13.3 The paper “Predicting Yolk Height, Yolk Width, Albumen Length, Eggshell Weight, Egg Shape Index, Eggshell Thickness, Egg Surface Area of Japanese Quails Using Various Egg Traits as Regressors” (International Journal of Poultry Science [2008]: 85–88) suggests that the simple linear regression model is reasonable for describing the relationship between y 5 eggshell thickness (in micrometers) and x 5 egg length (mm) for quail eggs. Suppose that the population regression line is y 5 0.135 1 0.003x and that s 5 0.005. Then, for a fixed x value, y has a normal distribution with mean 0.135 1 0.003x and standard deviation 0.005.

a. What is the mean eggshell thickness for quail eggs that are 15 mm in length? For quail eggs that are 17 mm in length?

b. What is the probability that a quail egg with a length of 15 mm will have a shell thickness that is greater than 0.18 mm?

c. Approximately what proportion of quail eggs of length 14 mm have a shell thickness of greater than .175? Less than .178? (Hint: The distribution of y at a fixed x is approximately normal.)