Chapter 13.1, Problem 4E

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 = Eggshell thickness (in micrometers) and x = Egg length (mm) for quail eggs. Suppose that the population regression line is y = 0.135 + 0.003x and that σ = 0.005. Then, for a fixed x value, y has a normal distribution with mean 0.135 + 0.003x and standard deviation 0.005.

1. 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?
2. 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 μm?
3. c. Approximately what proportion of quail eggs of length 14 mm have a shell thickness of greater than 0.175? Less than 0.178? (Hint: The distribution of y at a fixed x is approximately normal.)

To determine

Find the mean eggshell thickness that is 15 mm in length for quail eggs.

Find the mean eggshell thickness that is 17 mm in length for quail eggs.

Answer to Problem 4E

The mean eggshell thickness that is 15 mm in length for quail eggs is 0.18 micro meters.

The mean eggshell thickness that is 17 mm in length for quail eggs is 0.186 micro meters.

Explanation of Solution

Calculation:

The given information is that, the variable y denotes the eggshell thickness (in micro meters) and x denotes egg length (mm) for quail eggs. The population regression line is $y=0.135+0.003x$ and standard deviation is $\sigma =0.005$. For a fixed value if y and x that has the normal distribution with mean $0.135+0.003x$ with the standard deviation is $\sigma =0.005$.

Mean eggshell thickness for 15 mm:

Substitute $x=15$ in the regression equation of the mean.

$\begin{array}{c}{\mu }_y=0.135+0.003\left(15\right)\\ =0.135+0.045\\ =0.18\end{array}$

Hence, the mean eggshell thickness that is 15 mm in length for quail eggs is 0.18 micro meters.

Mean eggshell thickness for 17 mm:

Substitute $x=17$ in the regression equation of the mean.

$\begin{array}{c}{\mu }_y=0.135+0.003\left(17\right)\\ =0.135+0.051\\ =0.186\end{array}$

Hence, the mean eggshell thickness that is 17 mm in length for quail eggs is 0.186 micro meters.

To determine

Find the probability that a quail egg that has a length of 15 mm would have a shell thickness that was greater than 0.18$\mu \text{m}$.

Answer to Problem 4E

The probability that a quail egg that has a length of 15 mm would have a shell thickness that was greater than 0.18$\mu \text{m}$ is 0.5.

Explanation of Solution

Calculation:

The given information is that, the variable y denotes the eggshell thickness (in micro meters) and x denotes egg length (mm) for quail eggs. The quail egg has a length of 15 mm, that is $x=15$ and the mean thickness is ${\mu }_y=0.18$ and also shell thickness is greater than 0.18$\text{μm}$. For a fixed value if y and x that has the normal distribution with mean $0.135+0.03x$ with the standard deviation is $\sigma =0.005$.

z score using the normal distribution:

$z=\frac{x-\mu }{\sigma }$

In the formula, x denotes the data value, $\mu$ is the mean and $\sigma$ is standard deviation.

The probability is,

$\begin{array}{c}P\left(y>0.18\right)=P\left(\frac{y-{\mu }_y}{\sigma }>\frac{0.18-0.18}{0.005}\right)\\ =P\left(z>0\right)\\ =1-P\left(z\le 0\right)\end{array}$

From the “Standard Normal Probability (Cumulative z Curve Areas)”, the area to the left of $z\le 0$ is 0.5000.

$\begin{array}{c}\left(y>0.18\right)=1-P\left(z\le 0\right)\\ =1-0.5\\ =0.5\end{array}$

Thus, the probability that a quail egg that has a length of 15 mm would have a shell thickness that was greater than 0.18$\mu \text{m}$ is 0.5.

To determine

Find the proportion of quail eggs that has a length of 14 mm have a shell thickness of greater than 0.175.

Find the proportion of quail eggs that has a length of 14 mm have a shell thickness of less than 0.178.

Answer to Problem 4E

The proportion of quail eggs that has a length of 14 mm have a shell thickness of greater than 0.175 is 0.6554.

The proportion of quail eggs that has a length of 14 mm have a shell thickness of less than 0.178 is 0.5793.

Explanation of Solution

Calculation:

The given information is that, the variable y denotes the eggshell thickness (in micro meters) and x denotes egg length (mm) for quail eggs. The quail egg has a length of 15 mm, that is $x=14$. Also, shell thickness is greater than 0.175$\mu \text{m}$. For a fixed value if y and x that has the normal distribution with mean $0.135+0.03x$ with the standard deviation is $\sigma =0.005$.

Mean eggshell thickness for 14 mm:

Substitute $x=14$ in the regression equation of the mean.

$\begin{array}{c}{\mu }_y=0.135+0.003\left(14\right)\\ =0.135+0.042\\ =0.177\end{array}$

The mean eggshell thickness that is 15 mm in length for quail eggs is ${\mu }_y=0.177$ micro meters.

Proportion for shell thickness of greater than 0.175:

$\begin{array}{c}P\left(y>0.175\right)=P\left(\frac{y-{\mu }_y}{\sigma }>\frac{0.175-0.177}{0.005}\right)\\ =P\left(z>-0.4\right)\\ =1-P\left(z\le -0.4\right)\end{array}$

From the “Standard Normal Probability (Cumulative z Curve Areas)”, the area to the left of $z\le -0.4$ is 0.3446.

$\begin{array}{c}P\left(y>0.175\right)=1-P\left(z\le -0.4\right)\\ =1-0.3446\\ =0.6554\end{array}$

Hence, the proportion of quail eggs that has a length of 14 mm have a shell thickness of greater than 0.175 is 0.6554.

Proportion for shell thickness of greater than 0.178:

$\begin{array}{c}P\left(y<0.178\right)=P\left(\frac{y-{\mu }_y}{\sigma }<\frac{0.178-0.177}{0.005}\right)\\ =P\left(z<0.2\right)\end{array}$

From the “Standard Normal Probability (Cumulative z Curve Areas)”, the area to the left of $z\le 0.2$ is 0.5793.

$\begin{array}{c}P\left(y<0.178\right)=P\left(z<0.2\right)\\ =0.5793\end{array}$

Hence, the proportion of quail eggs that has a length of 14 mm have a shell thickness of greater than 0.178 is 0.5793.