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P6104: Introduction to Biostatistical Methods Autumn 2001 HW 1 Solutions 215 1 n xi = =8.6 days. The median 25 n i 1 hospital stay is 8 days. The Box plot for hospital stay indicates that the distribution is skewed to the right. Therefore, the median more accurately reflects central tendency (although there is not a very large difference between the median and the mean). 1) The mean hospital stay is x Box plot of hospital stay (in days): 40 30 7 20 10 0 N= 25 DURA TION 2) The standard deviation of hospital stay is 1 1 n (784) 5.7155. The range is 27 days (from a s ( xi x ) 2 n 1 i 1 24 minimum of 3 to a maximum of 30). 2 3) Stem-and-Leaf Plot for duration of hospital stay: Frequency Stem & Leaf 3.00 0 . 333 6.00 0 . 445555 3.00 0 . 677 5.00 0 . 88999 5.00 1 . 01111 .00 1. 1.00 1. 4 1.00 1. 7 1.00 Extremes (>=30) Stem width: 10 Each leaf: 1 case(s) Histogram of duration of hospital stay 14 12 10 8 6 4 Std. Dev = 5.72 2 Mean = 8 .6 N = 25.00 0 5.0 10.0 DURATION 15.0 20.0 25.0 30.0 4) Box-plots for men (1) and women (2) 40 7 30 DURATION 20 10 0 N= 11 14 1 2 SEX The distributions of duration of hospital stay are fairly similar for men and women. The median stay is slightly longer fro women (8.5 versus 7 days). The interquartile range (the difference between the 75th and 25th percentiles is slightly larger for men than for women (8 versus 6 days). There is one obviously outlying value (30 days) in the distribution for men. 5) E(Y) = 2E(X)+5 =2(100)+5 = 205. SD(Y)=2SD(X)=20. If we multiply a random variable by a constant, the mean of the new random variable is the mean of the original random variable times that constant. If we add a constant to a random variable, the mean of the new random variable is the mean of the original random variable plus that constant. If we add a constant to a random variable the standard deviation (and variance) is unchanged. If we multiply a random variable by a constant, the standard deviation of the new random variable is the standard deviation of the original random variable times the constant. The variance of the new random variable is the variance of the original random variable multiplied by the square of the constant. 6) Let y1,y2,...,yn be a sample of n observations. Prove that n (y i 1 i n (y y) 0 : i 1 i n n i 1 i 1 y ) y i y n y ny 0. n n n n n n n i 1 i 1 i 1 i 1 i 1 i 1 i 1 (c y) 2 (c 2 2c y y 2 ) c 2cy y 2 nc 2c y y 2 . Note that for the above we make use of the fact that if c is a n constant then: c nc , and i 1 that n n i 1 i 1 cxi c xi . We also used the fact n n n n i 1 i 1 i 1 i 1 (a b) a b , and the fact that nx xi .