STA3022F2018Test1.docx

# T stat p value intercept 21965 21195 1036 0300 x1

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t Stat P-value Intercept 21.965 21.195 1.036 0.300 x1 (fixed acidity) 0.025 0.026 0.963 0.336 x2 (volatile acidity) -1.084 0.121 -8.948 0.000 x3 (citric acid) -0.183 0.147 -1.240 0.215 x4 (residual sugar) 0.016 0.015 1.089 0.276 x5 (chlorides) -1.874 0.419 -4.470 0.000 x6 (free sulfur dioxide) 0.004 0.002 2.009 0.045 x7 (total sulfur dioxide) -0.003 0.001 -4.480 0.000 x8 (density) -17.881 21.633 -0.827 0.409 x9 (pH) -0.414 0.192 -2.159 0.031 x10 (sulphates) 0.916 0.114 8.014 0.000 x11 (alcohol) 0.276 0.026 10.429 0.000 --- Residual standard error: 0.648 on 1587 degrees of freedom Multiple R-squared: 0.6005, Adjusted R-squared 0.3561 F-statistic: 81.35 on 11 and 1587 DF, p-value: 1.791e-145 Question 2 [13 marks] Longley (1967) provides a macro-economic data set measuring 7 economic variables observed yearly from 1947 to 1962. The variables are GNP.deflator GNP implicit price deflator (1954 = 100) GNP Gross National Product Unemployed number of unemployed (1000’s) Armed.Forces number of people in the armed forces (1000’s) Population population aged 14 years and older (1000’s) Year the year when the measurements were made: 1947… 1962 Employed number of people employed (1000’s) > longley GNP.deflator GNP Unemployed Armed.Forces Population Year Employed 1947 83.0 234.289 235.6 159.0 107.608 1947 60.323 1948 88.5 259.426 232.5 145.6 108.632 1948 61.122 1949 88.2 258.054 368.2 161.6 109.773 1949 60.171 1950 89.5 284.599 335.1 165.0 110.929 1950 61.187 1951 96.2 328.975 209.9 309.9 112.075 1951 63.221 1952 98.1 346.999 193.2 359.4 113.270 1952 63.639 1953 99.0 365.385 187.0 354.7 115.094 1953 64.989 1954 100.0 363.112 357.8 335.0 116.219 1954 63.761 1955 101.2 397.469 290.4 304.8 117.388 1955 66.019 1956 104.6 419.180 282.2 285.7 118.734 1956 67.857 1957 108.4 442.769 293.6 279.8 120.445 1957 68.169 1958 110.8 444.546 468.1 263.7 121.950 1958 66.513

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1959 112.6 482.704 381.3 255.2 123.366 1959 68.655 1960 114.2 502.601 393.1 251.4 125.368 1960 69.564 1961 115.7 518.173 480.6 257.2 127.852 1961 69.331 1962 116.9 554.894 400.7 282.7 130.081 1962 70.551 (a) What is the main justification for standardising a data set before performing a principal component analysis (PCA). Use the given data as an example in your explanation. (2) (b) The following output was obtained from a PCA, excluding the variable Year. > summary(princomp(datalongley, cor = TRUE)) Importance of components: Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Standard deviation 2.12957 1.08935 0.50190 0.12328 0.10205 0.02656 > round(princomp(datalongley, cor = TRUE)\$loadings,3) Loadings: Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 GNP.deflator 0.467 0.666 -0.530 -0.232 GNP 0.467 0.164 -0.202 -0.186 0.824 Unemployed 0.306 -0.622 -0.672 0.244 Armed.Forces 0.212 0.774 -0.584 0.114 Population 0.466 -0.684 -0.227 -0.500
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