During the manufacture of certain electrical components, items go through a series of heat processes. The length of time spent in this heat treatment is related to the life expectation of the component. To find the nature of this relationship a sample of 10 components was selected from the process and tested to destruction. The results are presented in the table below:
Table 1 Data – Heat Treatment times/Life expectancy
|Heat Treatment time (minutes)||25||27||25||26||31||30||32||29||30||44|
|Length of Life (hours)||2005||2157||2347||2239||2889||2942||3048||3002||2943||3844|
|Adjusted R Square||0.817114395|
|Coefficients||Standard Error||t Stat||P–value||Lower 95%||Upper 95%|
|Heat Treatment Time||91.56229686||14.26295877||6.419586||0.000204818||58.67185496||124.4527388|
Figure 1 Excel output: Regression Analysis
(a) Identify and apply two different approaches to determine whether there is a correlation between Heat Treatment time and Length of Life. Explain your findings.
(b) Use the output in Figure 1 to identify the regression model and to predict the life expectancy of a component which spends 33 minutes in the Heat Treatment process.
(c) Determine the coefficient of determination, also referred to as R2 or RSquare for the regression model used in your answer in (b) and discuss its meaning in context of the example data.
A course administrator has collected data on students including the following variables: final unit mark in QMDA, final unit mark in Economics for Business, attendance rate of seminars, distance between student accommodation and University campus, and age. The course administrator then calculated a number of correlation coefficients between these variables last year’s student cohort. The results are as follows:
(i) Correlation coefficient between final unit mark in QMDA and final unit mark in Economics for Business was 0.47
(ii) Correlation coefficient between attendance rate of seminars and distance between student accommodation and University campus was -0.34
(iii) Correlation coefficient between age and distance between student accommodation and University campus was -0.07
(iv) Correlation coefficient between attendance rate of seminars and final unit mark in QMDA was 0.89
(v) Correlation coefficient between attendance rate of seminars and final unit mark in Economics for Business was -1.25
For each of these correlation coefficients, identify the type and strength of the correlation, explain the meaning and justify whether they are reasonable in the
Portsmouth City Council has decided to investigate the number of reported crimes in different areas of the city. Data for the following variables was investigated for Hampshire county:
– Number of reported crimes Fratton, Years: 2000 – 2018
– Number of reported crimes Southsea, Years: 2000 – 2018
a) Briefly explain the difference between the Median and the Mean. Outline – in theory – when the use of the Median would be more suitable compared to the
b) Using the statistical output provided in Figure 2, explain whether there is statistical evidence for the number of reported crimes being higher in Fratton than in Southsea. Hint: Outline the hypotheses for the corresponding 2-sample t-test and interpret the relevant p-value.
c) Explain the difference between an independent and a paired 2-sample t-test.
Give an example for an independent and a paired case in context of reported crimes.
|t-Test: Two-Sample Assuming Equal Variances|
|Hypothesized Mean Difference||0|
|t Critical one-tail||1.688297714|
|t Critical two-tail||2.028094001|
Figure 2 Excel Output – 2 sample t-test
A company in Newcastle produces two types of music stands, the first type is made from wood and the other one from metal. The marketing manager specified that he wants to satisfy the following requests:
at least 30 wooden music stands have been ordered by the Newcastle Rock Choir for the following week, given the forthcoming joint concert with a Flute Orchestra.
The wooden music stands are characterized by a small blue fabric decoration.
The company holds only enough blue fabric for 40 wooden music stands.
In addition, the 8 craftsmen that are producing the music stands cannot work more than 6 hours per day and cannot work more than 5 days per week. Each wooden music
stand requires 3 hours of manufacturing, while each metal music stands requires 2 hours.
The director of the company is concerned about operational costs and aims to minimise expenses. The cost for each wooden music stand is £12 while producing a
metal music stand is less expensive and equals to £5.50.
(a) Formulate this as a linear programming problem to find the music stand product mix for next week.
(b) Using Figure 3 and Table 2, find the music stand product mix for next week which minimises expenses.
(c) Outline whether this problem has a bounded or unbounded feasible region.
Explain what this means.
(d) List one binding and one non-binding constraint and explain your choice.
[8 marks] [10 marks]
[3 marks] [4 marks]
Figure 2: PhpSimplex output
Table 2 Corner points
|No. of music stands (Wood)||No. of music stands (Metal)||Expense|
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