7.6. Problems

1. The trip rate (y) and the corresponding household sizes (x) from a sample are shown in table below. Compute the trip rate if the average household size is 3.25 (Hint: use regression method). (5)

This problem presents a task where students need to calculate the trip rate based on household sizes provided. It encourages students to apply the regression method they have learned to find a relationship between household size and trip rates. In this case, the input data provides various household sizes (x) along with their corresponding trip rates (y). Students are instructed to compute the trip rate specifically for an average household size of 3.25 using the regression equation derived from the data.

Think about how different sizes of families impact their travel habits. For instance, a single person might travel differently compared to a family of four. This problem asks you to use data about how many trips households of different sizes take, to predict how many trips a family of a certain size (3.25 people) would take. It's like being a travel planner trying to figure out how often people would go shopping or to work based on how many people live in their home.

Solution Fit the regression equation as below.

Σx = 3 1+3 2+3 3+3 4=30 Σx2 = 3 (12)+3 (22)+3 (32)+3 (42)=90 Σy = 7+12+16+21=56 Σxy = 1 1+1 3+1 3+2 3+2 4+2 5+ 3 4+3 5+3 7+4 5+4 8+4 8=163

y¯ = 63/12=4.67 x¯ = 30/12=2.5

To fit the regression equation, key data calculations are summarized. $Σx$ represents the summation of all household sizes, $Σx^2$ represents the summation of the square of each household size, and $Σy$ is the summation of the trip rates. These mathematical operations help students understand how to prepare their data for analysis. The average values $ȳ$ and $x̄$ represent the mean of the trip rates and household sizes, respectively, and are essential for building the regression model.

Imagine you are gathering scores from a basketball game to figure out how taller players perform compared to those who are shorter. You're adding up the heights and the points scored by each player to calculate averages. Just like you do with your basketball data, in this problem, you’re summing household sizes and trip rates to understand the relationship between them.

nΣxy ΣxΣy b = − nΣx2 (Σx)2 −

((12 163) (30 56)) = −

((12 90) (30)2)

− a = y¯ bx¯=4.67 1.533 2.5=0.837

y = 0.837+1.533x

Here, we calculate the slope (b) and y-intercept (a) for the regression line. The slope indicates how much the trip rate changes for each unit increase in household size. The formula presents how to calculate these two crucial coefficients, allowing students to derive the linear equation for predicting trip rates based on household size. Once determined, the regression equation will allow estimation of trip rates for any household size.

Think about how your study hours might affect your grades. If you plot your study hours against your grades, finding a slope and y-intercept will help predict how much your grades will improve with extra study hours. Here, the problem is doing the same but for trips based on household size, indicating how changes in the number of people in a household (just like hours spent studying) could affect the number of trips they make.

When average household size =3.25, number of trips becomes y =0.837+1.533 3.25=5.819

Finally, this component applies the regression equation found earlier to estimate the number of trips based on the average household size. The resulting value indicates how many trips a household of 3.25 members is likely to make. This step effectively completes the problem by demonstrating the practical application of the regression model, translating theoretical knowledge into a usable prediction.

Imagine predicting how many groceries a family of 3.25 might buy using your earlier calculations. Based on average behavior observed, this step represents applying those observations to forecast what a 'typical' household size would do in reality. It's like making a forecast based on a trend you've spotted, ensuring your estimates are grounded in actual data.

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