Question

An ornithologist wants to look at the relationship between the breadth of peregrine falcon eggs and the mass of the falcon chicks that hatch from them. the data show a linear pattern with the summary statistics shown below: mean standard deviation \[x=\] egg breadth \[(\text{mm})\] \[\bar{x}=41\] \[s_x=2.3\] \[y=\] chick's mass \[(\text{g})\] \[\bar{y}=33.6\] \[s_y=4.6\] \[r=0.95\] find the equation of the least-squares regression line for predicting the chick's mass from the breadth of the egg. round your entries to the nearest hundredth.

Answer 1

The equation of the least-squares regression line for predicting the chick's mass from the breadth of the egg is: y = -24.9 + 1.90 * x, where y represents the chick's mass and x represents the egg breadth.

To find the equation of the least-squares regression line for predicting the chick's mass from the breadth of the egg, we can use the following steps:

1. Determine the slope of the regression line (b):
- The slope (b) can be calculated using the formula: b = r * (s_y / s_x), where r is the correlation coefficient,
`s_y` is the standard deviation of the chick's mass, and
`s_x` is the standard deviation of the egg breadth.
- Plugging in the given values, we have: b = 0.95 * (4.6 / 2.3) = 1.90.

2. Determine the y-intercept of the regression line (a):
- The y-intercept (a) can be calculated using the formula:
`a = \bar{y} - b * \bar{x}, \\ \bar{y}`is the mean of the chick's mass, and
`\bar{x}` is the mean of the egg breadth.
- Plugging in the given values, we have: a = 33.6 - 1.90 * 41 = -24.9.

3. Write the equation of the least-squares regression line:
- The equation of the regression line is given by: y = a + b * x, where y represents the chick's mass and x represents the egg breadth.
- Plugging in the values we calculated, we have: y = -24.9 + 1.90 * x.