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Chapter 5: Problem 75

The relationship between the depth of flooding and the amount of flood damage was examined in the paper "Significance of Location in Computing Flood Damage" \((\) Journal of Water 91影视 Planning and Management [1985]: 65-81). The following data on \(x=\) depth of flooding (feet above first-floor level) and \(y=\) flood damage (as a percentage of structure value) were obtained using a sample of flood insurance claims: $$ \begin{array}{rrrrrrrr} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ y & 10 & 14 & 26 & 28 & 29 & 41 & 43 \\ x & 8 & 9 & 10 & 11 & 12 & 13 & \\ y & 44 & 45 & 46 & 47 & 48 & 49 & \end{array} $$ a. Obtain the equation of the least-squares line. b. Construct a scatterplot, and draw the least-squares line on the plot. Does it look as though a straight line provides an adequate description of the relationship between \(y\) and \(x\) ? Explain. c. Predict flood damage for a structure subjected to \(6.5 \mathrm{ft}\) of flooding. d. Would you use the least-squares line to predict flood damage when depth of flooding is \(18 \mathrm{ft}\) ? Explain.

Short Answer

Expert verified
To answer the given questions: a) The equation of the least-squares line can be obtained following step 1; b) After constructing the scatterplot and drawing the line, we can answer whether the straight-line provides an adequate description based on the observation and dispersion of points. c) The flood damage for a structure subjected to \(6.5 \mathrm{ft}\) flood depth can be predicted using the least-squares line equation; d) The appropriateness of using the line to predict flood damage when depth is \(18 \mathrm{ft}\) depends on the range of data. Generally, extrapolating outside the range can lead to less reliable results.

Step by step solution

01

Computation of Least-Squares Line

First, list all the given pairs where each pair is a point (x, y). The x-values are the depth of flooding and corresponding y-values represent the flood damage. Then, calculate the mean of both x-coordinates and y-coordinates. Secondly, find the slope and y-intercept for the line of best fit by using formulas \[ b=\frac{\Sigma (x_i-\bar{x})(y_i-\bar{y})}{\Sigma (x_i-\bar{x})^2} \] for slope and \[ a=\bar{y}-b\bar{x} \] for the y-intercept where \(b\) is the slope, \(a\) is the y-intercept, \(\bar{x}\) and \(\bar{y}\) are the means of x-coordinates and y-coordinates respectively. Substituting these values in the straight line equation \( y=mx+c \) will give the equation of the least-squares line.
02

Scatter Plot and Line of Best Fit

Now plot the given pairs on a graph, where x-axis represents 'Depth of Flooding' and y-axis represents 'Flood Damage'. This will produce a scatterplot of the data. Next, plot the least-squares line that was calculated in step 1 on the same scatterplot. Analyze whether this line provides an adequate representation of the relationship between \(y\) and \(x\) by observing the dispersion of points around the line.
03

Predictions

Use the equation from Step 1 to predict flood damage for a structure subjected to \(6.5 \mathrm{ft}\) of flooding, by substituting \(x=6.5\) in the equation. Similarly, predict flood damage for \(x=18\) and analyze how reasonable the prediction is. If the value of \(x=18\) is outside the range of the data used to create the model, caution should be used as this prediction would be an extrapolation, which can potentially lead to unreliable results.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Modeling
Statistical modeling involves using mathematical formulas to represent real-world phenomena. In this exercise, the relationship between flooding depth and flood damage is examined through the lens of statistical modeling.
The primary goal here is to identify patterns between the depth of flooding and the resulting damage percentages.
  • We use a set of statistical techniques to build a model that best fits our data points.
  • The least-squares regression line is one such model that minimizes the sum of the squares of the differences between observed and predicted values.
Through statistical modeling, we aim to turn raw data into useful insights, allowing us to make predictions or decisions based on mathematical logic rather than intuition alone.
Scatter Plot
A scatter plot is a graphical representation of two variables as individual points on a Cartesian coordinate plane. It helps us visualize the relationship between two continuous variables at a glance.
For our data on flood damage:
  • The x-axis represents the depth of flooding.
  • The y-axis represents the flood damage percentage.
Plotting these points helps illustrate how flood damage changes relative to the depth of flooding.
The scatter plot allows us to see patterns, clusters, or trends within the data. This visual analysis is crucial when deciding whether a linear relationship exists and whether a straight line can adequately describe the data behavior.
Linear Prediction
Linear prediction uses the least-squares line to estimate values of the dependent variable (flood damage) based on the independent variable (flood depth). From the computed least-squares line,\[ y = b\cdot x + a \]where \(b\) is the slope and \(a\) is the intercept, we can derive predicted values.
Key steps include:
  • Substitute the known value of \(x\) (e.g., 6.5 ft) into the regression equation.
  • Calculate the predicted \(y\), flood damage percentage, based on this input.
However, predictions should remain within the data range used to create the model. Predicting beyond, as in the case of 18 ft, requires caution, as the relationship might not hold outside observed values.
Data Analysis
Data analysis refers to techniques used to inspect, clean, transform, and model data. It aids in drawing conclusions and supporting decision-making processes. In this exercise, the data analysis process involves several critical parts:
  • First, the collection of data regarding flood depth and related damage.
  • Secondly, plotting this data to detect observable trends or patterns.
  • Then, using statistical methods, like regression analysis, to model the data.
  • Finally, interpreting the model to make informed predictions about future data points.
By analyzing data thoroughly, we draw more accurate conclusions about the impacts of flooding at various depths, which can guide decision-making for flood prevention measures or insurance assessments.

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Most popular questions from this chapter

An accurate assessment of oxygen consumption provides important information for determining energy expenditure requirements for physically demanding tasks. The paper "Oxygen Consumption During Fire Suppression: Error of Heart Rate Estimation" (Ergonomics [1991]: \(1469-1474\) ) reported on a study in which \(x=\) oxygen consumption (in milliliters per kilogram per minute) during a treadmill test was determined for a sample of 10 firefighters. Then \(y=\) oxygen consumption at a comparable heart rate was measured for each of the 10 individuals while they performed a fire-suppression simulation. This resulted in the following data and scatterplot: $$ \begin{array}{lrrrrr} \text { Firefighter } & 1 & 2 & 3 & 4 & 5 \\ x & 51.3 & 34.1 & 41.1 & 36.3 & 36.5 \\ y & 49.3 & 29.5 & 30.6 & 28.2 & 28.0 \\ \text { Firefighter } & 6 & 7 & 8 & 9 & 10 \\ x & 35.4 & 35.4 & 38.6 & 40.6 & 39.5 \\ y & 26.3 & 33.9 & 29.4 & 23.5 & 31.6 \end{array} $$ a. Does the scatterplot suggest an approximate linear relationship? b. The investigators fit a least-squares line. The resulting MINITAB output is given in the following:. Predict fire-simulation consumption when treadmill consumption is 40 . c. How effectively does a straight line summarize the relationship? d. Delete the first observation, \((51.3,49.3)\), and calculate the new equation of the least-squares line and the value of \(r^{2}\). What do you conclude? (Hint: For the original data, \(\sum x=388.8, \Sigma y=310.3, \sum x^{2}=15,338.54, \sum x y=\) \(12,306.58\), and \(\sum y^{2}=10,072.41\).)

The paper cited in Exercise \(5.69\) gave a scatterplot in which \(x\) values were for anterior teeth. Consider the following representative subset of the data: $$ \begin{aligned} &\begin{array}{lllrrl} x & 15 & 19 & 31 & 39 & 41 \\ y & 23 & 52 & 65 & 55 & 32 \\ x & 44 & 47 & 48 & 55 & 65 \\ y & 60 & 78 & 59 & 61 & 60 \\ \sum x=404 & \sum x y=18,448 & \sum y=545 & \end{array}\\\ &\begin{array}{rl} \sum x=404 & \sum x y=18,448 \quad \sum y=545 \\ a=32.080888 & b=0.554929 \end{array} \end{aligned} $$ a. Calculate the predicted values and residuals. b. Use the results of Part (a) to obtain SSResid and \(r^{2}\). c. Does the least-squares line appear to give accurate predictions? Explain your reasoning.

The sample correlation coefficient between annual raises and teaching evaluations for a sample of \(n=353\) college faculty was found to be \(r=.11\) ("Determination of Faculty Pay: An Agency Theory Perspective," Academy of Management Journal [1992]: 921-955). a. Interpret this value. b. If a straight line were fit to the data using least squares, what proportion of variation in raises could be attributed to the approximate linear relationship between raises and evaluations?

The paper "Crop Improvement for Tropical and Subtropical Australia: Designing Plants for Difficult Climates" (Field Crops Research [1991]: 113-139) gave the following data on \(x=\) crop duration (in days) for soybeans and \(y=\) crop yield (in tons per hectare): $$ \begin{array}{rrrrrr} x & 92 & 92 & 96 & 100 & 102 \\ y & 1.7 & 2.3 & 1.9 & 2.0 & 1.5 \\ x & 102 & 106 & 106 & 121 & 143 \\ y & 1.7 & 1.6 & 1.8 & 1.0 & 0.3 \end{array} $$ $$ \begin{gathered} \sum x=1060 \quad \sum y=15.8 \quad \sum x y=1601.1 \\ a=5.20683380 \quad b=-0.3421541 \end{gathered} $$ a. Construct a scatterplot of the data. Do you think the least-squares line will give accurate predictions? Explain. b. Delete the observation with the largest \(x\) value from the sample and recalculate the equation of the least-squares line. Does this observation greatly affect the equation of the line? c. What effect does the deletion in Part (b) have on the value of \(r^{2}\) ? Can you explain why this is so?

The following table gives the number of organ transplants performed in the United States each year from 1990 to 1999 (The Organ Procurement and Transplantation Network, 2003): $$ \begin{array}{cc} & \begin{array}{l} \text { Number of } \\ \text { Transplants } \\ \text { Year } \end{array} & \text { (in thousands) } \\ \hline 1(1990) & 15.0 \\ 2 & 15.7 \\ 3 & 16.1 \\ 4 & 17.6 \\ 5 & 18.3 \\ 6 & 19.4 \\ 7 & 20.0 \\ 8 & 20.3 \\ 9 & 21.4 \\ 10 \text { (1999) } & 21.8 \\ \hline \end{array} $$ a. Construct a scatterplot of these data, and then find the equation of the least-squares regression line that describes the relationship between \(y=\) number of transplants performed and \(x=\) year. Describe how the number of transplants performed has changed over time from 1990 to 1999 . b. Compute the 10 residuals, and construct a residual plot. Are there any features of the residual plot that indicate that the relationship between year and number of transplants performed would be better described by a curve rather than a line? Explain.
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