Quantitative Applications project

QASS Midterm Exam Review
The Quantitative Applications in Security Sciences midterm exam assess your master level in CILO 1.1 (Assumptions), CILO 1.2 (Transformation), CILO 2.1 (Arithmetic), and CILO 2.2 (Graphs).
Q1 (CILO 1.1 Assumptions).
Students were given different memory supplement treatments before revising for their exams. Some were given memory supplements, some placebo supplements, and some no treatment. The exam scores (%) are shown below for the three different groups:
Memory Supplements
Placebo
No Treatment
70
37
3
77
43
10
83
50
17
90
57
23
97
63
30
Mean
83.40
50.00
16.67
Standard Deviation (StDev)
10.60
10.44
10.60
The distributions of the exam scores (%) are shown below using box and whisker plots for the three different groups:
Assume the head of department wishes to perform a one-way ANOVA on this data, to test the hypothesis that some groups are more successful than others.
Task: Identify multiple assumptions relevant to the one-way ANOVA test. Perform tests for these assumptions and fully discuss the results of the tests.
Sample Answer
There are three primary assumptions in ANOVA:
1. Observations are drawn from normally distributed populations.
2. Variances of the populations are equal.
3. Observations represent random samples from the populations.
1. As we can see from shown Box and Whisker Plots, the data is relatively symmetric, The box and Whisker plots shows no obvious violations from Normal, but we should proceed with caution.
2. The summary statistics for the three brands are presented in the given question:
Variable Mean Standard Deviation
Memory Supplements 83.40 10.60
Placebo 50.00 10.44
No Treatment 16.67 10.60
The smallest standard deviation is 10.44, and twice the value is 20.88. The largest standard deviation 10.60 is less than this value 20.88. Since the sample sizes are the same, it is safe to assume the standard deviations (and thus the variances) are equal.
3. The samples were taken independently, so there is no indication that this assumption is violated.
Q2 (CILO 1.1 Assumptions).
Students were given different memory supplement treatments before revising for their exams. Some were given memory supplements, some placebo supplements, and some no treatment. The exam scores (%) are shown below for the three different groups:
Memory Supplements
Placebo
No Treatment
57
54
22
70
57
28
83
48
30
70
57
40
70
69
30
Mean
70
57
30
Median
70
57
30
Standard Deviation (StDev)
9.19
7.65
6.48
Assume the head of department wishes to perform a one-way ANOVA on this data, to test the hypothesis that some groups are more successful than others.
Task: Identify multiple assumptions relevant to the one-way ANOVA test. Perform tests for these assumptions and fully discuss the results of the tests.
Sample Answer
There are three primary assumptions in ANOVA:
1. Observations are drawn from normally distributed populations.
2. Variances of the populations are equal.
3. Observations represent random samples from the populations.
1. As we can see from different memory supplements treatments data: Memory Supplements: Mean = 70, median = 70, Mode = 70 Placebo: Mean = 57, median = 57, Mode = 57 No Treatment: Mean = 30, median = 30, mode = 30
The normal distribution has mean = median = mode. We say that responses for each factor level have a normal distribution.
2. The summary statistics for the three brands are presented in given question:
Variable Mean Standard Deviation
Memory Supplements 70 9.19
Placebo 57 7.65
No Treatment 30 6.48
The smallest standard deviation is 6.48, and twice the value is 12.96. The largest standard deviation 9.19 is less than this value 12.96. Since the sample sizes are the same, it is safe to assume the standard deviations (and thus the variances) are equal.
3. The samples were taken independently, so there is no indication that this assumption is violated.
Q3 (CILO 1.2 Transformation) An investment analyst wanted to determine if average monthly income distribution (in SAR) differ among Riyadh, Taif, and Dammam. He randomly selected eleven employees’ annual income from each place obtained the following data:
Monthly Income in Riyadh (SAR)
Monthly Income in Taif (SAR)
Monthly Income in Dammam (SAR) 5000 4000 15,000 40,000 38,000 4500 6000 7000 11,000 8000 39,000 38,000 45,000 17,000 36,000 22,000 15,000 20,000 14,000 10,000 18,000 10,000 15,000 39,000 16,000 21,000 6000 11,000 40,000 41,000 42,000 18,000 9000
Visual inspection of the given data suggest that they do not meet the assumptions of parametric statistical tests. Assume that transformation is needed on untransformed data.
Task: Describe the most appropriate options for data transformations for monthly income in Riyadh, apply the selected transformation and explain why that particular choice of transformation was selected.
Sample Response:
The given data do not meet the assumptions of parametric statistical tests. Then sometimes it can be useful to transform the data so that the transformed data is more normal or at least roughly symmetric.
There are many transformations that we can use for statistical analysis of data; here are two of the most common:
(1) Log transformation
(2) Square-root transformation I will apply log transformation on the given monthly income in Riyadh untransformed data. This consists of taking the log of each observation. I can also use Square Root transformation for the given data because there are no negative numbers. If I have negative numbers, I can’t take the square root
Here are 11 numbers from the monthly income in Riyadh data set; the first column is the untransformed data, the second column is the base-10 logarithm of the number in the first column.
Monthly Income in Riyadh Untransformed Data
Monthly Income in Riyadh Log Transformation 5000 3.699 40,000 4.602 6000 3.778 8000 3.903 45,000 4.653 22,000 4.342 14,000 4.146 10,000 4 16,000 4.204 11,000 4.041 42,000 4.623
I do the statistics on the log transformed numbers. For example, the mean of the log transformed data is 4.181 and median is 4.146. The mean is approximately close to the median. The mean and median are close together. In this case, applying the log transformation makes the data more normal. A distribution that is symmetric or nearly so is often easier to handle and interpret than a skewed distribution.
Q4 (CILO 2.1 Arithmetic & CILO 2.2 Graphs or Tables)
Suppose a researcher decides to analyze the effects of the machine operator on the valve opening measurements of valves produced in a manufacturing plant. Suppose further that four different operators operate the machines. A researcher wants to identify “Is there a significant difference in the mean valve openings of 24 valves produced by the four operators”? Table below contains the valve opening measurements for valves produced under each operator.
(Note: SSE = 0.15492, SSC = 0.23698, dfC = C – 1, dfE = N – C)
Task
Complete all mathematical calculations needed to perform the ANOVA statistical test, showing all working in a clear and standardized format. Draw the table as appropriate, including all data points, all appropriate labels and providing any summary statistics.
Sample Response:
From these computations, an analysis of variance chart can be constructed, as shown in Table below. The observed F value is 10.18.