The famous iris dataset (the first sheet of the spreadsheet linked above) was first published in 1936 Fisher. The dataset contains 50 samples from each of 3 iris species: setosa, virginia, and versicolor. are measured, all in cm : sepal length, sepal width, petal length, and petal width.
What is the equation for the least square regression line where the independent or predictor variable length and the dependent or response variable is sepal width for iris versicolor?
[tex]$\hat{y}=\text { Ex } 1.234 x+\square \text { Round to three decimal places. }$[/tex]
What is the predicted sepal width for iris versicolor for a flower with a sepal length of 8.00 ?
[tex]$\square$ cm Round to three decimal places.[/tex]
Answer (2)
Calculate the y-intercept using the formula: b 0 = y ˉ − b 1 x ˉ = 2.770 − 1.234 ∗ 5.936 = − 4.554 .
Form the regression equation: y ^ = 1.234 x − 4.554 .
Substitute x = 8.00 into the regression equation: y ^ = 1.234 ( 8.00 ) − 4.554 .
Calculate the predicted sepal width: y ^ = 5.318 cm. The final answer is 5.318 cm.
Explanation
Problem Analysis We are given the slope of the least squares regression line as 1.234. We need to find the y-intercept and then use the equation to predict the sepal width for a sepal length of 8.00 cm.
Gathering Data To find the y-intercept, we need the average sepal length and average sepal width for iris versicolor. Let's assume (since the data is not provided) that the average sepal length (x) is 5.936 cm and the average sepal width (y) is 2.770 cm.
Formula for y-intercept The formula for the y-intercept ( b 0 ) is: b 0 = y ˉ − b 1 x ˉ , where y ˉ is the average sepal width, x ˉ is the average sepal length, and b 1 is the slope.
Substituting Values Substituting the values, we get: b 0 = 2.770 − 1.234 ∗ 5.936
Calculating y-intercept Calculating the y-intercept: b 0 = 2.770 − 7.324 = − 4.554 (rounded to three decimal places).
Regression Line Equation The equation of the least squares regression line is: y ^ = 1.234 x − 4.554
Predicting Sepal Width To predict the sepal width for a flower with a sepal length of 8.00 cm, we substitute x = 8.00 into the equation: y ^ = 1.234 ( 8.00 ) − 4.554
Calculating Predicted Value Calculating the predicted sepal width: y ^ = 9.872 − 4.554 = 5.318 cm (rounded to three decimal places).
Final Answer Therefore, the equation for the least square regression line is y ^ = 1.234 x − 4.554 , and the predicted sepal width for a flower with a sepal length of 8.00 cm is 5.318 cm.
Examples
Understanding the relationship between sepal length and sepal width in Iris flowers can help botanists predict the characteristics of new flowers. For example, if a botanist discovers a new Iris versicolor flower with a sepal length of 8 cm, they can use the regression equation to estimate its sepal width. This kind of prediction is useful in ecological studies, agriculture, and even in art, where understanding natural proportions can inspire designs. Regression analysis provides a powerful tool for making informed estimations based on observed trends in nature.
The least squares regression line for predicting sepal width from sepal length for iris versicolor is y ^ = 1.234 x − 4.556 . The predicted sepal width for a flower with a sepal length of 8.00 cm is approximately 5.316 cm.
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