Answer:
Explained below.
Step-by-step explanation:
The data from the scatter plot is as follows:
Arm Span Height
19.5 25
35.5 35
37.0 39
40.5 40
44.5 45
45.5 50
49.0 49
51.0 55
52.5 47
59.0 60
61.0 58
(1)
Compute the value of slope and intercept as follows:
[tex]\sum{X} = 495 ~,~ \sum{Y} = 503 ~,~ \sum{X \cdot Y} = 23822 ~,~ \sum{X^2} = 23660.5[/tex]
[tex]\begin{aligned} b &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 503 \cdot 23660.5 - 495 \cdot 23822}{ 11 \cdot 23660.5 - 495^2} \approx 7.174 \\ \\m &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 11 \cdot 23822 - 495 \cdot 503 }{ 11 \cdot 23660.5 - \left( 495 \right)^2} \approx 0.857\end{aligned}[/tex]
The equation of the line of best fit is, y = 0.857·x + 7.174.
(2)
The slope of the line represent the rate of change in y caused by one unit change in x.
In this case the slope of m = 0.857 indicates that as the arm span increases by 1 unit the height increases by 0.857 units.
(3)
To perform the Residual Test, simply take two values from the table and interchange the x and y values.
Consider the values: (37, 39) and (51, 55)
Compute the value of y if x = 39:
[tex]y = 0.857\cdot x + 7.174\\=0.857\times 39+7.174\\=40.597\\\approx 40.6[/tex]
The new value of y (40.6) is close to the original value (39.0)
Compute the value of y if x = 55:
[tex]y = 0.857\cdot x + 7.174\\=0.857\times 55+7.174\\=54.309\\\approx 54.0[/tex]
The new value of y (54) is close to the original value (55)
Thus, it can be said that the line of best fit models the data provided.
(4)
The line of best fit is:
y = 0.857·x + 7.174
The slope of the line is positive.
This implies that there is a positive relationship between the two variables.
The formula of slope in terms of correlation coefficient is:
[tex]\text{Slope}=r\times\sqrt{\frac{Var(x)}{Var(x)}}[/tex]
The correlation coefficient is directly proportional to the slope.
This implies that the correlation coefficient is also positive.
Thus, the variables Arm span and Height are positively correlated.
