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Published on: 23/06/2021
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Take MCQ Business Maths and Statistics Test
1.
The following data shows the value of sample mean (\(\bar{X}\)) and the range R for 10 samples of size 5 each. Calculate the control limits for : mean chart and range chart.
| Sample No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Mean \(\bar{X}\) | 11.2 | 11.8 | 10.8 | 11.6 | 11.0 | 9.6 | 10.4 | 9.6 | 10.6 | 10.0 |
| Range | 7 | 4 | 8 | 5 | 7 | 4 | 8 | 4 | 7 | 9 |
(Given for n = 5, A2 = .577, D3 = 0, D4 = 2.115)
2.
Construct the cost of living index for 2003 on the basis of 2000 from the following data using family budget method.
| Item | Price(Rs.) | Weights | |
| Food | 2000 | 2003 | 30 |
| Rent | 200 | 280 | 30 |
| Clothing | 150 | 120 | 20 |
| Fuel & lighting | 50 | 100 | 10 |
| Miscellaneous | 100 | 200 | 20 |
3.
Calculate the seasonal indices by the method of simple average for the following data.
| Year | I quarter | II quarter | III quarter | IV quarter |
| 1985 | 68 | 62 | 61 | 63 |
| 1986 | 65 | 58 | 66 | 61 |
| 1987 | 68 | 63 | 63 | 67 |
4.
Calculate the 3-yearlymoving averages of the production figures (in tonnes) for the following data.
| Year | 1973 | 1974 | 1975 | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 1984 | 1985 | 1986 | 1987 |
| Production | 15 | 21 | 30 | 36 | 42 | 46 | 50 | 56 | 63 | 70 | 74 | 82 | 90 | 95 | 102 |
5.
Using the method ofleast squares, fit a straight line trend for Σx = 10, Σy = 16.9, Σx2 = 30, Σxy = 47.4 and n = 7.
1.
\(\bar{\bar{X}}\) = \(\frac{11.2 + 11.8 + 10.8 + 11.6 + 11.0+ 9.6 + 10.4 + 9.6 + 10.6 + 10.0}{10}\)
\(=\frac{106.6}{10}=10.66\)
\(\bar{R}=\frac{7+4+8+5+7+4+8+4+7+9}{10}\)
\(=\frac{63}{10}=6.3\)
Control limits for mean chart
UCL = \(\bar{\bar{X}}\) + A2\(\bar{R}\)
= 10.66 + .577(6.3) = 14.295
CL = \(\bar{\bar{X}}\) = 10.66
Control limits for R-chart
UCL = D2\(\bar{R}\) = 2.115 \(\times\) 6.3
= 13.324
CL = \(\bar{R}\) = 6.3
LCL = D3\(\bar{R}\) = 0
2.
| Items | p0 | p1 | Weights V | \(P=\frac{p_1}{p_0}\times100\) | PV |
| Food | 200 | 280 | 30 | 140 | 4200 |
| Rent | 100 | 200 | 20 | 200 | 4000 |
| Clothing | 150 | 120 | 20 | 80 | 1600 |
| Fuel & Lighting | 50 | 100 | 10 | 200 | 2000 |
| Miscellaneous | 100 | 200 | 20 | 200 | 4000 |
| 100 | 15800 |
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.
3.
| Year | I quarter | II quarter | III quarter | IV quarter |
| 1985 | 68 | 62 | 61 | 63 |
| 1986 | 65 | 58 | 66 | 61 |
| 1987 | 68 | 63 | 63 | 67 |
| Total | 201 | 183 | 190 | 191 |
| Average | 67 | 61 | 63.33 | 63.67 |
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
4.
| Year | Production | 3-yearly moving total | 3-yearly moving average |
| 1973 | 15 | - | - |
| 1974 | 21 | 22.00 | |
| 1975 | 30 | 66 | 29.00 |
| 1976 | 36 | 87 | 36.00 |
| 1977 | 42 | 108 | 41.33 |
| 1978 | 46 | 124 | 46.00 |
| 1979 | 50 | 138 | 50.67 |
| 1980 | 56 | 152 | 56.33 |
| 1981 | 63 | 169 | 63.00 |
| 1982 | 70 | 189 | 69.00 |
| 1983 | 74 | 207 | 75.33 |
| 1984 | 82 | 226 | 82.00 |
| 1985 | 90 | 246 | 89.00 |
| 1986 | 95 | 267 | 95.67 |
| 1987 | 102 | 287 | - |
5.
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
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