Samples, populations and statistical claims

(a) 50 students picked at random from the whole school register

1. A random sample from the whole population gives every student the same chance of being chosen.

• B1 The random sample from the whole register.

(b) Students at homework club are likely to spend more time on homework than other students.

1. The sample is not typical: the reason they are in it is linked to what is being measured.

• C1 A reason linking the club members to more (or different) homework time than the whole school.

(a) People at a gym exercise more than typical adults.

1. Everyone asked is at a gym, so the sample is likely to over-represent people who exercise.

• C1 People at a gym are more likely to exercise than adults in general.

(b) Only people free on a Monday evening can be asked.

1. The time restricts who can be asked, for example people who work evenings are missed.

• C1 A second valid reason, such as the day and time, or the small sample size.

(a) $238$ students

1. $$14/50\times 850$$
2. The sample proportion who usually cycle is 14/50.
3. Apply this proportion to the population: 850 $\times$ 14/50 = 238.
4. The sample gives an estimate of 238 students; it does not establish the exact population total.

• P1 Establishing $14/50\times 850$ or an equivalent valid method.
• A1 Correct answer: $238$ students

(a) $234$ homes

1. $$17/60\times 360$$
2. $$22/90\times 540$$
3. Scale each sample to its own population.
4. For A, the estimated number is 360 $\times$ 17/60 = 102.
5. For B, the estimated number is 540 $\times$ 22/90 = 132.
6. Add the estimates: 102 + 132 = 234 homes.

• P1 Establishing $17/60\times 360$ or an equivalent valid method.
• P1 Establishing $22/90\times 540$ or an equivalent valid method.
• A1 Correct answer: $234$ homes

(a) $72$

1. Use the sample proportion as an estimate for the population proportion.
   $$9/30\times 240$$
2. Therefore $72$.

• M1 Use the sample proportion as an estimate for the population proportion.
• A1 Correct answer: $72$

(b) Random samples vary, so the sample proportion may differ from the population proportion even with an unbiased selection method.

1. Random samples vary, so the sample proportion may differ from the population proportion even with an unbiased selection method.

• C1 Correct conclusion with supporting reasoning: Random samples vary, so the sample proportion may differ from the population proportion even with an unbiased selection method.

(a) $120$

1. Scale the observed sample fraction to the population.
   $$10/30\times 360$$
2. Therefore $120$.

• M1 Scale the observed sample fraction to the population.
• A1 Correct answer: $120$

(b) Cycling-shop customers are likely to cycle more often than the whole population. The selection method is biased.

1. Cycling-shop customers are likely to cycle more often than the whole population. The selection method is biased.

• C1 Correct conclusion with supporting reasoning: Cycling-shop customers are likely to cycle more often than the whole population. The selection method is biased.

(a) A sample is quicker and cheaper, and asking every member may not be practical.

1. A sample is quicker and cheaper, and asking every member may not be practical.

• C1 Correct conclusion with supporting reasoning: A sample is quicker and cheaper, and asking every member may not be practical.

(b) The samples contain different members, so the sample means vary by chance.

1. The samples contain different members, so the sample means vary by chance.

• C1 Correct conclusion with supporting reasoning: The samples contain different members, so the sample means vary by chance.

(c) $55$ minutes

1. Find the total time for all 50 members, then divide by 50.
   $$\frac{25\times 52+25\times 58}{50}$$
2. Therefore $55$ minutes.

• M1 Find the total time for all 50 members, then divide by 50.
• A1 Correct answer: $55$ minutes

(a) 3780

1. Combine the samples: $12 + 30 = 42$ left-handed out of $50 + 150 = 200$.
2. $$\tfrac{42}{200} \times 18\,000 = 3780$$

• P1 Combining the samples: $\frac{42}{200}$.
• P1 Multiplying their proportion by 18 000.
• A1 Correct answer: 3780.

(b) It is based on 200 people rather than 50, so it is likely to be closer to the true proportion.

1. Larger random samples give estimates that vary less from the population value.

• C1 A reason about the combined sample being larger (200 versus 50).