11.1 Population, sample and informal inference — Practise

Work through these on your own. For each estimate, write a sentence that says which group your conclusion describes.

The questions and the answers below come from a set that is checked by computer every time the book is built.

A sample of 40 students spends a mean of $3.5$ hours per week on homework. Estimate the total number of hours spent on homework each week by the 1200 students in the school.
In a sample of 200 components, 12 are faulty. Estimate the number of faulty components in a batch of 5000.
In a sample of 80 shoppers, 52 prefer brand A. Estimate the proportion of all shoppers who prefer brand A.
A sample of six trees has heights, in metres, $14, 16, 18, 20, 22, 24$ . Find the sample mean.
Two samples of size 20 are taken from the same population. Their means are 24 and 27. Use the two sample means to estimate the population mean.
A sample of 40 adults in a village finds that 15 own a bicycle. The village has 840 adults. Estimate how many adults in the village own a bicycle.
A sample of 25 apples has a mean mass of 180 g. Estimate the total mass, in grams, of 400 apples.
In a sample of 250 people, 90 use the bus every day. Estimate the percentage of the population who use the bus every day.
A company claims a mean delivery time of 40 minutes. A sample of 30 deliveries has a mean of 43.2 minutes. Find the difference between the sample mean and the claim.
Find the mean of $4, 5, 6, 7, 8$ and the mean of $4, 5, 6, 7, 30$ .
A sample of 30 trees has a mean height of $8.4$ m. Estimate the total height of 150 trees.
A sample of 400 voters finds that 220 support a plan. Estimate the percentage of the population who support the plan. Say whether your answer is a mean, a proportion or a total.

Answers and concise solutions

1. The estimated total

Answer: Multiply the sample mean by the population size: $3.5 \times 1200 = 4200$ hours.

2. The estimated number of faulty components

Answer: The sample proportion is $\dfrac{12}{200} = 0.06$ , so the estimate is $0.06 \times 5000 = 300$ components.

3. The estimated proportion

Answer: Divide the count by the sample size: $\dfrac{52}{80} = \dfrac{13}{20} = 0.65$ . About $65\%$ of shoppers prefer brand A.

4. The sample mean

Answer: Add the values and divide by the sample size: $\dfrac{114}{6} = 19$ metres.

5. The estimate of the population mean

Answer: The samples are the same size, so take the mean of the sample means: $\dfrac{24 + 27}{2} = 25.5$ . The two samples disagree because of sampling variability.

6. The estimated number of adults

Answer: The sample proportion is $\dfrac{15}{40} = 0.375$ , so the estimate is $0.375 \times 840 = 315$ adults.

7. The estimated total mass

Answer: Multiply the sample mean by the number of apples: $180 \times 400 = 72000$ g, which is 72 kg.

8. The estimated percentage

Answer: The sample proportion is $\dfrac{90}{250} = 0.36$ , which is $36\%$ .

9. The difference

Answer: The difference is $43.2 - 40 = 3.2$ minutes. The sample mean is above the claim, but one sample cannot settle the question.

10. The two means

Answer: The first mean is $\dfrac{30}{5} = 6$ . The second is $\dfrac{52}{5} = 10.4$ . The single large value raises the mean, which is why an outlier matters.

11. The estimated total height

Answer: Multiply the sample mean by the population size: $8.4 \times 150 = 1260$ m.

12. The estimated percentage

Answer: This is a proportion, written as a percentage: $\dfrac{220}{400} = 0.55$ , which is $55\%$ . It is an informal inference about the population of voters.