State 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
   1. a normal distribution,
   2. 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).

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 \(\lambda\)

A random sample \(X_1, X_2, \ldots, X_{10}\) is taken from a normal population with mean 100 and standard deviation 14.

1. Write down the distribution of \(\overline{X}\), the mean of this sample. [2]
2. Find \(\text{Pr}(|\overline{X} - 100| > 5)\). [3]

The weights of pears, \(P\) grams, are normally distributed with a mean of 110 and a standard deviation of 8. Geoff buys a bag of 16 pears.

1. Write down the distribution of \(\overline{P}\), the mean weight of the 16 pears. [2]
2. Find P\((110 < \overline{P} < 113)\). [3]

A railway company is investigating operations at a junction where delays often occur. Delays (in minutes) are modelled by the random variable \(T\) with the following cumulative distribution function. $$F(t) = \begin{cases} 0 & t \leq 0 \\ 1 - e^{-\frac{1}{t}} & t > 0 \end{cases}$$

1. Find the median delay and the 90th percentile delay. [5]
2. Derive the probability density function of \(T\). Hence use calculus to find the mean delay. [5]
3. Find the probability that a delay lasts longer than the mean delay. [2]

You are given that the variance of \(T\) is 9.

1. Let \(\overline{T}\) denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of \(\overline{T}\). [3]
2. A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling? [3]

The length of time that registered customers spend on each visit to a supermarket's website is normally distributed with a mean of 28.5 minutes and a standard deviation of 7.2 minutes. Eight visitors to the site are selected at random and the length of time, \(T\) minutes, that each stays is recorded.

1. Write down the distribution of \(\overline{T}\), the mean time spent at the site by these eight visitors. [2 marks]
2. Find \(P(25 < \overline{T} < 30)\). [4 marks]