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3-3 example10

# 3-3 example10 - Student Grady Silnonton Course Math119...

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Unformatted text preview: Student: Grady Silnonton Course: Math119: Elementary Statistics - Spring 2010 - CRN: 49239 Instructor: Shawn Parvini - 16 weeks Date: 2/18/10 Book: Triola: Elementary Statistics, 11e Time: 11:54 ANI Listed below are the nicotine amounts (in mg per cigarette) for samples of ﬁltered and nonﬁltered cigarettes. Find the coefficient of variation for each of the two sets of data, then compare the variation. Nonﬁltered 1.1 1.? 1.5 1.3 0.9 1.7 1.0 1.5 1.0 1.4 1.2 1.0 1.1 1.3 0.9 2.1 1.5 1.0 1.4 1.6 1.3 1.1 1.2 1.0 1.3 Filtered 0.2 1.0 0.2 1.0 0.7 0.5 1.2 0.8 1.0 1.0 0.8 1.1 1.0 0.8 0.1 1.4 1.0 0.8 1.2 0.9 0.? 1.0 0.5 1.2 1.2 S The coefficient of variation is given as CV = = - 100%, where s is the sample standard deviation, and E is the sample x mean . First, calculate the sample means. The mean, i, of a sample is found with the formula below, where x is the variable used to represent the individual data values, and n is the number of data values in the sample. 2 x <— sum of all data values x = n <— number of data values Let x1 be the nicotine amounts (in mg per cigarette) for samples of nonﬁltered cigarettes. Calculate £1. £1 = 1.284 Let X; be the nicotine amounts (in mg per cigarette) for samples of ﬁltered cigarettes. Calculate £2. £2 = 0.852 Next, calculate the sample standard deviations. — 2 x — x — The standard deviation, s, of a sample is found by using the formula 5 = ﬂ—l) , where x is the mean of the n _ sample, and n is the size of the sample. For x1, calculate the ﬁrst value of (x — E). (1.1— 1.284) —0.184 (x—;) Calculate (x - ;) for the remaining data values. Page 1 Student: Grady Silnonton Course: Mathll9: Elementary Statistics - Spring 2.010 - CRN: 49239 Instructor: Shawn Parvini - 16 weeks Date: 2/18/10 Book: Triola: Elementary Statistics, lle Time: 11:54 AMI Calculate (x — i) 2 for the ﬁrst data value. (X—;)2 = (1.1—1.284)2 = ( — 0.184)2 = 0.0339 Calculate (x — i) 2 for the remaining data values. Now ﬁnd the sum of the values of (x — E) 2. 2(x — i) 2 = 2.0947 Finally, divide the sum of the values of (x — E) 2 by (11— l) to ﬁnd the variance, where n is the sample size. Then take the square root to ﬁnd the standard deviation, rounding to three decimal places. .1: [Zea—EV n-l 2.0947 25 — l = 0.295 Repeat the calculations for the other sample. Page 2 Student: Grady Silnonton Course: Math119: Elementary Statistics - Spring 2.010 - CRN: 49239 Instructor: Shawn Parvini - 16 weeks Date: 2/18/10 Book: Triola: Elementary Statistics, 11e Time: 11:54 ANI Recall that the mean of X; is 0.852. Calculate (x — E) for the data values. Now ﬁnd the sum of the values of (x — ;) 2. 2(x — E) 2 = 2.?223 Finally, divide the sum of the values of (x — i) 2 by (n— l) to ﬁnd the variance, where n is the sample size. Then take the square root to ﬁnd the standard deviation, rounding to three decimal places. 51: 2(X_;)2 n-l = f 27223 25 — 1 = 0.33? Now that s and K have been calculated for both data sets, the coefficient of variation can be found and the variations can be compared. 5 _ Recall that the coefﬁcient of variation is given as CV = = - 100%, where s is the sample standard deviation, and x is the x Page 3 Student: Grady Silnonton Course: Math119: Elementary Statistics - Spring 2.010 - CRN: 49239 Instructor: Shawn Parvini - 16 weeks Date: 2/18/10 Book: Triola: Elementary Statistics, 11e Time: 11:54 ANI sample mean. Calculate the coefﬁcient of variation for the two samples, rounding to three decimal places. 51 s2 CV1 = ——- 100% CV2 = ——- 100% X1 X2 0 295 0.337 = —- 100% = — 100% 1.284 0.852 = 2.975% = 39.554% When comparing variation in two different sets of data, the standard deviation should be compared only if the two sets of data use the same scale and units and they have the same mean. If the means are substantially different, or if the samples use different scales or measurement units, use the coefficient of variation to compare the variation in the data. The coefficient of variation for the nicotine amounts (in mg per cigarette) for samples of nonﬁltcred cigarettes is approximately 2.975%, and the coefﬁcient of variation for the nicotine amounts (in mg per cigarette) for samples of ﬁltered cigarettes is approximately 39.554%. By comparing the coefficients of variation, it can be concluded that the nicotine amounts of nonﬁltered cigarettes have considerably less variation than the nicotine amounts of ﬁltered cigarettes. Page 4 ...
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