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Consider the following sample of 44 obscrvations: 8.9; 12.4; 8.6; 11.3; 9.2; 8.8; 8.8; 6.2; .07; 7.1; 11.8; 10.7; 7.6;9.1; 9.2; 8.2; 9.0; 8.7;9.1; 10.9; 10.3; 9.6; 7.8; 11.5; 9.3; 7.9; 8.8; 12.7; 8.4; 7.8; 5.7; 10.5; 10.5;9.6; 8.9; 10.2; 10.3; 7.7; 10.6; 8.3; 8.8; 9.5; 8.8; 9.4. 1. Find the mcan and the standard deviation for the data given 2. Calculate the interval for y± ks for k E {1, 2, 3). C ount the number of mcasureents that fall within cach interval and compare this result with the mber that you would expect according to the empirical rule for normal approximated data

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(a) The formula for calculating the population mean is 89+12.4+. . +9.4) 44(39857) 9.058409 y 90584 Therefore, the sample mean is y9584 The formula for calculating the standard deviati on is i-1 4 44-11 9-9.058409)2+(124-9.058409)+.+94-9.058409)2) 1.2Σ(-0. 15841)2 + (3.341591)2 + +(0.34 1591)2) (0.025093+11.16623+...+0.116684) 43(175.9648 A 092204 2.02292 s, s 2.02292 Therefore, the sample stan dard deviation is s, 1 2.02292 (b) The population mean is y-9.058409 The standard deviation is s 2.02292. The interval for ke(1,2,3 The value of the interval is when k = 1 yks 9.058409+(1)2.02292 9.058409 +2.02292 11.08133 11.08133 The value of the interval is when k 1 y- 9.058409-(1)2.02292 = 9.058409-2.02292 7.035489 y-s 7035489 The number of measurements in the interval calculated is 36 According, to the empirical rule for the normal approximated data k-1. Approximately 68% of the measurements lies within the interval y ks,

(c) The population mean is y -9.058409 The standard deviation is s 2.02292. The interval for k e(1,2,3 The value of the interval is when k 2. yks 9.058409+(2) 2.02292 9.058409 +2 (2.02292) 9.058409+4.04584 13.10425 +s13.10425 The value of the interval is when k 2. y- ks 9.058409 (2)2.02292 9.058409-4.04584 5.01257 y501257 The number of measurements in the interval calculated is 43 rk-2. Approximately 95% of the measurements lies within the interval y ks (d) The population mean is y 9058409 The standard deviation is s 2.02292. The interval for k e(1,2,3 The value of the interval is when k-3 y+k 9.058409+(3) 2.02292 9.058409+3(2.02292) 9.058409 +6.068759 15.12717 15 12717 The value of the interval is when k 3 y- ks 9.058409 (3)2.02292 9.058409- 6.068759 2.98965 y- s2.98965 The number of measurements in the interval calculated is 43 If k-3. Approximately 99.7% of the measurements lies within the interval y ks

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