Concept explainers
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
- 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 μm? - 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.)
a.
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
Mean eggshell thickness for 15 mm:
Substitute
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
Hence, the mean eggshell thickness that is 17 mm in length for quail eggs is 0.186 micro meters.
b.
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
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
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
z score using the normal distribution:
In the formula, x denotes the data value,
The probability is,
From the “Standard Normal Probability (Cumulative z Curve Areas)”, the area to the left of
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
c.
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
Mean eggshell thickness for 14 mm:
Substitute
The mean eggshell thickness that is 15 mm in length for quail eggs is
Proportion for shell thickness of greater than 0.175:
From the “Standard Normal Probability (Cumulative z Curve Areas)”, the area to the left of
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:
From the “Standard Normal Probability (Cumulative z Curve Areas)”, the area to the left of
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.
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Chapter 13 Solutions
Introduction To Statistics And Data Analysis
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