Practice section 1 of 6

Identify the independent and dependent variables in each source situation:

1. Investigate whether increased calcium intake would help prevent osteoporosis.
2. Investigate how the concentration of an injected drug in the bloodstream varies with time.
3. Investigate the time taken to complete a chemical reaction when different amounts of catalyst are used.
4. Record air temperature and altitude at noon on a particular day at eight mountainous locations.
5. Tabulate the distances of seven cities from the capital and the corresponding railway fares.

The source records the time for blood sugar to return to normal after insulin was injected into a diabetic patient.

Insulin (milli-units)	Time (minutes)
40	4.10
60	4.07
95	3.98
100	3.90
125	3.81
160	3.75
180	3.72
210	3.77
240	3.63
260	3.57
500	3.30

Sketch a scatter diagram with insulin injected as the independent variable. Give a brief description of any observed pattern.

The data show the intensity $I$ of radiation from a substance after time $t$.

$t$	$I$
0.2	3
0.4	3.5
0.6	1.3
0.8	0.6
1.0	0.4
1.2	0.2

(i) Use a suitable diagram to explain why the product moment correlation coefficient between $I$ and $t$ is negative and not very close to $-1$.

(ii) By using the formula for $r$, calculate the product moment correlation coefficient between $t$ and $I$.

(iii) State the coordinates of a point in the diagram in (i) which does not seem to follow the linear trend. If it is removed, without any calculations, explain why $r$ is closer to $-1$.

Eight pairs of observations are given below.

$x$	$y$
16	7
13	17
16	12
15	11
16	18
12	10
17	17
15	12

(i) Find the regression line of $y$ on $x$.

(ii) Find the regression line of $x$ on $y$.

(iii) On a scatter diagram, sketch the two regression lines.

Use your answers in (i) and (ii) to find $\bar x$ and $\bar y$.

When $p$ and $q$ are known, the following data give the regression line of $y$ on $x$ as $y=2x-2$.

$x$	$y$
3	$p$
2	2
3	6
6	$q$
1	0

(i) Find $\bar y$.

(ii) Find $\sum xy$.

(iii) Find $p$ and $q$.

The average weekly study hours $x$ and corresponding test marks $y$ of ten students are recorded below.

Student	$x$	$y$
A	24	6
B	17	30
C	20	40
D	41	76
E	52	96
F	23	46
G	46	90
H	17	35
I	15	26
J	29	57

(i) Calculate the product moment correlation coefficient between $y$ and $x$.

(ii) Determine the least-squares regression line of $y$ on $x$.

(iii) Three marks were wrongly recorded: A should receive 35, I should receive 6, and E should receive 99. Using appropriate diagrams, comment on how these changes affect (a) the gradient of the regression of $y$ on $x$ and (b) the correlation coefficient.