Question

1. table shows the miles driven by a truck driver.

time (hours) 0 1 2 3 4 5 6 7
distance (miles) 0 43 87 110 120 150 203 216
a. determine a linear regression equation for the data. round the slope and y - intercept to the nearest whole number.
b. interpret the slope & y - intercept for this context.
c. identify the correlation coefficient, or r - value, of the line. what does this value tell you?
d. predict the distance traveled after 4 hours. show your work and explain your reasoning.

1. the table shows the population decline of a small town over a seven - year period.

year 2005 2006 2007 2008 2009 2010 2011
population (thousands) 9.4 8.3 8.9 8.0 6.9 6.3 6.6

1. determine a linear regression equation for the data. round the slope and y - intercept to the nearest whole number.

Explanation:

Step1: Recall linear - regression formula

The linear - regression equation is of the form $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. The formulas for $m$ and $b$ are:
\[m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\]
\[b=\frac{\sum_{i = 1}^{n}y_i - m\sum_{i = 1}^{n}x_i}{n}\]
For the first data set (truck driver):
Let $x$ be the time (hours) and $y$ be the distance (miles). $n = 8$.
\(\sum_{i = 1}^{8}x_i=0 + 1+2 + 3+4+5+6+7=28\)
\(\sum_{i = 1}^{8}y_i=0 + 43+87+110+120+150+203+216 = 929\)
\(\sum_{i = 1}^{8}x_i^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}+7^{2}=0 + 1+4+9+16+25+36+49 = 140\)
\(\sum_{i = 1}^{8}x_iy_i=0\times0+1\times43+2\times87+3\times110+4\times120+5\times150+6\times203+7\times216\)
\(=0 + 43+174+330+480+750+1218+1512=4507\)

Step2: Calculate the slope $m$

\[m=\frac{8\times4507-28\times929}{8\times140 - 28^{2}}\]
\[m=\frac{36056-26012}{1120 - 784}\]
\[m=\frac{10044}{336}\approx30\]

Step3: Calculate the y - intercept $b$

\[b=\frac{929-30\times28}{8}\]
\[b=\frac{929 - 840}{8}=\frac{89}{8}\approx11\]
The linear regression equation is $y = 30x+11$.

Step4: Interpret the slope and y - intercept

The slope $m = 30$ means that for every additional hour of driving, the truck driver travels approximately 30 more miles. The y - intercept $b = 11$ means that when the driving time is 0 hours, there is an initial distance of 11 miles (this could be an error in measurement or some pre - existing distance value).

Step5: Calculate the correlation coefficient $r$

The formula for the correlation coefficient $r$ is:
\[r=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{\sqrt{[n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}][n\sum_{i = 1}^{n}y_i^{2}-(\sum_{i = 1}^{n}y_i)^{2}]}}\]
First, calculate \(\sum_{i = 1}^{8}y_i^{2}=0^{2}+43^{2}+87^{2}+110^{2}+120^{2}+150^{2}+203^{2}+216^{2}\)
\(=0 + 1849+7569+12100+14400+22500+41209+46656=146283\)
\[r=\frac{8\times4507-28\times929}{\sqrt{(8\times140 - 28^{2})(8\times146283-929^{2})}}\]
\[r=\frac{10044}{\sqrt{336\times(1170264 - 863041)}}\]
\[r=\frac{10044}{\sqrt{336\times307223}}\]
\[r=\frac{10044}{\sqrt{103226928}}\approx0.99\]
The $r$ - value close to 1 indicates a strong positive linear correlation between time and distance.

Step6: Predict the distance after 4 hours

Substitute $x = 4$ into the equation $y = 30x+11$.
\[y=30\times4+11=120 + 11=131\]

For the second data set (population decline):
Let $x$ be the year (with $x = 0$ for 2005, $x = 1$ for 2006, etc.) and $y$ be the population (thousands). $n = 7$.
\(\sum_{i = 1}^{7}x_i=0 + 1+2+3+4+5+6 = 21\)
\(\sum_{i = 1}^{7}y_i=9.4+8.3+8.9+8.0+6.9+6.3+6.6=54.4\)
\(\sum_{i = 1}^{7}x_i^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}=0 + 1+4+9+16+25+36 = 91\)
\(\sum_{i = 1}^{7}x_iy_i=0\times9.4+1\times8.3+2\times8.9+3\times8.0+4\times6.9+5\times6.3+6\times6.6\)
\(=0 + 8.3+17.8+24+27.6+31.5+39.6=148.8\)
Calculate the slope $m$:
\[m=\frac{7\times148.8-21\times54.4}{7\times91-21^{2}}\]
\[m=\frac{1041.6-1142.4}{637 - 441}\]
\[m=\frac{-100.8}{196}\approx - 0.5\]
Calculate the y - intercept $b$:
\[b=\frac{54.4-(-0.5)\times21}{7}\]
\[b=\frac{54.4 + 10.5}{7}=\frac{64.9}{7}\approx9\]
The linear regression equation is $y=-0.5x + 9$.

Answer:

A. For the truck driver data: $y = 30x+11$; For the population data: $y=-0.5x + 9$
B. For the truck driver: Slope means 30 miles per hour, y - intercept is an initial 11 miles. For population data: Slope means population decreases by 0.5 thousand per year, y - intercept is initial population of 9 thousand.
C. For the truck driver data: $r\approx0.99$, strong positive linear correlation.
D. For the truck driver, when $x = 4$, $y = 131$ miles.