Stratified Sampling

First of all, we need to calculate the total population.

 

\text{Total population }=1,354+3,480+3,776+1,865+430=10,905

 

We can use the formula to see how big our sample from each group should be (if you don’t like having to remember formulae, then use some maths logic by simply working out what percentage of the total population a particular income group represents, and then calculate this percentage of your sample size of 200):

 

\text{number to be sampled from group }=\dfrac{\text{number of people in group}}{\text{size of population}}\times \text{sample size}

 

 

\text{0 - 14,999 group: }\dfrac{1,354}{10,905}\times 200=24.832...=25 \text{ people}

 

\text{15,000 - 24,999 group: }\dfrac{3,480}{10,905}\times 200=63.823...=64 \text{ people}

 

\text{25,000 - 34,999 group: }\dfrac{3,776}{10,905}\times 200=69.252...=69 \text{ people}

 

\text{35,000 - 49,999 group: }\dfrac{1,865}{10,905}\times 200=34.204...=34 \text{ people}

 

\text{50,000 group: }\dfrac{430}{10,905}\times 200=7.886...=8 \text{ people}

 

If you want to check that your calculations are correct, then add up the totals for each group and the total should be the desired total of 200.

25+64+69+34+8=200

First of all, we need to calculate how many students there are in total:

 

112+104+98+90+76 = 480\text{ students}

 

If the house leader would like a stratified sample of 80 students, then we can work this out as a fraction (or even percentage) of the whole student population:

 

\dfrac{80}{480} = \dfrac{1}{6} \text{ or } 16.6\%

 

(Seeing the sample as \frac{1}{6} will be easier to manage than a difficult % amount.)

 

Since the house leader would like a stratified sample of \frac{1}{6} of the whole school population, this means that she will need to survey \frac{1}{6} of each year group.  For each year group, the number of students to be surveyed per year group can be calculated as follows:

 

Year 7:  \dfrac{112}{6}=19\text{ students}

 

Year 8:  \dfrac{104}{6}=17\text{ students}

 

Year 9:  \dfrac{98}{6}=16\text{ students}

 

Year 10:  \dfrac{90}{6}=15\text{ students}

 

Year 11:  \dfrac{76}{6}=13\text{ students}

We have been given complete information for the football club, and we can see that there are 196 players in total, of which 28 have been selected for the sample.  In other words, 28 out of 196 are in the sample.  We can write this as a fraction:

 

\dfrac{28}{196}

 

This fraction can be simplified to:

\dfrac{1}{7}

 

(If this is not an easy fraction for you to simplify, remember that the line of a fraction means ‘divide’, so if you type into your calculator 28\div196 your calculator should give you the answer \frac{1}{7}.)

 

Therefore \frac{1}{7} of the football club is included in the sample.

 

Since Michael is conducting a stratified sample, this means that he is taking  a sample of \frac{1}{7} from each category.

 

Since the rugby club has a total of 91 members, the number included in the sample can be calculated as follows:

 

\dfrac{91}{7} = 13\text{ players}

For the basketball team, we have been given the sample size and not the total number of members in the club.  Since the sample size is \frac{1}{7} of the total, this means that 19 basketball players = \frac{1}{7} of the overall total.  Therefore the total number of basketball players is:

 

19\times7=133\text{ players}

 

Therefore, the completed table should look like this:

 

If we know that 81 students represent 30\% of the entire sixth form, then we can work out how many students there are in total in the sixth form.

 

If

30\% = 81\text{ students}

then

10\% = 81\div3 = 27\text{ students}

so

100\% = 27\times10=270\text{ students}

 

Therefore, if there are 270 students in the sixth form, of which 156 are in year 12, then we can easily work out the number of students in year 13:

 

270-156=114\text{ students}

First of all, we need to calculate the entire student population.  In total there are:

 

212+200+192+186+170=960\text{ students}

 

The head of maths would like a sample of 80 students, so we can work out what fraction of the entire student population this represents, and simplify the fraction:

 

\dfrac{80}{960} = \dfrac{1}{12}

 

Therefore, for his stratified sample, he will need to survey \frac{1}{12} of each year group.

 

We know that there is a total of 192 students in year 9, so the total number of students he needs to survey in year 9 can be calculated as follows:

\dfrac{192}{12} = 16\text{ students}

 

However, the head of maths can’t simply survey 8 boys and 8 girls because he is also stratifying by gender as well as age.

 

We know that there are 192 students in total in year 9, of which 72 are girls, so we need to work out how many boys there are:

192-72=120\text{ boys}

 

We now need to work out what fraction of the year group are boys.  Since 120 out of the 192 students are boys, we can write this as a fraction, and simplify:

 

\dfrac{120}{192} = \dfrac{5}{8}

 

If \frac{5}{8} of the year group are boys, this means that \frac{5}{8} of the 16 year 9 students he plans to survey have to be boys.

 

\dfrac{5}{8}\times16 = 10\text{ boys}

 

Therefore, the head of maths needs to survey 10 year 9 boys.