Practice section 2 of 6

For each dataset, find (i) the correlation coefficient $r$ and comment on how it relates to the shape of the scatter diagram, (ii) the regression of $y$ on $x$, and (iii) the regression of $x$ on $y$.

(a)

$x$	$y$
3	5
7	12
9	5
11	12
14	10
14	17
15	23
21	16
22	10
23	20
26	25

(b)

$x$	$y$
0.6	5
1	10
2	15
2.5	10
2.8	2.5
3.6	7.5
4	2.5
4	5
4	15
5	10

A random sample of five employees has ages $x$ years and monthly earnings $y$ thousand dollars.

Employee	$x$	$y$
A	19	3.16
B	21	3.21
C	26	3.34
D	28	4.03
E	32	3.99

(i) Alex uses $y=0.1x+1.8$. Calculate the sum of squared residuals for Alex’s line.

(ii) Bertrand uses another straight line whose sum of squared residuals is 2. State, with a reason, which model gives a better fit.

Eight observed pairs are:

$x$	$y$
3	7
4	8
6	9
10	11
11	20
12	19
15	21
17	22

It is given that $r=0.93$, the regression of $y$ on $x$ is $y=1.188x+3.042$, and the regression of $x$ on $y$ is $x=0.736y-1.009$.

(a) Taking $x$ as the independent variable, estimate (i) $y$ when $x=14$ and (ii) $x$ when $y=11$.

(b) Now taking $y$ as the independent variable, estimate (i) $y$ when $x=14$ and (ii) $x$ when $y=11$.

(c) With neither variable identified as independent, estimate (i) $y$ when $x=14$ and (ii) $x$ when $y=11$.

(d) Use an appropriate regression line to estimate $x$ if the independent variable $y$ is correctly determined as 30. Explain why this is not a good estimate.

Ten randomly selected students have these French test marks $y$ and English test marks $x$.

$x$ (English)	$y$ (French)
19	20
42	43
44	33
52	56
51	50
53	67
66	73
56	68
60	77
37	43

(i) Obtain the product moment correlation between $y$ and $x$ and comment on it.

(ii) Obtain the regression line of $y$ on $x$, correct to 3 decimal places.

(iii) Given that the regression coefficient for $x$ on $y$ is 0.66, show by calculation that this line is $x=0.66y+13.02$.

(iv) Tom missed the French test but scored 65 in English. Predict a suitable French mark and comment on its reliability.

After a school physical fitness test, ten pupils have the following masses $m$ (kg) and overall scores $s$.

Pupil	$m$	$s$
A	38.6	60
B	41.8	58
C	38.6	72
D	54.1	59
E	44.9	62
F	32.1	94
G	38.4	86
H	41.4	62
I	30.2	81
J	52.4	54

(i) Calculate the product moment correlation coefficient between $m$ and $s$ for this sample.

(ii) Given $u=10(m-30)$ and $v=s-50$, calculate the product moment correlation coefficient between $u$ and $v$ for this sample.

A researcher gave eight children lists of words of varying difficulty and asked each child to define them. The ages $x$ (years) and numbers of correctly defined words $y$ are:

Child	$x$	$y$
A	2.5	9
B	3.1	13
C	4.3	18
D	5.0	25
E	5.9	35
F	7.1	53
G	8.1	81
H	9.4	132

(i) Sketch the scatter diagram and comment on whether a linear regression of $y$ on $x$ is suitable.

(ii) Calculate the product moment correlation coefficient between $\ln y$ and $\ln x$. Comment on the suitability of $\ln y=a+b\ln x$.

(iii) Find the least-squares regression of $\ln y$ on $\ln x$. Sketch it on the scatter diagram of $\ln y$ against $\ln x$.