\(\bar{\bar{X}}\) = 634.5, \(\bar{R}\) = 226.4
Control limits for \(\overline { X } \) - chart are
UCL = \(\bar{\bar{X}}\) + A₂\(\bar{R}\)
= 634.5+ 0.483x 226.4
= 743.85
CL = 634.5
LCL = \(\bar{\bar{X}}\)- A₂\(\bar{R}\) = 525.15
Control limits of R-chart are
UCL = D₄\(\bar{R}\) = 2.004 x226.4
= 453.7056
CL = 226.4
LCL = D_3\(\bar{R}\) = 0
\(\overline { X } \)-chart

R - chart

Conclusion: Since one point in R-chart lie outside the control limits, the given system is not in control

Fisher's Ideal Index = \(\sqrt\frac{{\Sigma p_1q_0}\times{\Sigma p_1q_1}}{{\Sigma p_0q_0}\times{\Sigma p_0q_1}}\times100\)
\(= \sqrt\frac{1380\times1500}{510\times571}\times100\)
= 266.61
Time reversaltest:
\(P_{01}\times P_{10}=\sqrt\frac{{\Sigma p_1q_0}\times{\Sigma p_1q_1}\times{\Sigma p_0q_1}\times{\Sigma p_0q_0}}{{\Sigma p_0q_0}\times{\Sigma p_0q_1}\times{\Sigma p_1q_1}\times{\Sigma p_1q_0}}\)

= \(\sqrt{1}=1\)
Hence, time reversal test is satisfied.
Factor reversaltest:
\(P_{01}\times Q_{01}=\sqrt\frac{{\Sigma p_1q_0}\times{\Sigma p_1q_1}\times{\Sigma q_1p_0}\times{\Sigma q_1p_1}}{{\Sigma p_0q_0}\times{\Sigma p_0q_1}\times{\Sigma q_0p_0}\times{\Sigma q_0p_1}}\)

\(P_{01}\times Q_{01}=\frac{{\Sigma p_1q_1}}{{\Sigma p_0q_0}}\)
Hence, Fisher's ideal index satisfies factor reversal test also.

(i) Laspeyre's index number
\(P_{01}^{L} = \frac{\Sigma p_1q_0}{\Sigma p_0q_0}\times100\)
\(=\frac{200}{160}\times100=125\)
(ii) Paasche's Price index number
\(P_{01}^{P} = \frac{\Sigma p_1q_1}{\Sigma p_0q_1}\times100\)
\(=\frac{130}{103}\times100=126.21\)
(iii) Fisher's price index number
\(P_{01}^{F} =\sqrt {{P_{01}^{L}}\times{P_{01}^{P}}}=125.6\)

Grand average = \(\frac{42.4+36.2+37.8+40.2}{4}\)
= 39.15
Seasonal Index (S.1) = \(\frac{Quarterlyaverage}{Grand average}\times 100\)
Hence,
S.1 or I quarter = \(\frac{42.4}{39.15}\times100=108.30\)
S.1 or II quarter = \(\frac{36.2}{39.15}\times100=92.54\)
S.1 or III quarter = \(\frac{37.8}{39.15}\times100=96.55\)
S.1 or IV quarter = \(\frac{40.2}{39.15}\times100=102.68\)

Let the required equation of the straight line trend is
y = a + bX
Since Σx = 0. \(a = \frac{\Sigma y}{x} = \frac{16.9}{5}\)
\(b = \frac{\Sigma xy}{\Sigma x^2} = \frac{13.3}{10} = 1.33\)
Hence, the straight line trend is
y = 3.38 + 1.33 (x - 2002)
∴ The trend for 2007 is
y_t = 3.38 + 1.33 (2007 - 2002)
⇒  y_t = 3.38 + 1.33 (5)
⇒  y_t = 3.38 + 6.65
⇒  y_t = 10.03