Planning an investigation: Year 11 Standard sampling and relationships in data

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

From Lesson 5: data run from x = 4 to x = 20. Is a prediction at x = 12 interpolation or extrapolation?
Answer
Interpolation. 12 lies between 4 and 20.
From Lesson 3: a population of 300 has a group of 60. A stratified sample of 40 is taken. How many come from that group?
Answer
8. 60 ÷ 300 × 40 = 8.
From Year 9: the mean of 8, 14, 11 and 7 is ?
Answer
10. The total is 40 and 40 ÷ 4 = 10.

Words used in this lesson

investigation: a planned piece of work that answers a question using data. For example: a project on sleep and screen time.
variable: a quantity that can take different values. For example: hours studied each week.
half-way mean point: the mean point of the lower half or the upper half of data sorted by x. For example: the point found from the four smallest x values.
convenience sample: a sample chosen because it is convenient to reach, not by chance. For example: the people in the room.
justified conclusion: a statement the data actually support, with its limits named. For example: the last paragraph of a report, which a reader could check against the data.

Two good students, one question

A year group of 24 students wants to know whether more hours of study each week go with higher test scores. Two students each build a line of best fit, using a sample of eight. Asha takes the eight students picked evenly through the register, whatever their hours. Bo asks the eight students who stayed after school to volunteer.

Predict first: if each uses their line to predict the score of a student who studies 3 hours a week, will they agree? Which line would you trust further away from its own data? Test it with the first figure, where the “typical score” for a number of hours is the mean of the two students with that many hours. Then see how the line itself is built in the second.

Hours studied 6

y = 3x + 42equation of the line from this sample

60score the line gives

59.5typical score of the two students with this many hours

Spread sample, 6 hours: y = 3x + 42 gives 60. The typical score for 6 hours is 59.5, so the line is 0.5 above.

The figure needs JavaScript. In words: 24 students gave hours studied and test score. A sample of eight spread from 1 to 12 hours gives the line y = 3x + 42. A sample of eight volunteers who all studied 8 to 11 hours gives y = 2.5x + 45.25. At 6 hours the first line gives 60 and the second gives 60.3, both near the typical score of 59.5. At 1 hour the first gives 45 and the second 47.8, against a typical 42.5: the volunteers' line is further off.

Show me the steps

Spread sample, 6 hours: the line y = 3x + 42 gives 60. The typical score is 59.5, so the line is 0.5 above.
Volunteers' sample, 6 hours: y = 2.5x + 45.25 gives 60.3. It is still close, but 6 hours is outside the volunteers' range of 8 to 11.
Volunteers' sample, 1 hour: the line gives 47.8 and the typical score is 42.5. 5.3 above, because the sample never saw anyone who studied so little.
Spread sample, 1 hour: the line gives 45 and the typical score is 42.5, 2.5 above. A spread sample keeps the line honest across the whole range.

An investigation is a chain, and each link has a job.

Question and variables: what is x, what is y, in what units, and what do you want to find out?
Population and sample: who is the population, how is the sample chosen (at random, in proportion), and how will you avoid or note bias?
Display and describe: a scatter graph; direction, strength and any outliers.
Model: a line through the mean point. By hand, split the sorted data into a lower and an upper half and join their two mean points.
Conclude: predict inside the data, say what the sample allows, and name its limits. A link is not a cause.

Stage 1

not yetgradient of the line

not yetintercept of the line

Stage 1: eight dots, hours studied against score, sorted by hours.

The figure needs JavaScript. In words: eight dots are sorted by hours studied: 1, 2, 2, 3 and 5, 6, 9, 12. Stage 2 splits them into a lower half and an upper half of four. Stage 3 finds the mean point of each half: (2, 48) and (8, 66). Stage 4 joins them: the gradient is (66 − 48) ÷ (8 − 2) = 3 and the intercept is 48 − 3 × 2 = 42, so the line is y = 3x + 42. It also passes through the mean point of all eight dots, (5, 57).

Show me the steps

Stage 1: eight dots, sorted by hours studied.
Stage 2: split the sorted dots into the lower half (hours 1, 2, 2, 3) and the upper half (hours 5, 6, 9, 12).
Stage 3: the mean point of the lower half is (2, 48) and of the upper half is (8, 66).
Stage 4: join the two mean points. The gradient is 18 ÷ 6 = 3, the intercept is 48 − 3 × 2 = 42, and the line y = 3x + 42 passes through the mean point of all eight dots, (5, 57).

A tidy line from a weak sample is a tidy answer to the wrong question. The line does not know how the data were collected. Look back along the chain: if the sample cannot represent the population, or the x values cover only a narrow range, no amount of neat working repairs it.

Worked examples

Eight pairs of (x, y): (1, 12), (2, 15), (3, 17), (4, 20), (6, 26), (7, 27), (8, 33), (9, 34). Build a line by the two half-way mean points and predict y at x = 5.

The sorted data split into 1, 2, 3, 4 and 6, 7, 8, 9. The lower half's mean point is (2.5, 16), because (12 + 15 + 17 + 20) ÷ 4 = 16.
The upper half's mean point is (7.5, 30), because (26 + 27 + 33 + 34) ÷ 4 = 30.
The gradient is (30 − 16) ÷ (7.5 − 2.5) = 14 ÷ 5 = 2.8, and the intercept is 16 − 2.8 × 2.5 = 9. The line is y = 2.8x + 9.
At x = 5, inside the data: 2.8 × 5 + 9 = 23.

A report says: “We asked 30 students in the canteen and found that those who ate more fruit scored higher, so eating fruit raises scores.” What would you question?

Sample: students in the canteen are not a random sample of the school.
Range: were there students who ate no fruit at all, or only a narrow band?
Cause: a link is not proof. Something else, such as breakfast habits or family income, could drive both.

Your go. The first step is given. The eight pairs (1, 6), (3, 9), (5, 12), (7, 13), (9, 17), (11, 20), (13, 24), (15, 27) are already sorted by x. Build the line through the two half-way mean points and predict y at x = 10.

The lower half is the first four pairs: x = 1, 3, 5, 7. Their mean x is 4.
Find the mean y of the lower half, then the mean point of the upper half, then the gradient.

Check the answer

The lower mean point is (4, 10) and the upper mean point is (12, 22). The gradient is 12 ÷ 8 = 1.5 and the intercept is 10 − 1.5 × 4 = 4, so y = 1.5x + 4. At x = 10: 19. A common slip is to join the first and last dots instead of the half-way mean points.

Check before you go on

1. A group wants to investigate whether the age of a used car is linked to its asking price. Which plan is most likely to give a trustworthy scatter graph?

Use the five cars owned by friends' families, since their ages and prices are known exactly. Take a random sample of 40 adverts for one model of car, and record each one's age and price. Use only adverts for cars less than a year old, because their prices are the most up to date. Use every car advert on one street, so that all the cars in the sample come from the same area.

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. The mean point of the lower half of a data set is (2, 15) and the mean point of the upper half is (8, 39). What is the gradient of the line through them?

3. A line has gradient 6 and passes through the point (3, 20). It is y = 6x + c. What is c?

Level 2 · Building

4. The line built from data with x from 1 to 9 is y = 3x + 11. Predict y when x = 5.

5. A school of 900 students in 5 year groups wants to know whether travel time to school is linked to hours of sleep. It will ask 45 students. Which plan is best?

Ask the 45 students who arrive earliest in the morning, as they are already at the gate. Ask the first 45 names on the school register, which lists every student in the school. Take a stratified sample by year group, in proportion, choosing at random within each year. Ask the 45 students who volunteer in assembly, as they will answer the questions carefully.

Level 3 · Stretch

6. The five year groups of the school have 200, 180, 220, 160 and 140 students. A stratified sample of 45 is taken in proportion. How many come from the year group of 220?

7. A report says: “Thirty volunteers aged 15 to 17 show that students who exercise more score higher in maths (line y = 1.2x + 48, for 1 to 8 hours a week). So exercising raises scores, and a student who exercises 20 hours will score 72.” Which critique is best?

There is nothing to question, because the line fits the data well and the sample has thirty students in it. The only problem is that 30 is a small sample, so the report should be repeated with many more students. The line should pass through the origin, because a student who does no exercise should score no marks. The sample is volunteers, 20 hours is far beyond the data, and a link does not show a cause.

Something to think about

Two investigations find the same line from different samples of the same size. One sample was chosen at random and one was every student in the library at lunch. Does the line deserve the same trust in each case? What would you want to know?

One way to think about it

Not the same trust. The lines match, but only the random sample gives a reason to think the line describes the whole group. For the library sample you would want to know whether library users differ from other students in ways that could change the link, and whether the library sample covers the whole range of the x values.

Exit ticket

8. A random sample of 40 students gives a scatter graph of daily screen time against hours of sleep, with a strong negative correlation, for screen times from 1 to 6 hours. Which conclusion is best supported?

Screen time causes students to sleep less, because the graph shows a strong negative correlation between them. Every student with more screen time sleeps less, because the sample shows a strong negative link. In this sample, more screen time went with less sleep, and the line is reasonable for 1 to 6 hours. A student with 12 hours of screen time will get no sleep, because the line keeps falling at the same rate.