IBDP Maths AI: Topic: SL 4.4: Linear correlation of bivariate data: IB style Questions HL Paper 1

Detailed Solution

Given Data:

(a) (i) Find Pearson’s product-moment correlation coefficient

The formula for Pearson’s correlation coefficient is:

$$r = \frac{n \sum (t_i m_i) – \sum t_i \sum m_i}{\sqrt{[n \sum t_i^2 – (\sum t_i)^2] [n \sum m_i^2 – (\sum m_i)^2]}}$$

where:

• $t_i$

  and

  $m_i$

  are the given values,

• $n = 5$

  (number of data points).

Let’s calculate

.

The Pearson’s product-moment correlation coefficient

is approximately 0.995, indicating a very strong positive correlation between study time and marks obtained.

(a) (ii) Find the least squares regression line of

The least squares regression line is given by:

$$m = a + bt$$

where:

• $b = \frac{n \sum (t_i m_i) – \sum t_i \sum m_i}{n \sum t_i^2 – (\sum t_i)^2}$

  (slope)

• $a = \frac{\sum m_i}{n} – b \frac{\sum t_i}{n}$

  (intercept)

Let’s calculate

and

.

The least squares regression line of

on

is:

$$m = 43.88 + 10.60t$$

(b) Predicting Additional Marks for Fatima

If Fatima studied an extra 1.5 hours, the increase in marks can be estimated using the slope

from our regression equation:

$$\text{Additional Marks} = b \times 1.5$$

If Fatima had studied an extra 1.5 hours, she would have achieved approximately 15.90 more marks.

(c): One reason the value 15.9 might not be valid is that the linear regression model assumes a constant increase in marks per hour of study, but in reality, additional study time might yield diminishing returns due to factors like fatigue or saturation, and the model may not account for individual variability in Fatima’s study effectiveness.

…………………………………..Markscheme………………………………….

Solution: –

(a) (i) r = 0.995 (0.994705…)

(ii) m = 10.6t + 43.9 (10.6032…t + 43.8780…)

(b) EITHER
$10.6032…\times1.5$
OR
(10.6032…(t+1.5)+43.8780…)-(10.6032…(t)+43.8780…)
THEN

15.9 (marks) (15.9048…)

(c) Accept any valid reason

e.g:

The students in the sample might not be of equal ability / she has not controlled for ability. She might have originally obtained close to full marks so an extra 15.9 would not be possible.