To find the relationship between petal length and petal width for the iris setosa species, we use linear regression, which gives us the equation of a line that best fits the data points. The equation of a line in the context of linear regression is generally written as:
[tex]\[
\hat{y} = mx + b
\][/tex]
where:
- [tex]\( \hat{y} \)[/tex] is the predicted value (petal width in this case),
- [tex]\( x \)[/tex] is the independent variable (petal length in this case),
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the y-intercept.
Through the process, we determine that the slope ([tex]\( m \)[/tex]) is 0.109 and the y-intercept ([tex]\( b \)[/tex]) is 0.091. Thus, the equation for the least square regression line is:
[tex]\[
\hat{y} = 0.109x + 0.091
\][/tex]
Next, we need to predict the petal width for a flower with a petal length of 4.68 cm using the regression equation. Plugging [tex]\( x = 4.68 \)[/tex] cm into the equation:
[tex]\[
\hat{y} = 0.109(4.68) + 0.091
\][/tex]
Calculating this:
[tex]\[
\hat{y} = 0.109 \times 4.68 + 0.091
\][/tex]
[tex]\[
\hat{y} = 0.51012 + 0.091
\][/tex]
[tex]\[
\hat{y} = 0.601
\][/tex]
Thus, the predicted petal width for an iris setosa flower with a petal length of 4.68 cm is:
[tex]\[
0.601 \text{ cm}
\][/tex]
Summarizing:
1. The equation for the least square regression line is:
[tex]\[
\hat{y} = 0.109x + 0.091
\][/tex]
2. The predicted petal width for a petal length of 4.68 cm is:
[tex]\[
0.601 \text{ cm}
\][/tex]
