Sampling and scatter graphs – Year 11 GCSE maths worksheet with answers to download as PDF

A school has 800 pupils. A stratified sample of 80 pupils is to be chosen by year group.

1. Explain why a stratified sample is better than asking the first 80 pupils to arrive. [2 marks]
2. Work out the number to choose from each year. [4 marks]
3. Check that your numbers add to 80. [1 mark]

A random sample of 50 pupils is asked how they travel to school. 18 of them walk. There are 1 200 pupils in the school.

1. Estimate the number of pupils in the school who walk. [2 marks]
2. Explain why this is only an estimate. [1 mark]
3. A second random sample of 100 pupils has 41 who walk. Use both samples together to improve the estimate. [3 marks]

For each survey, say why the sample may be biased and suggest an improvement.

1. Asking customers outside a gym about how often people exercise. [2 marks]
2. Asking only the pupils in the school choir which music the school should buy. [2 marks]
3. Phoning people at 11 am on a weekday about their jobs. [2 marks]

A biologist wants to estimate the number of frogs in a pond. She catches 60 frogs, marks them and releases them. Later she catches 40 frogs, of which 8 are marked.

1. Write a proportion for the second sample. [1 mark]
2. Estimate the number of frogs in the pond. [3 marks]
3. State two assumptions you have made. [2 marks]

Eight pupils revised for a test. The table shows hours of revision, \(x\), and test mark, \(y\) (out of 50).

1. Describe the correlation. [1 mark]
2. Which points would you use to draw a line of best fit? Write the equation of the line through \((2,15)\) and \((8,41)\) to 1 decimal place in the gradient. [4 marks]
3. Use the line to estimate the mark for a pupil who revised for 5.5 hours. [2 marks]

A line of best fit passes through \((2,40)\) and \((10,80)\).

1. Work out the gradient. [2 marks]
2. Find the equation in the form \(y=mx+c\). [2 marks]
3. Use it to predict \(y\) when \(x=7\). [1 mark]
4. Explain what the gradient means if \(x\) is hours of revision and \(y\) is mark. [1 mark]

A park ranger catches 80 fish in a lake and puts a small tag on each one. A week later she catches 60 fish and 12 of them have tags.

1. Estimate the number of fish in the lake. [3 marks]
2. The ranger catches the second sample using a different net that cannot catch small fish. Explain how this affects the estimate. [2 marks]
3. State why it is important to wait a week between the samples. [1 mark]

A company has 150 office workers, 60 factory workers and 30 drivers. A stratified sample of 40 employees is asked about travel to work.

1. Work out how many from each group should be in the sample. [4 marks]
2. A random sample of 40 from the whole list could contain no drivers. Why might that matter? [2 marks]
3. Describe how to choose the 10 factory workers fairly. [2 marks]

45 deer in a forest are marked. Later, 30 deer are seen and 6 of them have marks.

1. Estimate the number of deer in the forest. [3 marks]
2. Give one reason why the true number may be different. [1 mark]
3. If 5 more marked deer had been seen in the second sample, how would the estimate change? [2 marks]

The table shows the height and arm span (in cm) of 8 pupils.

1. Describe the relationship. [2 marks]
2. One pupil has height 168 cm. Estimate their arm span. [2 marks]
3. Explain why it is unwise to use the graph to predict the arm span of a person 120 cm tall. [2 marks]
4. Is it correct to say that a longer arm span causes greater height? [1 mark]

An ice cream van records the temperature and sales. A line of best fit passes through \((15,80)\) and \((30,260)\).

1. Work out the gradient and explain what it means. [3 marks]
2. Write the equation of the line. [2 marks]
3. Predict the sales on a day of 27°C. [2 marks]
4. Why is a prediction for 40°C less reliable? [1 mark]

A school register has 600 names. A teacher takes a systematic sample by choosing every 15th name, starting with the 7th.

1. How many pupils are in the sample? [2 marks]
2. Write the position of the first four names chosen. [2 marks]
3. Give one advantage and one disadvantage of this method. [2 marks]

The value of a car, \(y\) thousand pounds, falls with age, \(x\) years. A line of best fit has the equation \(y=15-1.2x\).

1. What type of correlation is shown? [1 mark]
2. Write down the meaning of the gradient and the intercept. [2 marks]
3. Estimate the value of a 5-year-old car. [2 marks]
4. After how many years does the equation predict the value to be zero? Comment. [2 marks]

The 480 pupils in Years 10 and 11 are numbered 1 to 480. A calculator random number generator gives: 47, 312, 479, 505, 47, 188, 0, 260.

1. Which of these numbers cannot be used? Explain. [2 marks]
2. Which number is repeated and what should you do? [2 marks]
3. Write the numbers of the first five pupils chosen. [2 marks]
4. Explain why this sample is random. [1 mark]

Two researchers estimate the mean time pupils spend on homework each week. Sample A has 20 pupils with mean 3.5 hours. Sample B has 100 pupils with mean 4.1 hours.

1. Which sample is more likely to give a reliable estimate? Explain. [2 marks]
2. Work out the mean of the two samples combined. [3 marks]
3. The school has 900 pupils. Estimate the total hours of homework per week using your combined mean. [2 marks]

50 fish are tagged. A second sample of 80 contains 10 tagged fish.

1. Estimate the population. [3 marks]
2. Some tags fall off between the two samples. Does this make the estimate too high or too low? Explain. [3 marks]
3. After the second sample, 20 new fish are born and 10 die. What other issue arises? [2 marks]

Seven pupils have these values of \((x,y)\): \((1,12)\), \((2,15)\), \((3,18)\), \((4,20)\), \((5,23)\), \((6,40)\), \((7,28)\).

1. Which point is an outlier? [1 mark]
2. Draw a line of best fit that ignores the outlier through \((1,12)\) and \((7,28)\) and find its equation. [4 marks]
3. Use it to estimate \(y\) when \(x=4.5\). [2 marks]
4. Give two possible reasons for the outlier. [2 marks]

A school has 1 200 pupils: 720 boys and 480 girls. A stratified sample of 60 pupils is chosen by gender.

1. Work out the number of boys and girls in the sample. [3 marks]
2. Among the sample, 21 boys and 9 girls prefer football. Estimate the number of pupils in the school who prefer football. [4 marks]
3. Estimate the percentage of the girls in the school who prefer football. [2 marks]

A magazine asks its readers to text in if they think the high street is getting worse. 85% of the 400 replies say it is.

1. Give two reasons why the result may not represent all adults. [2 marks]
2. The magazine claims that "85% of adults think the high street is getting worse". Is this justified? [2 marks]
3. Suggest a better way to get the information. [2 marks]
4. A random sample of 1 000 adults finds 61% agree. Which result do you trust more and why? [2 marks]

50 birds are ringed. A second sample of 40 birds has 5 ringed. After this, 100 new unringed birds arrive.

1. Estimate the original population. [3 marks]
2. What is the new population? [1 mark]
3. A third sample of 60 birds is taken. How many ringed birds would you expect? [3 marks]
4. Explain what the capture-recapture formula would give if you used the first sample (50 ringed) and the third sample wrongly, ignoring the new birds. [3 marks]