Chapter 43, Problem 4TY

Summary Introduction

Introduction:

The mean number of progenies that a woman produces during her life span is called thenet reproductive rate $(R_0)$

. The reproductive rate is calculated by the given formula:

$R_0={\displaystyle \sum l_x}{m}_x$

Where,

$R_0$ = net reproductive rate,

S = sum of all age class,

$l_x$ = age-specific survivorship and

$m_x$ = age-specific fertility.

Answer to Problem 4TY

The correct answer is option (c) 1.75

Explanation of Solution

Explanation/justification for the correct answer:

Option (c) 1.75

Given table:

Age	No. alive at start of year $n_x$	Number dying during year, $d_x$	Age-specific survivorship rate, $l_x$	Age-specific fertility $m_x$	$l_x{m}_x$
0	100	35	1	0	0
1	65		0.65	0	0
2	45	15		3	1.35
3	30	20	0.30	1
4	10	10	0.10	1	0.10
5	0	0	0	1	0

Some values are missing in the given table. So, firstly calculate the missing values and put them in the new table:

Calculating the missing value of $d_x$ , at the 1^stage.

$d_x=n_x-n_{x+1}$

Where, $d_x=$ No. dying during year,

$n_x=$ No. alive at start of year,

$x$ = age

So,

65 individuals were alive at the start of 1^st year $=n_1=65$

And only 45 individuals were alive at the start of the 2^nd year $=n_{(x+1)}=n_{(1+1)}=n_2=45$, then

$n_1=65$ And $n_2=45$ respectively

Putting the values in above formula:

$d_x=65-45$

$=20$

Hence, the number of individual dying during year ( $d_x$ ) is 20.

Calculating the missing value of $l_x$ at the 2^ndage.

$l_x=n_x/n_0$

Where, $l_x$ = Age-specific survivorship rate,

$n_x=$ No. alive at start of year,

$x$ At subscript = age

Zero (0) at subscript = age

Individuals were alive at the start of 2^ndyear $=n_x=n_2=45$

Individuals were alive at the start of zero (0) year $=n_0=100$

Putting the given value in above formula:

$l_x=45/100$

$=0.45$

Hence the age-specific survivorship rate of the individual at the 2^nd age is 0.45

Calculating the missing value of $l_x{m}_x$ at the 3^rdage.

Given the values of $l_x$ and $m_x$ is 0.30 and 1 respectively.

Putting the given values in $l_x{m}_x$, $=0.30\times 1$

$=0.30$

Hence the value of $l_x{m}_x$ is 0.30.

Putting the calculated values of points (A), (B), and (c) and making refresh table again:

Age	No. alive at start of year $n_x$	Number dying during year, $d_x$	Age-specific survivorship rate, $l_x$	Age-specific fertility $m_x$	$l_x{m}_x$
0	100	35	1	0	0
1	65	20	0.65	0	0
2	45	15	0.45	3	1.35
3	30	20	0.30	1	0.30
4	10	10	0.10	1	0.10
5	0	0	0	1	0

Now finding the net reproductive rate $(R_0)$

from above fresh table:

$R_0={\displaystyle \sum l_x}{m}_x$

$=0+0+1.35+0.30+0.10+0$

$=1.75$

Hence, the net reproductive rate $(R_0)$ is 1.75

So, this is the correct option.

Explanation for incorrect answer:

Option (a) 5. As explained in option (c), the net reproductive rate is not matched to given data. So, this is an incorrect option.

Option (b) 2.5. As explained in option (c), the net reproductive rate is not matched to given data. So, this is an incorrect option.

Option (d) 1.45. As explained in option (c), the net reproductive rate is not matched to given data. So, this is an incorrect option.

Option (e) 0.80. As explained in option (c), the net reproductive rate is not matched to given data. So, this is an incorrect option.