Question

the following scatter plot shows the billions of dollars spent by travelers in the united states since 1993. write an equation for the line of best fit and then predict how much money travelers will spend in 2008. a. y = 2x + 280; travelers will spend about $410 billion in the year 2008. b. y = 2x + 320; travelers will spend about $350 billion in the year 2008. c. y = 20x + 320; travelers will spend about $620 billion in the year 2008. d. y = 20x + 380; travelers will spend about $680 billion in the year 2008.

Explanation:

Step1: Recall slope - intercept form

The equation of a line in slope - intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. The slope $m=\frac{y_2 - y_1}{x_2 - x_1}$ for two points $(x_1,y_1)$ and $(x_2,y_2)$. Given the points $(1.5,3.5)$ and $(4,4)$.
$m=\frac{4 - 3.5}{4 - 1.5}=\frac{0.5}{2.5}=\frac{1}{5}= 0.2$.

Step2: Find the y - intercept

We use the point - slope form $y - y_1=m(x - x_1)$ with the point $(1.5,3.5)$ and $m = 0.2$.
$y-3.5 = 0.2(x - 1.5)$.
Expand: $y-3.5=0.2x-0.3$.
Solve for $y$: $y=0.2x + 3.2$.
We can also check by substituting the other point $(4,4)$ into the equations.
For $y = 2x+380$, when $x = 1.5$, $y=2\times1.5 + 380=3 + 380=383$ (not correct).
For $y = 2x+320$, when $x = 1.5$, $y=2\times1.5+320 = 3+320=323$ (not correct).
For $y = 20x+320$, when $x = 1.5$, $y=20\times1.5+320=30 + 320=350$ (not correct).
For $y = 20x+380$, when $x = 1.5$, $y=20\times1.5+380=30+380 = 410$. When $x = 4$, $y=20\times4+380=80 + 380=460$.
We can estimate from the scatter - plot trend. If we assume a linear relationship and use the two given points to calculate the slope and y - intercept more precisely, we know that the general form of a line is $y=mx + b$.
Let's use the two points $(1.5,3.5)$ and $(4,4)$ to calculate the slope $m=\frac{4 - 3.5}{4 - 1.5}=\frac{0.5}{2.5}=0.2$.
Using the point - slope form $y - y_1=m(x - x_1)$ with $(x_1,y_1)=(1.5,3.5)$ gives $y-3.5 = 0.2(x - 1.5)$, or $y=0.2x+3.2$. But if we consider the units and the nature of the problem in a more practical sense (assuming the x - axis is years since 1993 and y - axis is billions of dollars), and we want to predict for 2008 (15 years since 1993).
We can also use the fact that if we assume a linear model and we know two points on the line, we can rewrite the equation in the form $y=mx + b$.
If we assume the line passes through $(1.5,3.5)$ and $(4,4)$ and we want to predict for $x = 15$ (2008 - 1993=15).
We first find the slope $m=\frac{4 - 3.5}{4 - 1.5}=0.2$. Then using $y - y_1=m(x - x_1)$ with $(x_1,y_1)=(1.5,3.5)$ we get $y=0.2x+3.2$. But if we consider the scale and the problem context, we can try to fit the line visually and by calculation.
If we assume a linear model and we know that when $x$ (years since 1993) increases, $y$ (billions of dollars) increases.
We can check each option by substituting a value of $x$ (years since 1993) and seeing if it is consistent with the trend of the scatter - plot.
For 2008 ($x = 15$):
a. $y=2x + 380$, when $x = 15$, $y=2\times15+380=30 + 380=410$.
b. $y=2x+320$, when $x = 15$, $y=2\times15 + 320=30+320=350$.
c. $y=20x+320$, when $x = 15$, $y=20\times15+320=300+320=620$.
d. $y=20x+380$, when $x = 15$, $y=20\times15+380=300+380=680$.
By looking at the scatter - plot and estimating the trend, we can see that as $x$ (years since 1993) increases, $y$ (traveler spending in billions of dollars) increases. The line should pass through points that are consistent with the general upward - sloping trend of the data points.
If we assume a linear fit and we know that the slope and y - intercept of the line that best fits the data can be estimated using the two given points on the line (or by visual inspection of the scatter - plot).
The answer is d. because when we consider the trend of the scatter - plot and the increase in traveler spending over the years, and we want to predict for 2008 (15 years since 1993), the equation $y = 20x+380$ gives a value that is most consistent with the upward - sloping trend of the data points on the scatter - plot.

Answer:

d. $y = 20x+380$