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TMA 03 See module website for the cut-off date.
Question 1, which covers topics in Unit 5, and Question 2, which covers topics in Unit 6, form M248 TMA 03. Question 1 is marked out of 28; Question 2 is marked out of 22.
Question 1 – 28 marks
You should be able to answer this question after working through Unit 5.
(a) In this part of the question, you should calculate the required probabilities without using Minitab, and show your working. (You may use Minitab to check your answers, if you wish.)
In England, the most serious emergency calls requesting an ambulance are classified as ‘Red 1’. According to data from NHS England, in March 2017, the London Ambulance Service (LAS) received a total of
1597 Red 1 emergency calls. Based on this number and adjusting for variations during the day, suppose that Red 1 calls arriving at the LAS in daylight hours may be modelled as a Poisson process with rate 3 per hour.
(i) (1) Write down the distribution of the number of Red 1 calls arriving at the LAS in a 30-minute period during daylight
hours, including the values of any parameters. [2] (2) Calculate and report the probability that three Red 1 calls
arrive at the LAS in 30 minutes during daylight hours. [2]
(ii) (1) Write down the distribution of the waiting time (in hours) between the arrivals of two successive Red 1 calls at the LAS during daylight hours, including the values of any
parameters. [2] (2) Calculate and report the probability that the gap between the
arrivals of two successive Red 1 calls at the LAS during
daylight hours will exceed 20 minutes. [4]
(b) This part of the question concerns data on the lengths of the 51 time intervals (in days) between successive earthquakes in California, starting from a major earthquake on 9 January 1857, up to an earthquake on
24 August 2014. (To qualify for inclusion in this dataset, earthquakes had to be single mainshocks with magnitude at least 4.9.) These time intervals are in the variable Interval in the worksheet
california-earthquakes.mwx. In this part of the question, you will
explore whether or not a Poisson process is a suitable model for these data.
(i) The intervals between successive events in a Poisson process are exponentially distributed. Using Minitab, find the mean and standard deviation of the intervals between earthquakes in California. Are these values consistent with the data being observations from an exponential distribution? Give a reason for
(ii) Using Minitab, obtain a histogram with the following properties:
❼ the ticks on the horizontal axis are at the cutpoints
❼ the bins have width 500 days
❼ the first bin starts at 0 days and the last bin finishes at
7500 days.
Include a copy of your histogram in your answer. Is the shape of the histogram consistent with the data being observations from an exponential distribution? Give a reason for your answer. [4]
(iii) The data are listed in the order in which they arose. Using Minitab, produce an appropriate graph to investigate whether, for the period of observation, the data are consistent with the rate at which earthquakes occur in California remaining constant. Include a copy of your graph in your answer. Explain why your graph suggests that the average rate at which earthquakes occurred in California was not constant over the period. How did the average
rate change over time? [6] (c) A certain form of ‘triangular’ distribution has c.d.f.
F (x) = 1 − (1 − x)2, 0 < x < 1,
which is plotted in Figure 1. (It is called a triangular distribution because its p.d.f. is a line which, together with the axes, forms a triangle.)
F(x)
1
0
0 1
x
Figure 1 The c.d.f. F (x)
(i) Calculate the value of the upper quartile for this distribution. [3] (ii) On a copy of, or very rough sketch based on, Figure 1, show the
values of α and its corresponding quantile qα for the upper quartile
that you calculated in part (c)(i). [2]
Question 2 – 22 marks
You should be able to answer this question after working through Unit 6.
(a) In this part of the question, you should calculate the required probabilities using tables, not Minitab, and show your working. (You may use Minitab to check your answers, if you wish.)
A model for normal human body temperature, X, when measured orally in ◦F, is that it is normally distributed, X ∼ N(98.2, 0.5184).
(i) According to the model, what proportion of people have a normal
body temperature of 99 ◦F or more? [3]
(ii) Find the normal body temperature such that, according to the
model, only 10% of people have a lower normal body temperature. [2]
9
(iii) Let W denote normal human body temperature, when measured orally in ◦C. Given that W = 5 (X − 32) and X ∼ N(98.2, 0.5184),
what is the distribution of W ? [3] (iv) According to the model that you just derived for W , what
proportion of people have a normal body temperature of between
36 ◦C and 36.8 ◦C? [4]
(b) The Minitab file body-temperature.mwx contains values of the normal body temperature, measured orally, of n = 130 people. The model for normal human body temperature used in part (a) of this question was obtained partly by consideration of these data. The data can be used to check whether or not it is appropriate to assume that normal human body temperature can be modelled by a normal distribution.
Suggest a suitable graph to investigate specifically whether or not a normal distribution might be a good model for the normal body temperature of people, measured orally. Using the Mean Rank (Herd-Johnson) method in Minitab (see Activity 7 of Computer Book B), produce this graph. Include a copy of your graph in your answer. On the basis of this graph, do you think that a normal
distribution is a plausible model for these data? Explain your answer. [5]
(c) Suppose that the mean weight of a particular type of ripe tomato is
155 g, and the variance of the weight of this type of ripe tomato is
576 g2. A random sample of n = 36 such ripe tomatoes is obtained.
(i) What is the approximate distribution of the sample mean weight of
the random sample of 36 ripe tomatoes? [2] (ii) Use Minitab to find the probability that the sample mean weight of
the sample of 36 ripe tomatoes lies between 150 g and 157.5 g. To
show that you used Minitab, write down the results of any intermediate calculations that you make in Minitab to the same
number of decimal places as given by Minitab. [3]

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