Question 15 Marks
Draw a random sample of $1$ percent students without replacement from 1500 students of a particular college for giving their opinion on college facilities. There are $600$ students in First year $(FY), 500$ students in Second year $(SY)$ and $400$ students of Third year $(TY)$ class in the college. Use the following $40$ three — digit random numbers :
$158, 092, 411, 745, 009, 724, 674, 550, 716, 359, 419, 969, 200, 458, 384, 019, 676, 631,$
$390, 557, 299, 786, 706, 206, 729, 344, 543, 309, 227, 483, 741 766,$
$027, 070, 648, 956, 238, 912, 480, 558.$
(Use the first $14$ random numbers for $FY$, next $14$ random numbers for $SY$ and remaining random numbers for $TY$)
$158, 092, 411, 745, 009, 724, 674, 550, 716, 359, 419, 969, 200, 458, 384, 019, 676, 631,$
$390, 557, 299, 786, 706, 206, 729, 344, 543, 309, 227, 483, 741 766,$
$027, 070, 648, 956, 238, 912, 480, 558.$
(Use the first $14$ random numbers for $FY$, next $14$ random numbers for $SY$ and remaining random numbers for $TY$)
Answer
View full question & answer→We shall first divide $1500$ students of the college into three groups (strata) First Year, Second Year and Third Year students.
Now we will select a random sample of $1%$ from each group.
$1%$ of $600$ of First year class i.e. $6, 1%$ of $500$ of Second year class
i.e. $5$ and $1%$ of $400$ of Third year class i.e. $4$ will be selected using simple random sampling method.
Selecting sample of size $6$ from $600$ $FY$ students :
Random numbers for $FY: 158, 092, 411, 745, 009, 724, 674, 550, 716, 359, 419, 969, 200, 458.$
Numbers $1$ to $600$ will be assigned to $600$ students of $FY.$
Since there are $600$ students in $FY$,
we shall ignore random numbers which are greater than $600$.
As the sample has to be selected without replacement, we shall also ignore the random numbers which are repeated more than once. We require a random sample of size $6$,
so we shall select first $6$ random numbers.
Thus, the selected random numbers are: $158, 092, 411, 009, 550, 359$.
Students with these numbers are selected from $FY$ in our sample.
Selecting sample of size $5$ from $500$ $SY$ students :
Random numbers for sy: $384, 019, 676, 631, 390, 557, 299, 786, 706, 206, 729, 344, 543, 309$
Numbers $1$ to $500$ will be assigned to $500$ students of $SY.$
Since there are $500$ students in $SY$,
we shall ignore random numbers which are greater than $500$.
As the sample has to be selected without replacement,
we shall also ignore the random numbers which are repeated more than once.
We require a random sample of size $5$. so we shall select first $5$ random numbers.
Thus the selected random numbers are: $384,019,390,299,206$.
Students with these numbers are selected from $SY$ in our sample.
Selecting sample of size $4$ from $400$ $TY$ students :
Random numbers for $TY : 227, 483, 741, 766, 027, 070, 648, 956, 238, 912, 480, 558$
Numbers $1$ to $400$ will be assigned to $400$ students of $TY$.
Since there are $400$ students in $TY$,
we shall ignore random numbers which are greater than $400$.
As the sample has to be selected without replacement,
we shall also ignore the random numbers which are repeated more than once.
We require a random sample of size $4$,
so we shall select first $4$ random numbers.
Thus the selected random numbers are: $227, 027, 070, 238$.
Students with these numbers are selected from $TY$ in our sample.
Stratified random sample ofsize $15$ is the combination ot‘ $6$ students selected from $FY$,
$5$ students selected from $SY$ and $4$ students selected from $TY$.
Now we will select a random sample of $1%$ from each group.
$1%$ of $600$ of First year class i.e. $6, 1%$ of $500$ of Second year class
i.e. $5$ and $1%$ of $400$ of Third year class i.e. $4$ will be selected using simple random sampling method.
Selecting sample of size $6$ from $600$ $FY$ students :
Random numbers for $FY: 158, 092, 411, 745, 009, 724, 674, 550, 716, 359, 419, 969, 200, 458.$
Numbers $1$ to $600$ will be assigned to $600$ students of $FY.$
Since there are $600$ students in $FY$,
we shall ignore random numbers which are greater than $600$.
As the sample has to be selected without replacement, we shall also ignore the random numbers which are repeated more than once. We require a random sample of size $6$,
so we shall select first $6$ random numbers.
Thus, the selected random numbers are: $158, 092, 411, 009, 550, 359$.
Students with these numbers are selected from $FY$ in our sample.
Selecting sample of size $5$ from $500$ $SY$ students :
Random numbers for sy: $384, 019, 676, 631, 390, 557, 299, 786, 706, 206, 729, 344, 543, 309$
Numbers $1$ to $500$ will be assigned to $500$ students of $SY.$
Since there are $500$ students in $SY$,
we shall ignore random numbers which are greater than $500$.
As the sample has to be selected without replacement,
we shall also ignore the random numbers which are repeated more than once.
We require a random sample of size $5$. so we shall select first $5$ random numbers.
Thus the selected random numbers are: $384,019,390,299,206$.
Students with these numbers are selected from $SY$ in our sample.
Selecting sample of size $4$ from $400$ $TY$ students :
Random numbers for $TY : 227, 483, 741, 766, 027, 070, 648, 956, 238, 912, 480, 558$
Numbers $1$ to $400$ will be assigned to $400$ students of $TY$.
Since there are $400$ students in $TY$,
we shall ignore random numbers which are greater than $400$.
As the sample has to be selected without replacement,
we shall also ignore the random numbers which are repeated more than once.
We require a random sample of size $4$,
so we shall select first $4$ random numbers.
Thus the selected random numbers are: $227, 027, 070, 238$.
Students with these numbers are selected from $TY$ in our sample.
Stratified random sample ofsize $15$ is the combination ot‘ $6$ students selected from $FY$,
$5$ students selected from $SY$ and $4$ students selected from $TY$.
