Download Chapter 6, Number 15

Chapter 6, Number 15
Hens usually begin laying eggs when they are about 6 months old. Young hens tend to solve smaller
eggs, often weighing less than the desired minimum weight of 51 grams.
a) The average weight of the eggs produces by the young hens is 50.6 grams, and only 29% of their
eggs exceed the desired minimum weight. If a Normal model is appropriate, what would the
standard deviation of the egg weights be?
Solution: Draw a picture of this scenario and mark the graph with the known information:
Make the center of the distribution with 50.6 because this is the mean weight.
Write down the minimum desired weight of 51.
Since 29% of the eggs exceed the minimum weight, this means that the area to the right of 51 grams
represents 29% of the data. Since the whole graph is 100%, then 100 – 29 = 71% of the data is to the left
of 51 grams.
Find the value closest to .7100 in the Normal distribution table. This will help us find the z-score that
corresponds to 51 grams:
The closest value is .6985, which corresponds to z-value = 0.52
To find the standard deviation, use the formula for the z-value:
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b) By the time these hens have reached the age of one year, the eggs they produce average 67.8
grams, and 95% of them are above the minimum weight. What is the standard deviation for
these older hens?
Solution: Draw a picture of this scenario and mark the graph with the known information:
Make the center of the distribution with 67.8 because this is the mean weight.
Write down the minimum desired weight of 51.
Since 95% of the eggs exceed the minimum weight, this means that the area to the right of 51 grams
represents 95% of the data. Since the whole graph is 100%, then 100 – 95 = 5% of the data is to the left
of 51 grams.
Find the value closest to .0500 in the Normal distribution table. This will help us find the z-score that
corresponds to 51 grams:
.0500 falls exactly halfway between .0495 and .0505, which corresponds to z-value = -1.645
To find the standard deviation, use the formula for the z-value:
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c) Are the egg sizes more consistent for the younger hens or the older hens?
Answer: The eggs sizes are more consistent for the younger hens because their standard deviation is
smaller.
d) A certain poultry farmer finds that 9% of his eggs are underweight and that 10% weigh over 73
grams. Estimate the mean and standard deviation of the eggs.
Solution: Draw a picture of this scenario and mark the graph with the known information:
Find the z-scores that correspond with each of these weights.
For the weight of 51, look for the value closest to .0900 in the Normal distribution table.
The closest value is 0.0901, which corresponds to a z-value of -1.34
For the weight of 73 grams, first subtract 100 – 10 to get 90% to the left of the value of 73. In the
table find the value closest to 0.9000.
The closest value is 0.8997, and this corresponds to the z-value of 1.28
To find the mean and standard deviation, use the formula for the z-value:
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For the first z-value, we have:
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For the second z-value, we have:
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Multiply both sides by s for both equations:
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Multiply the first equation by -1:
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Add the two equations:
Now plug this value into one of the equations:
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