The eggs of a species of bird have an average diameter of 23 mm and an SD of 0.45 mm. The weights of the chicks that hatch from these eggs have an average of 6 grams and an SD of 0.5 grams. The correlation between the two variables is 0.75 and the scatter diagram is roughly football shaped.
Find the regression estimate of the weight of a chick that hatches from an egg 24 mm in diameter.
Find the regression estimate of the weight of a chick that hatches from an egg 22.5 mm in diameter.
no idea..
Eggs(x): mean = 23, sd = 0.45
Weights of chicks(y): mean = 6, sd = 0.5
Correlation: r = 0.75
Regression equation is in this format:
predicted y = a + bx
...where a = intercept and b = slope.
To find the equation, you need to substitute the information given in the problem into a workable formula:
predicted y = (rSy/Sx)X - (rSy/Sx)xbar + ybar
...where r = correlation, Sy = sd of y, Sx = sd of x, and X is the variable in 'a + bx' equation.
Note: xbar = mean of x; ybar = mean of y.
I'll let you take it from here. (Once you calculate the predicted y formula, substitute 24 and 22.5 for x in the formula to predict weights of the chicks.)
How do you find the intercept?
-13.17
To find the regression estimate of the weight of a chick that hatches from an egg of a given diameter, you can use the formula for simple linear regression. Let's denote the diameter of the egg as X and the weight of the chick as Y.
Given information:
X (diameter of the egg): average = 23 mm, SD = 0.45 mm
Y (weight of the chick): average = 6 grams, SD = 0.5 grams
Correlation (r) between X and Y = 0.75
Step 1: Convert the diameter of the eggs to a standardized Z-score.
To find the Z-score, use the formula:
Z = (X - μ) / σ
where X is the value of the variable (diameter of the egg), μ is the population mean (average diameter), and σ is the standard deviation.
For an egg with 24 mm diameter:
Z1 = (24 - 23) / 0.45
For an egg with 22.5 mm diameter:
Z2 = (22.5 - 23) / 0.45
Step 2: Use the regression equation to estimate the weight of the chick.
The regression equation is given by:
Ŷ = a + bX
where Ŷ is the predicted (estimated) value of Y, a is the intercept (Y when X=0), b is the slope (the change in Y for each unit change in X), and X is the value of the independent variable.
Step 3: Calculate the regression estimate for the weight of the chick.
For an egg with 24 mm diameter:
Ŷ1 = a + bZ1
For an egg with 22.5 mm diameter:
Ŷ2 = a + bZ2
The intercept a and slope b can be calculated using the following formulas:
b = r * (σY / σX)
a = μY - b * μX
where r is the correlation coefficient, σY is the standard deviation of Y (weight of the chick), σX is the standard deviation of X (diameter of the egg), μY is the average of Y, and μX is the average of X.
Step 4: Plug in the values.
You have the values for X (diameter of the egg), μX, μY, σX, σY, and r. Using these values, calculate a, b, and then substitute them in the regression estimation formula to find Ŷ.
Note: The shape of the scatter diagram being described as "roughly football-shaped" suggests that the relationship between X (diameter of the egg) and Y (weight of the chick) is nonlinear. In that case, a linear regression may not provide the best estimate, and a different type of regression analysis, such as polynomial regression, would be more appropriate. However, for the given information and using linear regression, we can proceed with the explanation above.