The table gives the median household income (in 2005 year) for two cities in various years. Let x represent the median household income for citizens of city A and y represent the corresponding median household income for citizens of city B. Find a linear model that expresses the median household income for citizens in city B as a function of the median household income for citizens of city B. Interpret the slope of the linear model. Round numerical values in your answer to three decimal places.
A) Slope: median household income is growing at the same rate for city A as it is for city B.
B) Slope: median household income is growing at the same rate for city A as it is for city B.
C) Slope: median household income is growing at the same rate for city A as it is for city B.
D) Slope: median household income is growing faster for city A than it is for city B.
E) Slope: median household income is growing slower for city A than it is for city B.

Question

The table gives the median household income (in 2005 year) for two cities in various years. Let x represent the median household income for citizens of city A and y represent the corresponding median household income for citizens of city B. Find a linear model that expresses the median household income for citizens in city B as a function of the median household income for citizens of city B. Interpret the slope of the linear model. Round numerical values in your answer to three decimal places. 
A) Slope: median household income is growing at the same rate for city A as it is for city B. 
B) Slope: median household income is growing at the same rate for city A as it is for city B. 
C) Slope: median household income is growing at the same rate for city A as it is for city B. 
D) Slope: median household income is growing faster for city A than it is for city B. 
E) Slope: median household income is growing slower for city A than it is for city B.

Quizplus · Accepted Answer

We want to find a linear model that expresses the median household income for citizens in city B as a function of the median household income for citizens of city A. This means we are looking for the equation y = mx + b, where y represents the median household income for citizens in city B and x represents the corresponding median household income for citizens of city A. Since we are looking for the relationship between city A and city B, we want to express y as a function of x, which means solving for y. To do this, we can use the data from the table to find the slope (m) and y-intercept (b) of the linear model.

Using the formula for the slope of a line, we can find that 
m = (y2 - y1) / (x2 - x1)
= (67,357 - 52,361) / (50,735 - 39,094)
= 0.944

To find the y-intercept, we can use one of the data points and the slope formula: 
y = mx + b 
b = y - mx

We can use the data from 1990: 
y = 52,361 and x = 39,094

So we have 
b = y - mx 
= 52,361 - 0.944(39,094) 
= 15,659.872

Therefore, the linear model that expresses the median household income for citizens in city B as a function of the median household income for citizens of city A is: 
y = 0.944x + 15,659.872

Interpretation of the slope: The slope of the linear model is 0.944. This means that for every increase in $1,000 for the median household income of city A, the median household income of city B increases by $944, on average. This also means that city A is growing faster than city B in terms of median household income.

The Table Gives the Median Household Income (In 2005 Year)