Distribution of sample mean

2 A population has mean 7 and standard deviation 3. A random sample of size \(n\) is chosen from this population.

1. Write down the mean and standard deviation of the distribution of the sample mean.
2. Under what circumstances does the sample mean have
   (a) a normal distribution,
   (b) an approximately normal distribution?

2 The random variable \(X\) has mean 372 and standard deviation 54 .

1. Describe fully the distribution of the mean of a random sample of 36 values of \(X\).
2. The distribution in part (i) might be either exact or approximate. State a condition under which the distribution is exact.

2

1. A random variable \(X\) has mean \(\mu\) and variance \(\sigma ^ { 2 }\). The mean of a random sample of \(n\) values of \(X\) is denoted by \(\bar { X }\). Give expressions for \(\mathrm { E } ( \bar { X } )\) and \(\operatorname { Var } ( \bar { X } )\).
2. The heights, in centimetres, of adult males in Brancot are normally distributed with mean 177.8 and standard deviation 6.1. Find the probability that the mean height of a random sample of 12 adult males from Brancot is less than 176 cm .
3. State, with a reason, whether it was necessary to use the Central Limit Theorem in the calculation in part (ii).

4 A set of observations of a random variable \(W\) can be summarised as follows: $$n = 14 , \quad \Sigma w = 100.8 , \quad \Sigma w ^ { 2 } = 938.70 .$$

1. Calculate an unbiased estimate of the variance of \(W\).
2. The mean of 70 observations of \(W\) is denoted by \(\bar { W }\). State the approximate distribution of \(\bar { W }\), including unbiased estimate(s) of any parameter(s).

1 The standard deviation of a random variable \(F\) is 12.0. The mean of \(n\) independent observations of \(F\) is denoted by \(\bar { F }\).

1. Given that the standard deviation of \(\bar { F }\) is 1.50 , find the value of \(n\).
2. For this value of \(n\), state, with justification, what can be said about the distribution of \(\bar { F }\).

3 The mean of a sample of 80 independent observations of a continuous random variable \(Y\) is denoted by \(\bar { Y }\). It is given that \(\mathrm { P } ( \bar { Y } \leqslant 157.18 ) = 0.1\) and \(\mathrm { P } ( \bar { Y } \geqslant 164.76 ) = 0.7\).

1. Calculate \(\mathrm { E } ( Y )\) and the standard deviation of \(Y\).
2. State
   (a) where in your calculations you have used the Central Limit Theorem,
   (b) why it was necessary to use the Central Limit Theorem,
   (c) why it was possible to use the Central Limit Theorem.

4 A timber supplier cuts wooden fence posts from felled trees. The posts are of length \(( k + X ) \mathrm { cm }\) where \(k\) is a constant and \(X\) is a random variable which has probability density function $$f ( x ) = \begin{cases} 1 + x & - 1 \leqslant x < 0
1 - x & 0 \leqslant x \leqslant 1
0 & \text { elsewhere } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\).
2. Write down the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
3. Write down, in terms of \(k\), the approximate distribution of \(\bar { L }\), the mean length of a random sample of 50 fence posts. Justify your choice of distribution.
4. In a particular sample of 50 posts, the mean length is 90.06 cm . Find a \(95 \%\) confidence interval for the true mean length of the fence posts.
5. Explain whether it is reasonable to suppose that \(k = 90\).

1. The continuous random variable \(D\) is uniformly distributed over the interval \([ x - 1 , x + 5 ]\) where \(x\) is a constant.

A random sample of \(n\) observations of \(D\) is taken, where \(n\) is large.

1. Use the Central Limit Theorem to find an approximate distribution for \(\bar { D }\) Give your answer in terms of \(n\) and \(x\) where appropriate. The \(n\) observations of \(D\) have a sample mean of 24.6
   Given that the lower bound of the \(99 \%\) confidence interval for \(x\) is 22.101 to 3 decimal places,
2. find the value of \(n\) Show your working clearly.

6. The number of toasters sold by a shop each week may be modelled by a Poisson distribution with mean 4 A random sample of 35 weeks is taken and the mean number of toasters sold per week is found.

1. Write down the approximate distribution for the mean number of toasters sold per week from a random sample of 35 weeks. The number of kettles sold by the shop each week may be modelled by a Poisson distribution with mean \(\lambda\) A random sample of 40 weeks is taken and the mean number of kettles sold per week is found. The width of the \(99 \%\) confidence interval for \(\lambda\) is 2.6
2. Find an estimate for \(\lambda\) A second, independent random sample of 40 weeks is taken and a second \(99 \%\) confidence interval for \(\lambda\) is found.
3. Find the probability that only one of these two confidence intervals contains \(\lambda\)
   

7 The volume of water, \(V\), used by a guest in an en suite shower room at a small guest house may be modelled by a random variable with mean \(\mu\) litres and standard deviation 65 litres. A random sample of 80 guests using this shower room showed a mean usage of 118 litres of water.

1. 1. Give a numerical justification as to why \(V\) is unlikely to be normally distributed.
   2. Explain why \(\bar { V }\), the mean of a random sample of 80 observations of \(V\), may be assumed to be approximately normally distributed.
2. 1. Construct a \(98 \%\) confidence interval for \(\mu\).
   2. Hence comment on a claim that \(\mu\) is 140 .
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