes explained variation, not how often predictions are right); "83.5% of the canopy is related to temperature." (the statement must be about variation in temperature)

   Model response

   i. For each additional percentage point of tree canopy, the predicted average midday July surface temperature of a block decreases by 0.367 degrees Fahrenheit.

   ii. About 83.5% of the variation in average midday surface temperature among these blocks is explained by the linear relationship with the percent of tree canopy.

3. C.

   One of the sampled blocks, Block P, has 35% tree canopy and an average midday surface temperature of 113.9 degrees Fahrenheit.

   i. Calculate the residual for Block P. Show your work.

   ii. Interpret the residual for Block P in context.

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   Score part C

   • 1 point Calculates the predicted temperature, $123.89-0.367(35)=111.045$ degrees, and the residual, $113.9-111.045\approx 2.86$ degrees Fahrenheit. A residual of 2.85 from rounding the predicted value also earns the point, and an arithmetic error can be ignored if correct work is shown.Earns it: "Predicted $=123.89-0.367(35)=111.045$. Residual $=113.9-111.045\approx 2.86$ degrees."Does not: "Residual $=111.045-113.9=-2.86$" (a residual is observed minus predicted); "111.045" (the predicted value, not the residual)
   • 1 point Interprets the residual in context: Block P's actual average midday surface temperature is about 2.86 degrees Fahrenheit higher than the temperature the regression line predicts for a block with 35% canopy, so the line underestimates Block P's temperature. A correct interpretation of an incorrect residual from (i) also earns the point.Earns it: "Block P was 2.86 degrees F hotter than the line predicts for 35% canopy. The line underpredicts its temperature."Does not: "The line overpredicts Block P's temperature." (a positive residual means the line underpredicts); "The residual is the error in the prediction." (not interpreted for Block P in context)

   Model response

   i. Predicted temperature: $123.89-0.367(35)=111.045$ degrees. Residual $=113.9-111.045\approx 2.86$ degrees Fahrenheit.

   ii. Block P's actual average midday surface temperature was about 2.86 degrees Fahrenheit higher than the regression line predicts for a block with 35% tree canopy. The line underestimates Block P's temperature.

4. D.

   A neighborhood association wants to use the regression line to predict the average midday surface temperature of a block next to a park that has 85% tree canopy. Explain whether this prediction would be reliable.

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   Score part D

   • 1 point Explains that the prediction would not be reliable because 85% is well outside the range of canopy values in the data (4% to 60%), so the prediction is an extrapolation and the linear pattern may not continue.Earns it: "Not reliable. The data only go up to 60% canopy, so predicting at 85% is extrapolation."Does not: "Reliable, because the correlation is strong." (a strong correlation within the data says nothing about blocks outside the range of the data); "Not reliable, because $r^2$ is less than 100%." (does not identify extrapolation)

   Model response

   The prediction would not be reliable. The canopy values in the sample range only from 4% to 60%, so 85% is far outside the data. Using the line there is extrapolation, and we have no evidence that the linear pattern continues beyond 60% canopy.

5. E.

   The planner concludes from these data that adding tree canopy to a block will cause the block's surface temperature to decrease.

   i. Explain why the planner's study does not support this cause-and-effect conclusion.

   ii. Identify a possible confounding variable, and explain how it could be associated with both tree canopy and surface temperature.

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   Score part E

   • 1 point Explains that the study is observational: the planner recorded canopy and temperature without randomly assigning tree canopy to blocks, so the association alone cannot show cause and effect. A response may add that blocks with more canopy could differ from other blocks in ways that affect temperature.Earns it: "This is an observational study. Canopy was not randomly assigned to blocks, so something else that differs between leafy and bare blocks could cause the temperature difference."Does not: "Correlation does not imply causation." (does not refer to how the data were collected); "The sample of 18 blocks is too small." (a larger observational study still could not show cause and effect); "The blocks were randomly selected, so the results apply only to this city." (random selection concerns generalizing to a population, not cause and effect)
   • 1 point Identifies a plausible confounding variable and explains how it is associated with both variables, for example the amount of pavement on a block: blocks with more parking lots and wide roads have less room for trees (less canopy), and pavement absorbs heat (higher surface temperature). Both associations must be explained.Earns it: "Amount of pavement. Blocks with big parking lots have little room for trees, and asphalt heats up in the sun, so those blocks have less canopy and higher temperatures."; "Neighborhood income. Wealthier blocks tend to have more trees, and they also tend to have watered lawns instead of parking lots, which keeps the ground cooler."Does not: "Sunlight." (not explained, and every block was measured on clear afternoons); "Pavement makes blocks hotter." (explains the link to temperature but not the link to canopy); "Air pollution." (no explanation of either association)

   Model response

   i. This is an observational study. The planner recorded the canopy and temperature of randomly selected blocks but did not randomly assign trees to blocks, so blocks with more canopy may differ from blocks with less canopy in other ways that affect temperature. The association alone does not show that canopy causes the temperature difference.

   ii. The amount of pavement on a block, such as parking lots and wide roads, could be a confounding variable. Blocks with more pavement have less room for trees, so they have less canopy, and pavement absorbs heat, so those blocks have higher surface temperatures. Pavement could explain part of the association instead of the trees.

6. F.

   To test the planner's claim, the city randomly selects 40 residential blocks that have less than 10% tree canopy and randomly assigns 20 of them to receive new street trees. The other 20 blocks receive no new trees. After five years, the city will record each block's change in average midday July surface temperature (later temperature minus earlier temperature).

   i. Identify the appropriate inference procedure for determining whether new street trees lower surface temperatures on blocks like these.

   ii. State the null and alternative hypotheses for the procedure in (i), defining any symbols you use.

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   Score part F

   • 1 point Identifies the procedure as a two-sample t-test for a difference between two population means (the mean change in temperature for blocks that receive new trees and for blocks that do not).Earns it: "two-sample t-test for a difference between two population means"Does not: "matched pairs t-test" (the two groups are different blocks, not pairs); "two-sample z-test for a difference between two population proportions" (the response, change in temperature, is quantitative); "two-sample t-interval" (the question asks whether the trees lower temperatures, which calls for a test)
   • 1 point States $H_0:{\mu }_T-{\mu }_N=0$ and $H_a:{\mu }_T-{\mu }_N<0$ (or the equivalent $H_0:{\mu }_T={\mu }_N$ and $H_a:{\mu }_T<{\mu }_N$), and defines ${\mu }_T$ and ${\mu }_N$ as the true mean changes in average midday July surface temperature for blocks like these that do and do not receive new trees. Hypotheses that define the change as earlier minus later, with $H_a:{\mu }_T>{\mu }_N$, also earn the point.Earns it: "$H_0:{\mu }_T={\mu }_N$, $H_a:{\mu }_T<{\mu }_N$, where ${\mu }_T$ and ${\mu }_N$ are the true mean changes in July surface temperature for blocks like these with and without new trees"Does not: "$H_a:{\mu }_T\ne {\mu }_N$" (the city wants to know whether trees lower temperatures, which is a one-sided alternative); "$H_0:{\bar{x}}_T={\bar{x}}_N$" (hypotheses are about population means, not sample means); Correct hypotheses with ${\mu }_T$ and ${\mu }_N$ not defined

   Model response

   i. A two-sample t-test for a difference between two population means.

   ii. $H_0:{\mu }_T-{\mu }_N=0$ and $H_a:{\mu }_T-{\mu }_N<0$, where ${\mu }_T$ is the true mean change in average midday July surface temperature (later minus earlier) for blocks like these that receive new street trees, and ${\mu }_N$ is the true mean change for blocks like these that receive no new trees.

Your score: 0 of 10. Each point is earned on its own, the way the exam scores it.