At a Children's Party Each Child Was Blindfolded and Asked to Pin a Tail on a Cardboard

January practise Qs #1

Question 1

At a children's party each child was blindfolded and asked to pin a tail on a cardboard

donkey. the distance, in cm, of the pin from the correct position was measured and the results

are recorded below

17, 15, 5, 9, 13, 42, 8, 24, 34, 38, 29, 6.

(a)Find the mean and the standard deviation for this set of numbers.

A statistics student, who was helping at this party, attempts to model the distance, in cm, a

child places the pin from the correct position using the continuous uniform distribution over

the interval [0, 50].

(b)Use the formulae in the formula booklet to find the mean and the standard deviation of

this distribution.

(c)Comment on the suitability of this distribution as a model in the present situation.

The student attempts to refine the model and considers two distributions with probability

density functions f(x) and g(x) illustrated below.

(d)Explain, giving a reason, which of these two probability density functions the student

should choose. [12]

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Question 2

Two friends make regular telephone calls to each other. The duration, in minutes, of their

telephone conversation is modelled initially by the random variable T, having probability

density function

(a)Sketch the probability density function of T.

(b)For all values of t, find the cumulative distribution function of T.

(c)Find the probability that a telephone conversation lasts longer than 12 minutes.

(d)Show that the median duration of a telephone conversation is given by (25 - 5)

minutes.

(e)Give a reason why this initial model may not be realistic for the distribution of durations

of telephone conversations.

(f)Sketch the probability density function of a more realistic model.

[16]

Question 3

The total number of radio taxi calls received at a control centre in a month is modelled by a

random variable X (in tens of thousands of calls) having the probability density function

(a)Show that the value of c is 1.

(b)Write down the probability that X≤ 1.

(c)Show that the cumulative distribution function of X is

(d)Find the probability that the control centre receives between 8000 and 12 000 calls in a

month.

A colleague criticises the model on the grounds that the number of radio calls must be

discrete, while the model used for X is continuous.

(e)State briefly whether you consider that it was reasonable to use this model for X.

(f)Give two reasons why the probability density function in the diagram above might be unsuitable as a model.

(g)Sketch the shape of a more suitable probability density function.

[20]

Question 4

The continuous random variable T has cumulative distribution function

(a)Find the probability density function of T.

(b)Sketch the probability density function of T for all t.

(c)Calculate E(T) and show that Var(T) =

(d)Given that a = 60, find P(20 T 40).

A hospital accident unit wishes to model the times between the arrivals of consecutive patients.

(e)Give two reasons why the above distribution might be a reasonable model for these times.

(f)Sketch an alternative probability density function which might be a more refined model to

represent these times.

[15]

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Question 5

The owners of a Chinese take-away kept records for many years of their daily takings and were concerned that their mid-week takings were lower than those for the rest of the week. The takings for Wednesdays followed the normal distribution with mean £130.00 and standard deviation £27.60. The owners tried to boost their mid-week takings by having special offers available to customers on Wednesdays for a period of 30 weeks. The mean takings for these 30 Wednesdays was £145.67. Treating these 30 Wednesdays as a random sample, the owners decided to test whether or not there was evidence that the special offers increased takings on Wednesdays.

(a)Starting your hypotheses clearly, carry out this test using a 1% level of significance.

(b)State an assumption you have made about the effect of the special offers on the standard

deviation of the takings on Wednesdays.

[8]

Total = 71