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Answer :
To solve this problem, we need to find the missing survivorship rates, denoted as [tex]\(I_x\)[/tex], for the ages 3 and 4 in the table. The survivorship rate is the proportion of the original population that is still alive at the start of each year. We are given the following information:
- At age 1, 100 individuals are alive, and the survival rate [tex]\(I_1 = 1.0\)[/tex].
- At age 2, 50 individuals are alive, and the survival rate [tex]\(I_2 = 0.5\)[/tex].
The formula for survivorship [tex]\(I_x\)[/tex] is:
[tex]\[
I_x = \frac{\text{Number Alive at Start of Year}}{\text{Initial Number Alive at Age 1}}
\][/tex]
### Step-by-step Calculation:
1. Calculate [tex]\(y\)[/tex] for Age 3:
- Number Alive at Start of Year for Age 3 = 30
- Initial Number Alive at Age 1 = 100
Using the formula, we find:
[tex]\[
I_3 = \frac{30}{100} = 0.3
\][/tex]
2. Calculate [tex]\(z\)[/tex] for Age 4:
- Number Alive at Start of Year for Age 4 = 10
- Initial Number Alive at Age 1 = 100
Using the formula, we find:
[tex]\[
I_4 = \frac{10}{100} = 0.1
\][/tex]
Based on the calculations, the missing values for [tex]\(I_x\)[/tex] are:
- [tex]\(y = 0.3\)[/tex]
- [tex]\(z = 0.1\)[/tex]
This corresponds to the option: [tex]\(y = 0.3, z = 0.1\)[/tex].
- At age 1, 100 individuals are alive, and the survival rate [tex]\(I_1 = 1.0\)[/tex].
- At age 2, 50 individuals are alive, and the survival rate [tex]\(I_2 = 0.5\)[/tex].
The formula for survivorship [tex]\(I_x\)[/tex] is:
[tex]\[
I_x = \frac{\text{Number Alive at Start of Year}}{\text{Initial Number Alive at Age 1}}
\][/tex]
### Step-by-step Calculation:
1. Calculate [tex]\(y\)[/tex] for Age 3:
- Number Alive at Start of Year for Age 3 = 30
- Initial Number Alive at Age 1 = 100
Using the formula, we find:
[tex]\[
I_3 = \frac{30}{100} = 0.3
\][/tex]
2. Calculate [tex]\(z\)[/tex] for Age 4:
- Number Alive at Start of Year for Age 4 = 10
- Initial Number Alive at Age 1 = 100
Using the formula, we find:
[tex]\[
I_4 = \frac{10}{100} = 0.1
\][/tex]
Based on the calculations, the missing values for [tex]\(I_x\)[/tex] are:
- [tex]\(y = 0.3\)[/tex]
- [tex]\(z = 0.1\)[/tex]
This corresponds to the option: [tex]\(y = 0.3, z = 0.1\)[/tex].
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