The line of best fit is Y = aX + b
The normal equation are 
\( a \Sigma X^{2}+b \Sigma X =\Sigma X Y \)
\(a \Sigma X+n b =\Sigma Y \)
30a + 10b = 90 ..... (1)
10a+ 5b = 25 ...... (2)
Solving (1) and (2) we get
a = 4, b = -3
The line of best fit Y = 4X - 3 and y intercept = -3.

The line of best fet is Y = aX+b
\( \Sigma Y a= \Sigma X+n b \)
\(\Sigma X Y =a \Sigma X^{2}+b \Sigma X\)
115 75a + 6b ..............(1)
1875 1375a + 75b ................(2)
Solving (1) and (2)
a = 1, b = 6.6667
Y = X + 6.6667

The line of best fit is Y = aX+ b
The normal equations are
\( \Sigma Y =a \Sigma X+n b \)
\(\Sigma X Y =a \Sigma X^{2}+b \Sigma X\)
10a +7b = 16.9 .......... (1)
30a + 10b = 47.4 .........(2)
Solving (1) and (2) we get,
a = 1.48, b = 0.3
The line of best fit is Y = 1.482x + 0.3.
The x intercept of the line of best fit is \(-\frac{0.3}{1.48}\)

The line of best fit is Y = aX + b
The normal equations are
\( \sum Y =a \sum X+n b \)
\(\sum X Y =a \sum X^{2}+b \sum X \)
10a + 5b = 19 ............(1)
30a + 10b = 53 ..............(2)
Solving (1) and (2) we get
a = 1.5 and b = 0.8
The line of best fit is Y = 1.5X + 0.8.

 Grand average \( =\frac{252.33}{4}=63.08 \)
Seasonal Index (S.I) \( =\frac{\text { Quarterly Average }}{\text { Grand Average }} \times 100 \)
S.I for I Quarter \( =\frac{67}{63.08} \times 100 =106.21 \% \)
S.I for II Quarter  \( =\frac{58.33}{63.08} \times 100 =92.7 \% \)
S.I for III Quarter \( =\frac{63.33}{63.08} \times 100 =100.4 \% \)
S.I for IV Quarter  \( =\frac{63.67}{63.08} \times 100 =100.94 \% \)

Grand average = \(\frac{255}{4}=63.75\)
Seasonal Index (S.I) \( =\frac{\text { Quarterly Average }}{\text { Grand Average }} \times 100 \)
S.I for I Quarter \( =\frac{67}{63.75} \times 100=105.01 \)
S.I for II Quarter  \( =\frac{61}{63.75} \times 100=95.68 \)
S.I for III Quarter \( =\frac{6.3 .333}{6.3 .75} \times 100=99.35 \)
S.I for IV Quarter  = \(\frac{63.67}{63.75} \times 100=99.87 \)

\(\bar{\bar{X}}\) = \(\frac{11.2 + 11.8 + 10.8 + 11.6 + 11.0+ 9.6 + 10.4 + 9.6 + 10.6 + 10.0}{10}\)
\(=\frac{106.6}{10}=10.66\)
\(\bar{R}=\frac{7+4+8+5+7+4+8+4+7+9}{10}\)
\(=\frac{63}{10}=6.3\)
Control limits for mean chart
UCL = \(\bar{\bar{X}}\) +  A_{2^\(\bar{R}\)}
= 10.66 + .577(6.3) = 14.295
CL = \(\bar{\bar{X}}\) = 10.66
Control limits for R-chart
UCL =   D_{2^\(\bar{R}\)} = 2.115 \(\times\) 6.3
= 13.324
CL = _^\(\bar{R}\) = 6.3
LCL = D₃\(\bar{R}\) = 0

Cost of living index (C.L.I) = \(\frac{\Sigma PV}{\Sigma V}\)
= \(\frac{15800}{100}\) = 158
Hence, there is 58% increase in cost of living in 2003 compared to 2000.

C.L.I = \(\frac{\Sigma p_1q_0}{\Sigma p_0q_0}\times100\)

C.L.I = \(\frac{4362.50}{3505}\times100\) = 124.46

Grand average = \(\frac{67 + 61 + 63.33 + 63.37}{4}\)
= \(\frac{255}{4}=63.75\)
Seasonal index (S.I) = \(\frac{Quarterly average}{Grand average}\times100\)
Hence, S.I for I quarter = \(\frac{67}{63.75}\times100\) = 105.01
S.I for II quarter = \(\frac{61}{63.75}\times100\) = 95.68
S.I for III quarter = \(\frac{63.33}{63.75}\times100\) = 99.35
S.I for IV quarter = \(\frac{63.67}{63.75}\times100\) = 99.87

Let the straight line of best fit be y = ax + b.
The normal equations are
Σy = a Σx + nb
Σxy = a Σx² + bΣx
⇒ 10a + 7b    = 16.9 ....(1)
30a + 10b  = 47.4.... (2)

Substituting b = 0.3 in (2) we get
30a + 3 = 47.4 ⇒ 30a = 44.4
\(a=\frac{44.4}{30}=1.48\)
∴ The straight line trend is y = 1.48x + 0.3