Discussion questions · Section 1 of 2

A student obtains eight pairs of sample data $(x,y)$ and summarises them as follows:

$$\sum x=360,\quad\sum x^2=21830,\quad\sum y=140,$$ $$\sum y^2=3500,\qquad\sum xy=3985.$$

Calculate the product moment correlation coefficient. What can you say about the relationship between $x$ and $y$?

The monthly incomes of eight former classmates of 2T03 in CJC are tabulated against their Advanced Level Examination aggregates.

Aggregate $x$	Monthly income $y$ (dollars)
72	4400
55	3550
65	4000
45	3000
60	3800
50	3200
42	2850
62	3950

(i) Obtain a scatter diagram and calculate the linear correlation coefficient.

(ii) Comment on the claim: “The strong correlation between a person’s aggregate in his Advanced Level Examination and his subsequent monthly income shows that good results determine one’s income.”

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

An engineering company makes cranes. The numbers sold $x$ in each three-month period for two years and the profits $y$ (thousand dollars) on those sales are:

$x$	$y$
15	290
17	350
13	270
21	430
16	340
22	410
14	300
18	360

(i) Sketch the scatter diagram as shown on your calculator.

(ii) Find $\bar x$ and $\bar y$ and mark their point on the scatter diagram.

(iii) Calculate the regression of $y$ on $x$ and draw it on the scatter diagram.

(iv) Calculate the product moment correlation coefficient and comment on it in relation to the scatter.

(v) Estimate the profit corresponding to a target of 20 cranes for the next three-month period.

(vi) Comment on the validity of using the regression from (iii) to predict profit if 40 cranes were sold in a three-month period.

(a) Eight pairs of measured values $(x,y)$ are given. Sketch a possible scatter diagram for each case: (i) correlation approximately zero; (ii) correlation approximately $-0.8$.

(b) Monthly earnings $y$ thousand dollars of seven workers of ages $x$ years are:

$x$	$y$
18	2.55
20	2.65
22	2.85
27	3.15
35	4.76
45	5.45
55	6.26

(i) Sketch the scatter diagram as shown on your calculator.

(ii) Find the product moment correlation coefficient.

(iii) Find the regression of $y$ on $x$ as $y=mx+c$, with $m,c$ correct to 4 decimal places.

(iv) Estimate monthly earnings of a 40-year-old worker. State why you would expect the estimate to be reliable.

(v) Every worker receives an increase of $N$ thousand dollars per month. Without further calculations state the expected changes in $m$ and $c$.

The table gives world-record one-mile times, measured in seconds above 3 minutes 30 seconds, as at 1 January in various years.

Year $x$	Excess time $t$ (seconds)
1930	40.4
1940	36.4
1950	31.3
1960	24.5
1970	21.1
1980	19.0
1990	16.3
2000	13.1

(i) Draw a scatter diagram.

(ii) Refer to both the scatter and context to comment on whether a linear model is appropriate.

(iii) Explain why a quadratic model would probably be inappropriate for long-term predictions.

(iv) Fit $\ln t=a+bx$ and predict the world-record time as at 1 January 2010. Comment on reliability.

(a) Traffic engineers compare traffic flow on a busy main road with air pollution at a nearby monitoring station. Each hour, sensors report the average traffic flow $x$ (vehicles per hour) for the preceding hour; the station supplies a pollution reading $y$ in a suitable unit, with higher readings indicating more pollution. A random sample of eight hours gives:

$x$	$y$
1796	1.0
1918	2.2
2120	3.5
2315	4.2
2368	4.3
2420	4.5
2588	4.9
2850	5.5

(i) Draw the scatter diagram, labelling the axes.

Two proposed models are $y=a+bx$ and $y^2=c+dx$.

(ii) Find the correlation coefficient between (a) $x$ and $y$, and (b) $x$ and $y^2$.

(iii) Use (i) and (ii) to explain which model is better.

(iv) Find a suitable regression equation and use it to estimate $y$ when $x=2000$.

(v) Comment on the validity of the headline “Heavy traffic causes air pollution” in light of these results.

(b) A partly destroyed research paper retains these statements: the mean of $x$ is 4.4; the regression of $y$ on $x$ is $y=2.5x+3.8$; the regression of $x$ on $y$ begins $x=1.5y-\ldots$. Calculate the missing constant at the end of the second equation.