Ask 50 people, learn about 5,000
Tonight's idea
You want to know something about a big group, such as how many people in a town swim. The whole group is the population. Asking everyone takes too long, so you ask a few: a sample.
In this town the swimmers mostly live near the pool, on the right. The corner sample was taken beside the pool, so it is packed with swimmers. A sample that leans one way is biased. It tells you about that corner, not the town.
The fix is to choose at random, so every person has the same chance of being picked. Names out of a hat will do. Chance does not know where the pool is. A random sample is usually a small copy of the population: close to the truth, though rarely exact.
If the sample is fair, the fraction in the sample is a good estimate of the fraction in the population.
Watch one: how many walk to school?
A school has 600 students. Fifty names are drawn from a hat, and 18 of those students walk to school.
Fraction in the sample: \(18 \div 50 = 0.36\), which is 36%.
Estimate for the school: 36% of 600 is \(0.36 \times 600 = 216\).
So about 216 students walk. Say "about": another draw might give 16 or 20 walkers. Asking the first 50 through the front gate would be worse, because bus riders use another entrance.
Draw sample after sample and see how close each lands: mathdetours.com/demos/sample.html.
Your turn
- A school has 800 students. In a random sample of 40, there are 14 who own a pet. (a) What percentage of the sample own a pet? (b) Estimate how many students in the school own a pet.
- Each sample is biased. Say who is counted too often. (a) "What is your favourite sport?", asked of the crowd at a football match. (b) "How long do you spend on homework?", asked of students in the library after school. (c) A phone survey of adults, made at 11 in the morning on a weekday.
- A factory makes 5,000 light bulbs a day. A tester picks 125 at random and finds 3 faulty. (a) Estimate the number of faulty bulbs made that day. (b) One day she tests the first 125 bulbs made that morning instead. Why is that sample less trustworthy?
- A town of 20,000 is thinking about a new skate park. Survey A is a form on the skate shop's website: 2,000 people fill it in and 90% say yes. Survey B asks 100 people picked at random from the whole town: 38 say yes. (a) Which survey should the town trust? Why? (b) Estimate how many people in the town want the park. (c) Survey A is twenty times bigger. Why does that not rescue it?
Stretch: grade by grade
A school has 300 students in grade 7 and 200 in grade 8. You want a sample of 50 with the grades in the same proportions. How many from each? In it, 12 of the grade 7s and 14 of the grade 8s walk to school. Estimate the walkers in each grade, then in the school.
Answers
- (a) \(14 \div 40 = 0.35\), so 35%. (b) 35% of 800 is \(0.35 \times 800 = 280\) students.
- (a) Football fans. Football will come out far ahead. (b) Students who study a lot. The ones who go straight home are missed. (c) People at home on a weekday morning, such as retired people. Most people at work or school are missed.
- (a) \(3 \div 125 = 0.024\), which is 2.4%. Then \(0.024 \times 5000 = 120\) bulbs. (b) It is not random: bulbs made later have no chance of being picked. If the machine behaves differently once it warms up or wears during the day, the first 125 will not show it.
- (a) Survey B. Everyone in town had the same chance of being asked. Survey A reached people who visit a skate shop's website and care enough to fill in a form. (b) \(38 \div 100 = 38\%\), and 38% of 20,000 is 7,600 people. (c) A bigger lopsided sample is still lopsided. It measures skate shop visitors more exactly, not the town.
Stretch. Grade 7 is \(\tfrac{300}{500} = \tfrac35\) of the school, so take \(\tfrac35\) of 50, which is 30, and 20 from grade 8. Grade 7: \(\tfrac{12}{30} = 40\%\) walk, and 40% of 300 is 120. Grade 8: \(\tfrac{14}{20} = 70\%\) walk, and 70% of 200 is 140. School: \(120 + 140 = 260\) walkers.