\(\bar{\bar{X}}\) = \(\frac{11.2 + 11.8 + 10.8 + 11.6 + 11.0+ 9.6 + 10.4 + 9.6 + 10.6 + 10.0}{10}\)
\(=\frac{106.6}{10}=10.66\)
\(\bar{R}=\frac{7+4+8+5+7+4+8+4+7+9}{10}\)
\(=\frac{63}{10}=6.3\)
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

The statistical tool applied in process control is the Control Chart. Control Charts are the devices to describe the patterns of variation. It is an instrument to be used in specitication, production and inspection and is the core of statistical quality control. It is essentially a graphic device, simple to construct and easy to interpret.

(i) It indicates whether the real wages of workers are rising or falling for a given time.
(ii) It is used by the administrators for regulating dearness allowance or grant of bonus to the workers.

Fisher's price index number is the geometric mean of Laspeyre's and Paasche's price index number. Hence it is weighted index number.
Fisher's price index number = \(\sqrt {\frac {\sum p_{1}q_{0}}{\sum p_{0}q_{0} }}{\times}{\frac {\sum p_{1}q_{1}}{\sum p_{0}q_{1}} \times {100}}\)

It is a general tendency of time series to increase or decrease or stagnates during a long period of time. An upward tendency is usually observed in population of a country, production, sales, prices in industries, income of individuals etc., A downward tendency is observed in deaths,epidemics, prices of electronic gadgets, water sources, mortality rate etc. It is not necessarily that the increase or decrease should be in the same direction throughout the given period of time. This feature is known as secular trend.

(i) It helps in the analysis of the past behavior
(i) It helps in forecasting and for future plans
(ii) It helps in the evaluation of current achievements
(iv) It helps in making comparative studies between one time period and others

Fisher's price index number
\(P^{F}_{01}\) =  \(\sqrt \frac {\sum p_{1}q_{0}\times \sum p_{1}q_{1}}{{\sum p_{0}q_{0}\times \sum p_{0}q_{1}}}\) × 100
=  \(\sqrt \frac {97\times117}{116\times140} \times 100\)
= \(\sqrt \frac {11349}{16240} \times 100\)
= \(\sqrt {0.6988}\times100\)
\(P^{F}_{01}\)   =  83.59