2.04h Select appropriate distribution

6 The lengths of body feathers of a particular species of bird are modelled by a normal distribution. A researcher measures the lengths of a random sample of 600 body feathers from birds of this species and finds that 63 are less than 6 cm long and 155 are more than 12 cm long.

1. Find estimates of the mean and standard deviation of the lengths of body feathers of birds of this species.
2. In a random sample of 1000 body feathers from birds of this species, how many would the researcher expect to find with lengths more than 1 standard deviation from the mean?

8 A market gardener records the masses of a random sample of 100 of this year's crop of plums. The table shows his results.

1. Explain why the normal distribution might be a reasonable model for this distribution. The market gardener models the distribution of masses by \(\mathrm { N } \left( 47.5,10 ^ { 2 } \right)\).
2. Find the number of plums in the sample that this model would predict to have masses in the range:
   1. \(35 \leq m < 45\)
   2. \(m < 25\).
3. Use your answers to parts (b)(i) and (b)(ii) to comment on the suitability of this model. The market gardener plans to use this model to predict the distribution of the masses of next year's crop of plums.
4. Comment on this plan.

5 A survey of all the households on an estate is undertaken to provide information on the number of children per household. The results, for the 99 households with children, are shown in the table.

1. For these 99 households, calculate values for:
   1. the median and the interquartile range;
   2. the mean and the standard deviation.
2. In fact, 163 households were surveyed, of which 64 contained no children.
   1. For all 163 households, calculate the value for the mean number of children per household.
   2. State whether the value for the standard deviation, when calculated for all 163 households, will be smaller than, the same as, or greater than that calculated in part (a)(ii).
   3. It is claimed that, for all 163 households on the estate, the number of children per household may be modelled approximately by a normal distribution. Comment, with justification, on this claim. Your comment should refer to a fact other than the discrete nature of the data.
      
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2. (a) Explain briefly what is meant by a statistical model.
(b) State, with a reason, whether or not the normal distribution might be suitable for modelling each of the following:

1. The number of children in a family;
2. The time taken for a particular employee to cycle to work each day using the same route;
3. The quarterly electricity bills for a particular house.

1. Find the value of the constant \(a\) such that
2. Find the value of the constant \(a\) such that

$$\mathrm { P } \left( a < F _ { 8,10 } < 3.07 \right) = 0.94$$

1. (a) Find the value of the constant \(a\) such that

$$\mathrm { P } \left( 1.690 < \chi _ { 7 } ^ { 2 } < a \right) = 0.95$$ The random variable \(Y\) follows an \(F\)-distribution with 6 and 4 degrees of freedom.
(b) (i) Find the upper \(1 \%\) critical value for \(Y\).
(ii) Find the lower \(1 \%\) critical value for \(Y\).

5 James plays an arcade game. Each time he plays, he puts a \(\pounds 1\) coin in the slot to start the game. The possible outcomes of each game are as follows: James loses the game with a probability of 0.7 and the machine pays out nothing, James draws the game with a probability of 0.25 and the machine pays out a \(\pounds 1\) coin, James wins the game with a probability of 0.05 and the machine pays out ten \(\pounds 1\) coins. The outcomes can be modelled by a random variable \(X\) representing the number of \(\pounds 1\) coins gained at the end of a game.

1. Construct a probability distribution table for \(X\).
2. Show that \(\mathrm { E } ( X ) = - 0.25\) and find \(\operatorname { Var } ( X )\). James starts off with \(10 \pounds 1\) coins and decides to play exactly 10 games.
3. Find the expected number of \(\pounds 1\) coins that James will have at the end of his 10 games.
4. Find the probability that after his 10 games James will have at least \(10 \pounds 1\) coins left.

The small village of Tornep has a preservation society which is campaigning for a new by-pass to be built. The society needs to measure

1. the strength of opinion amongst the residents of Tornep for the scheme and
2. the flow of traffic through the village on weekdays. The society wants to know whether to use a census or a sample survey for each of these measures.
   1. In each case suggest which they should use and specify a suitable sampling frame. [4] For the measurement of traffic flow through Tornep,
   2. suggest a suitable statistic and a possible statistical model for this statistic. [2]

Briefly describe what is meant by

1. a statistical model, [2 marks]
2. a refinement of a model. [2 marks]