Sampling and Populations

A council estimates the travel habits of everyone aged 16 or over living in its town. It studies 200 of these residents. What is the population?

1. The 200 residents studied.
2. Everyone aged 16 or over living in the town.
3. All the journeys made by the 200 residents.
4. Everyone who works for the council.

A manufacturer measures 50 batteries selected from a delivery of 4000 batteries. Which group is the sample?

1. All 4000 batteries in the delivery.
2. Every battery the manufacturer has ever made.
3. The 50 batteries measured.
4. Only batteries with a below-average life.

A test destroys each component that is tested. Why would the manufacturer usually test a sample rather than the whole delivery?

1. Testing everything would destroy the whole delivery.
2. A sample always gives the exact result for the delivery.
3. A small sample cannot be biased.
4. Every component in the delivery is guaranteed to be identical.

A random sample of 80 components contains 6 faulty components. Estimate the number of faulty components in a batch of 2000.

An estimate for a whole batch is calculated from the faulty proportion in a sample. Which assumption is needed?

1. The batch must contain exactly the same number of faulty and good items.
2. The sample must include every faulty item.
3. The sample must have no faulty items.
4. The sample faulty proportion is representative of the batch.

A club has 500 members. Which method best selects a random sample of 40 distinct members?

1. Choose the first 40 members who arrive.
2. Number all members 1 to 500 and use random numbers, rejecting repeats and numbers outside that range.
3. Ask the committee to choose their friends.
4. Choose the 40 members who attend most often.

A school estimates how its pupils travel to school by counting only pupils who arrive at the bicycle racks. What is the main problem?

1. The sample must be too large.
2. Pupils arriving by bicycle are less likely to be counted.
3. The method over-represents cyclists and can miss other travel methods.
4. Counting pupils can never produce useful data.

A random sample of five parcels has masses 104, 99, 97, 102 and 103 grams. Use the sample mean to estimate the mean mass of all parcels from the same production run.

A random sample of bottles has mean volume 1.8 litres. Estimate the total volume, in litres, of a delivery of 250 similar bottles.

A representative sample of journey times has median 12 minutes. The whole population is ten times the sample size. Which is a suitable estimate for the population median?

1. 12 minutes.
2. 120 minutes.
3. 1.2 minutes.
4. 1200 minutes.

Of 250 randomly selected devices, 23 fail a test. Use this sample to estimate the probability that a device from the same batch fails. Give your answer as a fraction or a decimal.

Two independent random samples are taken from the same large population using the same method. One has size 20 and the other size 200. Which statement is most reasonable?

1. The sample of 200 is guaranteed to reproduce the population exactly.
2. The sample of 20 must have a lower mean.
3. Both samples must give the same results.
4. The sample of 200 will usually give a more reliable estimate.

A school has 240 pupils in Year 10 and 360 pupils in Year 11. A sample of 50 of these pupils is to be taken. The numbers of Year 10 and Year 11 pupils in the sample are to be in the same proportion as in the two year groups. How many Year 10 pupils should be in the sample?

A warehouse sends out 480 parcels in a day. Each parcel is first class or second class. Each day a sample of 24 parcels is checked. The proportion of first-class parcels in the sample is the same as the proportion of first-class parcels sent out that day. On Friday the sample contained 9 first-class parcels. Work out the number of first-class parcels sent out on Friday.

A sample of 100 seeds is tested. Exactly 62 germinate, 20 rot, and the remaining 18 do neither. The three groups do not overlap. Estimate how many seeds will not rot in a batch of 450 similar seeds.

A tennis club has 900 members. Each member is a junior or a senior. A sample of 45 members, with juniors and seniors in the same proportion as in the club, contains 18 juniors. Later, 180 new junior members join and no members leave. A new sample of 42 members is taken in the same way. How many juniors should be in the new sample?

A shipment contains 1000 items from factory A and 1200 from factory B. A random sample from A has 3 faulty items out of 50; a separate random sample from B has 2 faulty items out of 80. Estimate the total number of faulty items in the shipment.

A website asks visitors to volunteer their views about a transport service. It receives many more responses than an earlier random sample. Which statement is justified?

1. The larger number of volunteers guarantees an unbiased estimate.
2. The voluntary sample may still be biased because people with strong views may be more likely to respond.
3. Every voluntary response must be false.
4. The earlier random sample must be exactly correct.

A cinema has 1200 members. 75 members are asked which day of the week they prefer to watch films. 28 of them say Saturday.

Estimate how many of the 1200 members prefer Saturday.

The 75 members were all asked at a Saturday evening screening. Which statement is correct?

1. The estimate in (a) is likely to be too low, because only 75 members were asked.
2. The estimate in (a) is exact, because everyone asked was a member.
3. The estimate in (a) is likely to be too high, because members at a Saturday screening are more likely to prefer Saturday.
4. The estimate in (a) is not affected, because the members were chosen on one evening.

Two separate random samples are used to estimate use of a service by 600 residents. Before a change, 24 of 120 sampled residents used it. After the change, 54 of 180 sampled residents used it.

Estimate the increase in the number of residents using the service.

Which conclusion is justified?

1. The estimated increase must be exactly the increase in the number of users.
2. The service must now be used by every resident.
3. The samples suggest an increase, but do not establish the exact population change.
4. The number of residents in the area must have increased.