Respuesta :
Answer:
a) Figure attached
b) [tex]y=1.31 x +98.57[/tex]
c) The correlation coefficient would be r =0.47719
d) [tex]y=1.31 x +98.57=1.31*21 + 98.57 =126.08[/tex]
Step-by-step explanation:
(a) Draw a scatter diagram for the data.
See the figure attached
(b) Find x, y, b, and the equation of the least-squares line. (Round your answers to three decimal places.) x =__ y =__ b =__ y =__ + __x
[tex]m=\frac{S_{xy}}{S_{xx}}[/tex]
Where:
[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}[/tex]
[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}[/tex]
With these we can find the sums:
[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=6576-\frac{300^2}{14}=147.429[/tex]
[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i){n}}=38186-\frac{300*1773}{14}=193.143[/tex]
And the slope would be:
[tex]m=\frac{193.143}{147.429}=1.31[/tex]
Nowe we can find the means for x and y like this:
[tex]\bar x= \frac{\sum x_i}{n}=\frac{300}{14}=21.429[/tex]
[tex]\bar y= \frac{\sum y_i}{n}=\frac{1773}{14}=126.643[/tex]
And we can find the intercept using this:
[tex]b=\bar y -m \bar x=126.643-(1.31*21.429)=98.571[/tex]
So the line would be given by:
[tex]y=1.31 x +98.57[/tex]
(c) Find the sample correlation coefficient r and the coefficient of determination r?2. (Round your answers to three decimal places.)
n=14 [tex] \sum x = 300, \sum y = 1773, \sum xy=38186, \sum x^2 =6576, \sum y^2 =225649[/tex]
And in order to calculate the correlation coefficient we can use this formula:
[tex]r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}[/tex]
[tex]r=\frac{14(38186)-(300)(1773)}{\sqrt{[14(6576) -(300)^2][14(225649) -(1773)^2]}}=0.9534[/tex]
So then the correlation coefficient would be r =0.47719
What percentage of variation in y is explained by the least-squares model? (Round your answer to one decimal place.)
The % of variation is given by the determination coefficient given by [tex]r^2[/tex] and on this case [tex]0.47719^2 =0.2277[/tex], so then the % of variation explaines is 22.8%.
(d) If a female baby weighs 21 pounds at 1 year, what do you predict she will weigh at 30 years of age? (Round your answer to two decimal places.) ___ lb
So we can replace in the linear model like this:
[tex]y=1.31 x +98.57=1.31*21 + 98.57 =126.08[/tex]
