Stat 428 Autumn 2006 Homework 2 Solutions

Transcription

1 Section 6.3 (5, 8) Here is the Minitab output for the service time data set. Descriptive Statistics: Service Times Service Times Variable Q3 Maximum Service Times The center of the distribution of the service time can be described by the sample mean of X = seconds, the trimmed mean of seconds, or the sample median of 66 seconds. The sample standard deviation is s=17.59 seconds, and the service times range from the smallest observation of seconds to the largest observation of 186 seconds. The lower sample quartile is 61 seconds and the upper quartile is 76 seconds. In the boxplot, the service times appear slightly positively skewed and Minitab has flagged 186, 177, 143,135,5, and 28 as outliers. 0 Boxplot of Service Times 0 Service Times

2 6.3.8 Some sample statistics for paving slab weights are given below. Descriptive Statistics: Paving Slab Weights Paving Slab Weig Variable Q3 Maximum Paving Slab Weig The sample mean of the weights is X = 1. 15kg while the trimmed mean is kg and the median is 1.10kg. The three summary statistics for the center of the distribution are very similar. The spread of the distribution of slab weights can be summarized by the sample standard deviation, s=0.05kg. The smallest observation is kg and the largest observation is 1.70kg. The lower sample quartile is 1.08kg and the upper quartile is kg The boxplot shows that the distribution of the slab weights is almost symmetric and the smallest observation kg and the largest observation 1.70kg are considered as outliers. 1.3 Boxplot of Paving Slab Weights 1.2 Paving Slab Weights

3 Section 6.5 (2, 5) First, one may want to see if there is any year to year difference in the number of accidents. For this comparison, some summary statistics and side-by-side boxplots are obtained. The boxplots suggest that the distribution for Year 1 is roughly symmetric except for an outlier while that for Year 2 is positively skewed. So, there seems to be a change in the shape of the distributions. In Year 1, there were more accidents than Year 2 on average as summarized by sample statistics,19.17 vs in the sample means or vs in the sample medians. In terms of spread of the number of accidents, the sample standard deviations 6.83 and 5.35 for Year 1 and 2, respectively suggest that there was more variability in Year 1 than Year 2. However, this may be due to the influence of the outlier in Year 1. As a robust measure of the spread of a distribution, the sample IQRs were computed for the two distributions, and they turned out to be the same (7.75) for this data set. Descriptive Statistics: Year 1, Year 2 Q3 Year Year Variable Maximum IQR Year Year Boxplot of Year 1, Year 2 35 Data Year 1 Year 2 3

4 Additionally, one may be also interested in whether there is any seasonal effect or not. For this, we calculate the averaged number of accidents for each month in the two years and create a new variable named Average for Each Month. A bar chart for monthly average number of accidents is given below. Chart of Average for Each Month vs Month Average for Each Month 5 0 January February March April May June July Month August September October November December There does not appear to be any notable seasonal effects although there may possibly be a correlation from one month to the next Some numerical summaries of the distribution of the heights are given below. The sample mean of the heights is X = and the sample standard deviation is s= The heights of the bamboo shoots range from 11.8 to 43.9 with the lower sample quartile,.4 and the upper quartile, Descriptive Statistics: Bamboo Shoot Heights Bamboo Shoot Hei Variable Q3 Maximum IQR Bamboo Shoot Hei Also, a stem and leaf plot and a boxplot as graphical summaries of the data are given as follows. 4

5 Stem-and-Leaf Display: Bamboo Shoot Heights Stem-and-leaf of Bamboo Shoot Heights N = 40 Leaf Unit = (22) Boxplot of Bamboo Shoot Heights 40 Bamboo Shoot Heights 35 Both the stem-and-leaf plot and the boxplot show that the distribution is roughly symmetric or mildly negatively skewed, and there are some potential outliers (11.8, 14.1, 22.2, and 43.9) as indicated by the asterisks in the box plot. 5