CEG2002: Statistics for Civil Engineers

Practical 2

 

Instructions

 

1.    Answer all questions. 

2.    This practical is not assessed – but you should work through all questions, saving your work at the end of the session for future use. 

 

 

You should write your solutions to the following questions in a Microsoft WORD document (this is how the computing part of the Statistics assignment will have to be written up).    Remember to include a title at the top of the page – e.g. “CEG2002 Practical 2”.  Don’t copy and paste the entire Minitab output into your solutions – just the relevant bits!  And don’t forget to save your work!

 

  1. In Minitab, click on Calc – Probability Distributions – Binomial. Select Probability, Enter 11 in Number of trials, and enter 0.9 in Event probability.  Select Input constant and enter 8. Click OK and then you can find the probability P(X=8) at the output in the Session window for X~Binomial(11,0.9).  Repeat the above steps but select Cumulative probability this time, and enter 5 in Input constant. You will get the probability of . How to find the value of  by using Minitab?
  2. In Minitab, click on Calc – Probability Distributions – normal. Select Cumulative probability, Enter 40 in Mean, and enter 12 in Standard deviation. Select Input constant and enter 30. Click OK and then you can find the probability P(X<30) at the output in the Session window for X ~ N(40, 122). Compare this value with the one you have got in Tutorial 1 for Question 8. How to find the values of P(X>55) and P(15<X<45) by using Minitab?

 

  1. Visit the following website for this course (or click the link from Blackboard)

http://www.staff.ncl.ac.uk/j.q.shi/teaching/CEG2002/

Click the name Barcelona_gothic.MTW to download the data file (or open the data file directly). In Minitab, click on File and Open Worksheet, and then open the data file you just saved.

In Column C1 of the Minitab worksheet you should now see the data, which show the lead content (in mg per litre) of 10 jars of water taken from the public supply in the Gothic Quarter of Barcelona (it will also be discussed in Tutorial 2).

  

(a)  Test the null hypothesis that the average lead content in this part of the city is 50 mg per litre, by clicking on Stat – Basic Statistics – 1-Sample t.  Enter C1 in Samples in columns, click Perform hypothesis test and enter 50 as the Hypothesized mean.  Click OK.  Look at the output in the Session window, and then answer the following questions. 

 

                                          i.    What is the test statistic as calculated by Minitab? 

                                        ii.    What is the exact p-value for this hypothesis test?  Interpret this p-value and form your conclusions. 

                                       iii.    Write down the 95% confidence interval for the population mean lead content of jars of water in collected in the Gothic Quarter of Barcelona. 

 

(b)  Recall the assumptions of independence and Normality which underlie the test you performed in part (a). 

 

                                          i.    Produce a histogram and a Box-and-Whisker plot of the data in column C1, and use these to comment on the assumption of Normality. 

                                        ii.    In small samples, it is often difficult to check for Normality by looking at a Histogram or a Box-and-Whisker plot.  We often prefer to use a Normal Probability Plot.    Click on Graph – Probability Plot – Single, and then click OK.  Enter C1 in the Graph variables box and click OK.  The graph you see is a Normal Probability Plot, which plots theoretical probabilities from the Normal distribution against observed probabilities from your data – the diagonal line represents equality between these two quantities, and so the closer the points lie to this line, the more plausible the assumption of Normality.  Comment on the assumption of Normality using this plot. 

 

  1. Enter the sample of 15 jars of water taken from the Example district of Barcelona in column C2.  These data are given as follows

 

40

42

49

31

43

52

50

45

44

39

33

34

36

35

42

 

 

Test the null hypothesis that there is no difference between the average lead content of water in the two areas of Barcelona by clicking on Stat – Basic Statistics – 2-Sample t.  Select Samples in different columns, and enter C1 in First and C2 in Second.  Tick the box which says Assume equal variances (one of the assumptions for this test) and then click OK. You may record the output in the Session window and answer the following questions after Tutorial 2.

 

(a)  What is the test statistics as calculated by Minitab?  Does this match up with the test statistic you calculate by hand for these data in tutorial?

(b)  What is the exact p-value for this hypothesis test?  Does this match up with the range for the p-value you obtained by hand in tutorial?  Interpret this p-value and form your conclusions. 

(c)   Minitab also gives the 95% confidence interval for the mean difference.  If this contains zero, then this implies there is no significant difference between the two population means.  Interpret this confidence interval – does this match up with your conclusions in part (b)? 

(d)  Use Minitab to check the assumption that both samples are drawn from a Normal distribution. 

 

 

  1. In a study of automobile collision insurance costs, a random sample of 80 body repair costs for a particular kind of damage gave

 

 = £542       and       = £70. 

 

Construct a 90% confidence interval for the average cost of collisions of this kind by clicking on Stat – Basic Statistics – 1-Sample t.  Select Summarized data and enter the Sample size, Mean and Standard Deviation.  Untick Perform hypothesis test, click Options, and change the Confidence level to 90.  Click OK twice.  Save the output and compare this to the interval you will calculate by hand in Tutorial 2, and repeat these steps to calculate a 95% and 99% interval in Minitab.