Statistics for Managerial Decisions Academic Essay

Question 1 4 Marks
(a) List all the values in a table and then construct a stem-and-leaf display for the data. 1 mark
(b) Construct a relative frequency histogram for these data with equal class widths, the first class being “\$4 to less than \$6”. 1 mark
(c) Briefly describe what the histogram and the stem-and-leaf display tell you about the data. What effects would there be if the class width is doubled, which means the first class will be “\$4 to less than \$8”? 1 mark
(d) What proportion of stock prices were above \$10? 1 mark

Question 2 4 Marks
The following table provides the median weekly rents of a 3-bedroom house of a few randomly selected suburbs in four capital cities of Australia – Sydney, Melbourne, Brisbane and Perth – for March 2016. The data is obtained from the website https://www.realestate.com.au/neighbourhoods/. From the data answer the questions below for the capital cities.

(a) Compute the mean, median, first quartile, and third quartile for each capital city (with only the data provided for that city, do not add/delete values for any new/given suburb of question 2) using the exact position, (n+1)f, where n is the number of observations and f the relevant fraction for the quartile. 1 mark
(b) Compute the standard deviation, range and coefficient of variation from the sample data for each city. 1 mark
(c) Draw a box and whisker plot for the median weekly rents of each city and put them side by side on the same scale so that the prices can be compared. 1mark
(d) Compare the box plots and comment on the distribution of the data. 1 mark

Question 3 4 Marks
The Table below is taken from the Australian Bureau of Statistics website. It provides data on energy use of households – almost all houses use mains electricity but some use additional energy sources such as gas and solar. (You can get the data from Table 1 from the URL: http://www.abs.gov.au/AUSSTATS/abs@.nsf/DetailsPage/4602.0.55.001Mar%202014?OpenDocument.) Missing cells indicate data not published, and totals may be higher since all possibilities may not be listed. The totals are not incorrect.).

Based on the information available in the table above –
(a) What is the probability that an Australian household, randomly selected, uses solar as a source of energy? 1 mark
(b) What is the probability that an Australian household, randomly selected, uses mains gas and is located in Victoria? 1 mark
(c) Given that a household uses LPG/bottled gas, what is the probability that the household is located in South Australia? 1 mark
(d) Is the percentage of Australian households using mains gas independent of the state? 1 mark

Question 4 4 Marks
(a) The following data collected from the Australian Bureau of Meteorology Website (http://www.bom.gov.au/climate/data/?ref=ftr) gives the daily rainfall data for the year 2015 in Brisbane. The zero values indicate no rainfall and the left-most column gives the date. Assuming that the weekly rainfall event (number of days in a week with rainfall) follows a Poisson distribution (There are 52 weeks in a year and a week is assumed to start from Monday. The first week starts from 29 December 2014 – you are expected to visit the website and get the daily values which are not given in the table below. Make sure you put the correct station number. Ignore the last few days of 2015 if it exceeds 52 weeks.):
(i) What is the probability that on any given week in a year there would be no rainfall? 1 mark
(ii) What is the probability that there will be 2 or more days of rainfall in a week? 1 mark

(Question 4 continued next page)
(Question 4 continued)
(b) Assuming that the weekly total amount of rainfall (in mm) from the data provided in part (a) has a normal distribution, compute the mean and standard deviation of weekly totals.
(i) What is the probability that in a given week there will be between 5 mm and 10 mm of rainfall? 1 mark
(ii) What is the amount of rainfall if only 13% of the weeks have that amount of rainfall or higher? 1 mark

Question 5 4 Marks
The following data is taken from the UCI machine learning data repository (https://archive.ics.uci.edu/ml/datasets/Wine+Quality). It lists a few attributes of red wine, randomly sampled from thousands of bottles, which can be classified as of good, medium and poor quality.
(a) Test for normality of all the variables separately for good wine using normal probability plot. 2 marks
(b) Construct a 95% confidence interval for each of the variables for good wine. 1 mark
(c) Find the mean of each of the variables for medium quality red wine. Do the same for the poor quality red wine. 1/2 mark
(d) Check if the means calculated for the medium and poor quality red wines fall within the corresponding confidence intervals of the good quality wine. For those attributes whose means lie outside the confidence interval, the attributes are significant in determining the quality. This assumption is, however, partially compromised if the attribute fails the normality test. Identify the significant and non-significant variables, and comment. 1/2 mark

Good quality red wine
Alcohol Residual sugar Chlorides Total sulfur dioxide Density pH Sulphates Citric acid
10 1.2 0.065 21 0.9946 3.39 0.47 0
9.5 2 0.073 18 0.9968 3.36 0.57 0.02
10.5 1.8 0.092 103 0.9969 3.3 0.75 0.56
9.7 2.1 0.066 30 0.9968 3.23 0.73 0.28
9.5 1.9 0.085 35 0.9968 3.38 0.62 0.16
10.5 1.8 0.065 16 0.9962 3.42 0.92 0.16
13 1.2 0.046 93 0.9924 3.57 0.85 0.08
10.3 1.4 0.056 24 0.99695 3.22 0.82 0.47
10.8 2.6 0.095 28 0.9994 3.2 0.77 0.74
10.8 2.6 0.095 28 0.9994 3.2 0.77 0.74
10.5 2.1 0.054 19 0.998 3.31 0.88 0.58
12.2 1.6 0.054 106 0.9927 3.54 0.62 0.04
9.2 2.2 0.075 24 1.00005 3.07 0.84 0.44
9.2 2.2 0.075 24 1.00005 3.07 0.84 0.44
10.5 2.6 0.085 33 0.99965 3.36 0.8 0.47
10.2 1.8 0.071 10 0.9968 3.2 0.72 0.52
12.8 3.6 0.078 37 0.9973 3.35 0.86 0.46
12.6 6.4 0.073 13 0.9976 3.23 0.82 0.45
10.5 5.6 0.087 47 0.9991 3.38 0.77 0.32
9.9 3.5 0.358 10 0.9972 3.25 1.08 0.68
10.5 5.6 0.087 47 0.9991 3.38 0.77 0.32
10.6 2.5 0.091 49 0.9976 3.34 0.86 0.09
10.6 2.5 0.091 49 0.9976 3.34 0.86 0.09
11.5 3.2 0.083 59 0.9989 3.37 0.71 0.39
11.5 3.2 0.083 59 0.9989 3.37 0.71 0.39
11.5 3.65 0.121 14 0.9978 3.05 0.74 0.66
11.7 2.5 0.078 38 0.9963 3.34 0.74 0.01
12.2 3.4 0.128 21 0.9992 3.17 0.84 0.53
9.8 2.3 0.082 29 0.9997 3.11 1.36 0.54
12.3 2.7 0.072 34 0.9955 3.58 0.89 0.02
11.7 2.95 0.116 29 0.997 3.24 0.75 0.66
10.4 3.1 0.109 23 1 3.15 0.85 0.66
10 5.8 0.083 42 1.0022 3.07 0.73 0.66
10 5.8 0.083 42 1.0022 3.07 0.73 0.66
12 2.4 0.074 18 0.9962 3.2 1.13 0.53
11.8 4.4 0.124 15 0.9984 3.01 0.83 0.71
12 2.4 0.074 18 0.9962 3.2 1.13 0.53
10 2.5 0.096 49 0.9982 3.19 0.7 0.31
12.9 1.4 0.045 88 0.9924 3.56 0.82 0.05
13 4.2 0.066 38 1.0004 3.22 0.6 0.76
10.8 3 0.093 30 0.9996 3.18 0.63 0.66
11.7 6.7 0.097 19 0.9986 3.27 0.82 0.53
11.8 2.4 0.089 67 0.9972 3.28 0.73 0.33
12.3 2.3 0.059 48 0.9952 3.52 0.56 0.03
11 2.1 0.066 24 0.9978 3.15 0.9 0.47
12.3 2.3 0.059 48 0.9952 3.52 0.56 0.03
11 2.1 0.066 24 0.9978 3.15 0.9 0.47
9.8 2.2 0.072 29 0.9987 2.88 0.82 0.72
11.2 3.7 0.1 43 1.0032 2.95 0.68 0.76
11.6 2.7 0.077 19 0.9963 3.23 0.63 0.49
12.5 1.7 0.054 27 0.9934 3.57 0.84 0.01
11.2 2.8 0.084 22 0.9998 3.26 0.74 0.63
13.4 5.2 0.086 19 0.9988 3.22 0.69 0.67
11.2 2.8 0.084 22 0.9998 3.26 0.74 0.63
11.7 2.8 0.08 17 0.9964 3.15 0.92 0.56
10.8 2.8 0.081 67 1.0002 3.32 0.92 0.55
13.3 2.5 0.055 25 0.9952 3.34 0.79 0.5
13.4 2.6 0.052 27 0.995 3.32 0.9 0.51
11 2.6 0.07 16 0.9972 3.15 0.65 0.53
11 2.6 0.07 16 0.9972 3.15 0.65 0.53
12 6.55 0.074 76 0.999 3.17 0.85 0.73
12 6.55 0.074 76 0.999 3.17 0.85 0.73
10.9 1.9 0.078 24 0.9976 3.18 1.04 0.47
10.8 1.8 0.077 22 0.9976 3.21 1.05 0.42
12.5 2.9 0.072 26 0.9968 3.16 0.78 0.63
10.8 1.8 0.075 21 0.9976 3.25 1.02 0.46
11.4 2.8 0.084 43 0.9986 3.04 0.68 0.75
11.8 2.4 0.107 15 0.9973 3.09 0.66 0.64
11.8 2.4 0.107 15 0.9973 3.09 0.66 0.64

Medium quality red wine
alcohol residual sugar chlorides total sulfur dioxide density pH sulphates citric acid
9.4 1.9 0.076 34 0.9978 3.51 0.56 0
9.8 2.6 0.098 67 0.9968 3.2 0.68 0
9.8 2.3 0.092 54 0.997 3.26 0.65 0.04
9.4 1.9 0.076 34 0.9978 3.51 0.56 0
9.4 1.8 0.075 40 0.9978 3.51 0.56 0
9.4 1.6 0.069 59 0.9964 3.3 0.46 0.06
10.5 6.1 0.071 102 0.9978 3.35 0.8 0.36
9.2 1.8 0.097 65 0.9959 3.28 0.54 0.08
10.5 6.1 0.071 102 0.9978 3.35 0.8 0.36
9.9 1.6 0.089 59 0.9943 3.58 0.52 0
9.1 1.6 0.114 29 0.9974 3.26 1.56 0.29
9.2 3.8 0.176 145 0.9986 3.16 0.88 0.18
9.2 3.9 0.17 148 0.9986 3.17 0.93 0.19
9.3 1.7 0.368 56 0.9968 3.11 1.28 0.28
9.7 2.3 0.082 71 0.9982 3.52 0.65 0.31
9.5 1.6 0.106 37 0.9966 3.17 0.91 0.21
9.4 2.3 0.084 67 0.9968 3.17 0.53 0.11
9.3 1.4 0.08 23 0.9955 3.34 0.56 0.16
9.5 1.8 0.08 11 0.9962 3.28 0.59 0.24
9.5 1.6 0.106 37 0.9966 3.17 0.91 0.21
9.4 1.9 0.08 35 0.9972 3.47 0.55 0
10.1 2.4 0.089 82 0.9958 3.35 0.54 0.07
9.8 2.3 0.083 113 0.9966 3.17 0.66 0.12
9.2 1.8 0.103 50 0.9957 3.38 0.55 0.25
10.5 5.9 0.074 87 0.9978 3.33 0.83 0.36
10.5 5.9 0.074 87 0.9978 3.33 0.83 0.36
10.3 2.2 0.069 23 0.9968 3.3 1.2 0.22
9.5 1.8 0.05 11 0.9962 3.48 0.52 0.02
9.2 2.2 0.114 114 0.997 3.25 0.73 0.43
9.5 1.6 0.113 37 0.9969 3.25 0.58 0.52
9.2 1.6 0.066 12 0.9958 3.34 0.56 0.23
9.2 1.4 0.074 96 0.9954 3.32 0.58 0.37
9.2 1.7 0.074 23 0.9971 3.15 0.74 0.26
9.4 3 0.081 119 0.997 3.2 0.56 0.36
9.5 3.8 0.084 45 0.9978 3.34 0.53 0.04
9.6 3.4 0.07 10 0.9971 3.04 0.63 0.57
9.4 5.1 0.111 110 0.9983 3.26 0.77 0.12
10 2.3 0.076 54 0.9975 3.43 0.59 0.18
9.2 2.2 0.079 52 0.998 3.44 0.64 0.4
9.3 1.8 0.115 112 0.9968 3.21 0.71 0.49
9.8 2 0.081 54 0.9966 3.39 0.57 0.05
10.9 4.65 0.086 11 0.9962 3.41 0.39 0.05
10.9 4.65 0.086 11 0.9962 3.41 0.39 0.05
9.6 1.5 0.079 39 0.9968 3.42 0.58 0.11
10.7 1.6 0.076 15 0.9962 3.44 0.58 0.07
10.7 2 0.074 65 0.9969 3.28 0.79 0.57
9.5 2.1 0.088 96 0.9962 3.32 0.48 0.23
9.5 1.9 0.084 94 0.9961 3.31 0.48 0.22
9.6 2.5 0.094 83 0.9984 3.28 0.82 0.54
10.5 2.2 0.093 42 0.9986 3.54 0.66 0.64
10.5 2.2 0.093 42 0.9986 3.54 0.66 0.64
10.1 2 0.086 80 0.9958 3.38 0.52 0.12
9.2 1.6 0.069 15 0.9958 3.41 0.56 0.2
9.4 1.9 0.464 67 0.9974 3.13 1.28 0.7
9.1 2 0.086 73 0.997 3.36 0.57 0.47
9.4 1.8 0.401 51 0.9969 3.16 1.14 0.26

Poor quality red wine
alcohol residual sugar chlorides total sulfur dioxide density pH sulphates citric acid
9 2.2 0.074 47 1.0008 3.25 0.57 0.66
8.4 2.1 0.2 16 0.9994 3.16 0.63 0.49
10.7 4.25 0.097 14 0.9966 3.63 0.54 0
9.9 1.5 0.145 48 0.99832 3.38 0.86 0.42
11 3.4 0.084 11 0.99892 3.48 0.49 0.02
10.9 2.1 0.137 9 0.99476 3.5 0.4 0
9.8 1.2 0.267 29 0.99471 3.32 0.51 0
10.2 5.7 0.082 14 0.99808 3.4 0.52 0.05
9.95 1.8 0.078 12 0.996 3.55 0.63 0.02
9 4.4 0.086 29 0.9974 3.38 0.5 0.08
9.8 1.5 0.172 19 0.994 3.5 0.48 0.09
9.3 2.8 0.088 46 0.9976 3.26 0.51 0.3
13.1 2.1 0.054 65 0.9934 3.9 0.56 0.15
9.2 2.1 0.084 43 0.9976 3.31 0.53 0.26
9.1 1.5 0.08 119 0.9972 3.16 1.12 0.2
10.5 1.4 0.045 85 0.9938 3.75 0.48 0.04
9.4 3.4 0.61 69 0.9996 2.74 2 1
9.2 1.3 0.072 20 0.9965 3.17 1.08 0.02
9 1.6 0.072 42 0.9956 3.37 0.48 0.03
9.1 1.8 0.058 8 0.9972 3.36 0.33 0.03
11.4 2.1 0.061 31 0.9948 3.51 0.43 0.06
10.4 2 0.089 55 0.99745 3.31 0.57 0.36
9.4 2 0.087 67 0.99565 3.35 0.6 0.04
9.8 3.3 0.096 61 1.00025 3.6 0.72 0
9.6 4.5 0.07 49 0.9981 3.05 0.57 0.49
9.6 2.1 0.07 47 0.9991 3.3 0.56 0.49
10 2.3 0.103 14 0.9978 3.34 0.52 0.24
10 2.1 0.088 23 0.9962 3.26 0.47 0.27
11.3 3.4 0.105 86 1.001 3.43 0.64 0.22
11 2.2 0.07 14 0.9967 3.32 0.58 0.01
11 4.4 0.096 13 0.997 3.41 0.57 0.02
9.6 2.6 0.073 84 0.9972 3.32 0.7 0.48
9.7 1.6 0.078 14 0.998 3.29 0.54 0.04
11.2 3.1 0.086 12 0.9958 3.54 0.6 0.1
11.4 2.1 0.102 7 0.99462 3.44 0.58 0.24
10.9 2.5 0.058 9 0.99632 3.38 0.55 0.07
9.9 1.6 0.147 51 0.99836 3.38 0.86 0.44

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