Survey and Data Analysis - Assignment Example

Research Methods BUSM027 Assessment 2 December 2009 Survey and Data Analysis Standard Deviation The closest value for the standard deviation is 8.4. The reason for this is that all of the data values lie within 2 standard deviation of the mean (25 ± 8.4). According to Chebyshev’s Theorem , for any data set, no matter how it is distributed, the percentage of observations that lie within 2, and 3 standard deviations of the mean must be at least 75%, and 88.9% , respectively. 2) Singers a) We might expect approximately a normal distribution of heights for a mixed voice (ie, men and women) choir. Table 1: The heights (in inches) of 130 members of a choir Height 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 Count 2 6 9 7 5 20 17 7 10 5 14 8 9 4 2 4 1 The data for four extra members who have joined the choir are as below 62 + x = 62 + 4 = 66 62 + y = 62 + 0 = 62 66 + z = 66 + 2 = 68 66 + x = 66 + 4 = 70 b) The amended frequency table is given below Table 2: The heights (in inches) of 134 members of a choir Height (x) 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 Count (f) 2 6 10 7 5 20 18 7 11 5 15 8 9 4 2 4 1 Cumulative Count 2 8 18 25 30 50 68 75 86 91 106 114 123 127 129 133 134 The 5 Number Summary Minimum = 60 Maximum = 76 The first quartile, Q1 = the value of the = 33.75th observation From table 2 it can be seen that the 33rd and 34th observations have value 65 inches. Therefore, Q1 = 65 inches Median, Q2 = the value of the = 67.5th observation = the mean value of 67th and 68th observations From table 2 it can be seen that the 67th and 68th observations have value 66 inches. Therefore, Q2 = 66 inches The third quartile, Q3 = the value of the = 101.25th observation From table 2 it can be seen that the 101st and 102nd observations have value 70 inches. Therefore, Q3 = 70 inches The five number summary is (60, 65, 66, 70, 76). c) Figure 1 shows the distribution (histogram) of the heights of 134 members of a choir. Figure 1: Distribution (Histogram) of the heights of 134 members of a choir c) As shown in figure1, there appears that the distribution of the heights of 134 members of a choir is approximately normally distributed (slightly positively skewed). The five number summary (60, 65, 66, 70, 76) also suggest that the distribution of heights is slightly positively skewed. 3) Drug Abuse a) Figure 2 shows the scatterplot of the percent of teens who have used marijuana and who have used other drugs. Figure 2: Scatterplot of teens who have used marijuana against other drugs The correlation between the percent of teens who have used marijuana and who have used other drugs is 0.9341. b) The data suggests that there is a strong positive linear association between the percent of teens who have used marijuana and who have used other drugs. There is no outlier in data as all data points falls around linear trend line. Therefore, there is no need of exclusion of any particular country. c) Theses results confirm that marijuana is a ‘gateway drug’ - that is that marijuana use leads to the use of other drugs as there is strong positive association between the percent of teens who have used marijuana and who have used other drugs. d) We might seek data for most of the countries for the percent of teens who have used marijuana and who have used other drugs for a correlation hypothesis test (or hypothesis test of slope) as additional evidence if we were asked to carry out further investigation of whether marijuana is a ‘gateway drug’. 4) Repair Shop a) Labelling the population average level of emissions of cars as μ and the stipulated level as μ0, we set μ = μ0 as the null hypothesis (H0) and test it against an alternative (H1) that μ < μ0. The null and alternate hypotheses are H0: μ = μ0 H1: μ < μ0 The research question is stated in alternate hypothesis. b) Type I error: false positive- the average level of emissions of cars is not less than the stipulated level, however, test indicates less than the stipulated level. Type II error: false negative- the average level of emissions of cars is less than the stipulated level, however, test indicates not less than the stipulated level. c) The power of the test would be greater if they tested a sample of 40 instead of 20 cars. The reason of this is that if all other values are fixed than power of a test increases as sample size increases. A larger sample gives us a more accurate estimate, and thus a greater chance of detecting differences. d) If the repair shop is only a little out of compliance (opposite to the alternative direction) than the power of test will be greater (the probability of Type II error β will be small) as compared to when the repair shop is a lot out of compliance (opposite to the alternative direction). 5) Road Accidents a) Figure 3 shows the numbers killed in road accidents in London in the period 2000-2003. The percentage of pedestrian, cyclists, and motorcyclists killed were about 60%, 10%, and 30%, respectively. The type of vehicle involved in accident was HGV-19%, LGV-10%, and Car-71%. The type of person killed in road accidents (pedestrian, cyclists, and motorcyclists) and the type of vehicle involved in accident are associated, χ2(4, N = 633) = 63.87, p < .001. Figure 3: Numbers killed in road accidents in London in the period 2000-2003 The data suggest that most of the time car is involved in accident (killing) and majority of the casualty is of pedestrian. Therefore, it is advice to pedestrian to take more care from cars when they are walking on a road or crossing a road. This does not mean that they ignore HGV and LGV. b) Figure 4 shows the numbers killed in road accidents by goods vehicles (both HGH and LGV) in London in the period 2000-2003. Figure 4: Numbers killed in road accidents by goods vehicles in London, 2000-2003 The percentage of pedestrian, cyclists, and motorcyclists killed by good vehicles were about 54%, 22%, and 24%, respectively. Majority (64%) of the times the type of goods vehicle involved in accident was HGV. c) The sample size taken for the partial analysis of numbers killed in road accidents in London in the period 2000-2003 is large (n = 633). Therefore, the data is sufficient to interpret and make any conclusion. However, the data is taken from a particular geographical region. It would be better if the data is recent and taken (cluster sample) from all geographical regions to make any conclusion for present trend. Chi-square Test of Independence The null and alternate hypotheses are H0: The type of person killed in road accidents (pedestrian, cyclists, and motorcyclists) and the type of vehicle involved in accident are independent. H1: The type of person killed in road accidents (pedestrian, cyclists, and motorcyclists) and the type of vehicle involved in accident are dependent. Using a 5% level of significance (α = 0.05), the critical value of χ2 with 4 df is 9.488, therefore, we reject H0 if observed χ2 > 9.488. Table 3 shows the Chi-square Contingency Table Test for Independence. The value of observed χ2 is 63.87, so we reject H0 in favour of H1. In conclusion, the type of person killed in road accidents (pedestrian, cyclists, and motorcyclists) and the type of vehicle involved in accident are dependent. Table 3: Chi-square Contingency Table Test for Independence Pedestrian Cyclist Motor Cyclist Total HGV Observed 64 35 20 119 Expected 71.06 12.22 35.72 119.00 (O - E)² / E 0.70 42.47 6.92 50.09 LGV Observed 35 6 25 66 Expected 39.41 6.78 19.81 66.00 (O - E)² / E 0.49 0.09 1.36 1.94 Car Observed 279 24 145 448 Expected 267.53 46.00 134.47 448.00 (O - E)² / E 0.49 10.52 0.82 11.84 Total Observed 378 65 190 633 Expected 378.00 65.00 190.00 633.00 (O - E)² / E 1.69 53.08 9.10 63.87 63.87 chi-square 4 df Read More