Scatter graphs and predictions

(a) Negative

1. As age increases, value tends to decrease.

• B1 Correct answer: Negative.

(b) No correlation

1. For adults, shoe size and spelling have no link, so expect no correlation.

• B1 Correct answer: No correlation.

(a) Older bicycles tend to have lower resale prices.

1. Negative correlation means that larger values of one variable tend to go with smaller values of the other.
2. Older bicycles tend to have lower resale prices. Correlation alone does not establish the cause of that relationship.

• B1 Correct answer: Older bicycles tend to have lower resale prices.

(a) $33$

1. Find the gradient from two points on the line.
   $$\frac{45-13}{8}$$
2. Move five hours along the line from x = 2.
   $$13+5\times 4$$
3. Therefore $33$.

• M1 Find the gradient from two points on the line.
• M1 Move five hours along the line from x = 2.
• A1 Correct answer: $33$

(b) 25 hours lies outside the observed range, so this is extrapolation; the relationship may change beyond the recorded data.

1. Therefore 25 hours lies outside the observed range, so this is extrapolation; the relationship may change beyond the recorded data.

• C1 Correct conclusion with supporting reasoning: 25 hours lies outside the observed range, so this is extrapolation; the relationship may change beyond the recorded data.

(a) $5$

1. Divide change in score by change in study time.
   $$\frac{55-15}{10-2}$$
2. Therefore $5$.

• M1 Divide change in score by change in study time.
• A1 Correct answer: $5$

(b) No. An observed association does not prove causation; other factors could affect both study time and scores.

1. No. An observed association does not prove causation; other factors could affect both study time and scores.

• C1 Correct conclusion with supporting reasoning: No. An observed association does not prove causation; other factors could affect both study time and scores.

(a) $-2$

1. Divide the change in value by the change in age.
   $$\frac{4-16}{7-1}$$
2. Therefore $-2$.

• M1 Divide the change in value by the change in age.
• A1 Correct answer: $-2$

(b) The value of a car falls by about £2000 for each extra year of age.

1. The value of a car falls by about £2000 for each extra year of age.

• C1 Correct conclusion with supporting reasoning: The value of a car falls by about £2000 for each extra year of age.

(c) £$8000$

1. Move 4 years along the line from x = 1: the value falls by 2 thousand pounds each year.
   $$16-2\times 4=8$$
2. $y$ is in thousands of pounds, so the value is £8000.

• M1 Move 4 years along the line from x = 1.
• A1 Correct answer: £$8000$

(a) On sunnier days the shop tends to sell fewer umbrellas.

1. On sunnier days the shop tends to sell fewer umbrellas.

• C1 Correct conclusion with supporting reasoning: On sunnier days the shop tends to sell fewer umbrellas.

(b) No. Correlation does not show cause; rain on days with little sunshine is a more likely reason for umbrella sales.

1. No. Correlation does not show cause; rain on days with little sunshine is a more likely reason for umbrella sales.

• C1 Correct conclusion with supporting reasoning: No. Correlation does not show cause; rain on days with little sunshine is a more likely reason for umbrella sales.

(c) It is an outlier. Check it is not a recording error, and do not let it pull the line of best fit away from the pattern of the other points.

1. It is an outlier. Check it is not a recording error, and do not let it pull the line of best fit away from the pattern of the other points.

• C1 Correct conclusion with supporting reasoning: It is an outlier. Check it is not a recording error, and do not let it pull the line of best fit away from the pattern of the other points.

(a) For each extra 1 m/s of speed, the stopping distance increases by about 3.2 m.

1. The gradient is the change in $y$ for each increase of 1 in $x$.

• C1 Stopping distance goes up about 3.2 m for each 1 m/s of speed.

(b) No: the line does not pass through the origin (intercept $-15$).

1. Direct proportion means $y = kx$: a straight line through $(0, 0)$.
2. This line has intercept $-15$, so doubling the speed does not double the distance.

• C1 "No", because the line does not pass through the origin (for example 10 m/s gives 17 m but 20 m/s gives 49 m, not 34 m).

(c) 49 m

1. $3.2 \times 20 - 15 = 49$.

• B1 Correct answer: 49 m.

(a) $56$ g

1. $$\frac{66-42}{29-17}$$
2. $$42+2\times (24-17)$$
3. The gradient of the line is (66 $-$ 42)/(29 $-$ 17) = 24/12 = 2 g per cm.
4. The length 24 cm is 7 cm above 17 cm, so the predicted mass rises by 2 $\times$ 7 = 14 g.
5. The estimated mass is 42 + 14 = 56 g. This is interpolation because 24 cm is within the observed range.

• P1 Establishing $\frac{66-42}{29-17}$ or an equivalent valid method.
• P1 Establishing $42+2\times (24-17)$ or an equivalent valid method.
• A1 Correct answer: $56$ g