Heat Treatment time and Length of Life – Electrical components

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
SUMMARY OUTPUT      
       
Regression Statistics     
Multiple R0.915114757     
R Square0.837435018     
Adjusted R Square0.817114395     
Standard Error237.340091     
Observations10     
       
 CoefficientsStandard Errort StatPvalueLower 95%Upper 95%
Intercept3.887323944433.01647510.0089770.993057069-994.65045831002.425106
Heat Treatment Time91.5622968614.262958776.4195860.00020481858.67185496124.4527388

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.

[10 marks]

(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.

[10 marks]

(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.

[5 marks]

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

particular context.

Question 3:

[25 marks]

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

Mean.

[7 marks]

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.

[10 marks]

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.

[8 marks]

t-Test: Two-Sample Assuming Equal Variances 
   
 FrattonSouthsea
Mean6892.1684215232.747368
Variance1443345.5191081795.18
Observations1919
Pooled Variance1262570.35 
Hypothesized Mean Difference0 
df36 
t Stat4.551877912 
P(T<=t) one-tail0.00002928 
t Critical one-tail1.688297714 
P(T<=t) two-tail0.00005856 
t Critical two-tail2.028094001 

Figure 2 Excel Output – 2 sample t-test

Question 4:

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]

                                A  

E                                  D

A

Figure 2: PhpSimplex output

Table 2 Corner points

 No. of music stands (Wood)No. of music stands (Metal)Expense
A3020470
B4010535
C400480
D300360

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