5.05b Unbiased estimates: of population mean and variance

4

1. A large survey in the USA establishes that 60 per cent of its residents own a smartphone. A survey of 250 UK residents reveals that 164 of them own a smartphone.
   Assuming that these 250 UK residents may be regarded as a random sample, investigate the claim that the percentage of UK residents owning a smartphone is the same as that in the USA. Use the 5\% level of significance.
2. A random sample of 40 residents in a market town reveals that 5 of them own a 4 G mobile phone. Use an exact test to investigate, at the \(5 \%\) level of significance, the belief that fewer than 25 per cent of the town's residents own a 4 G mobile phone.
3. A marketing company needs to estimate the proportion of residents in a large city who own a 4 G mobile phone. It wishes to estimate this proportion to within 0.05 with a confidence of 98\%. Given that the proportion is known to be at most 30 per cent, estimate the sample size necessary in order to meet the company's need.
   [0pt] [5 marks]

1. A museum is open to the public for six hours a day from Monday to Friday every week. The number of visitors, \(V\), to the museum on ten randomly chosen days were as follows:

$$\begin{array} { l l l l l l l l l l } 182 & 172 & 113 & 99 & 168 & 183 & 135 & 129 & 150 & 108 \end{array}$$

1. Calculate an unbiased estimate of the mean of \(V\). Assuming that \(V\) is normally distributed with a variance of 130 ,
2. find a 95\% confidence interval for the mean of \(V\).
   

1. A diabetic patient records her blood glucose readings in \(\mathrm { mmol } / \mathrm { l }\) at random times of day over several days. Her readings are given below.

$$\begin{array} { l l l l l l l } 5.3 & 5.7 & 8.4 & 8.7 & 6.3 & 8.0 & 7.2 \end{array}$$ Assuming that the blood glucose readings are normally distributed calculate

1. an unbiased estimate for the variance \(\sigma ^ { 2 }\) of the blood glucose readings,
2. a \(90 \%\) confidence interval for the variance \(\sigma ^ { 2 }\) of blood glucose readings.
3. State whether or not the confidence interval supports the assertion that \(\sigma = 0.9\). Give a reason for your answer.
   

3. A population has mean \(\mu\) and variance \(\sigma ^ { 2 }\). A random sample of size 3 is to be taken from this population and \(\bar { X }\) denotes its sample mean. A second random sample of size 4 is to be taken from this population and \(\bar { Y }\) denotes its sample mean.

1. Show that unbiased estimators for \(\mu\) are given by
   1. \(\hat { \mu } _ { 1 } = \frac { 1 } { 3 } \bar { X } + \frac { 2 } { 3 } \bar { Y }\),
   2. \(\hat { \mu } _ { 2 } = \frac { 5 \bar { X } + 4 \bar { Y } } { 9 }\).
2. Calculate Var \(\left( \hat { \mu } _ { 1 } \right)\)
3. Given that \(\operatorname { Var } \left( \hat { \mu } _ { 2 } \right) = \frac { 37 } { 243 } \sigma ^ { 2 }\), state, giving a reason, which of these two estimators should be
   used. used.

5. (a) Explain briefly what you understand by

1. an unbiased estimator,
2. a consistent estimator.
   of an unknown population parameter \(\theta\). From a binomial population, in which the proportion of successes is \(p , 3\) samples of size \(n\) are taken. The number of successes \(X _ { 1 } , X _ { 2 }\), and \(X _ { 3 }\) are recorded and used to estimate \(p\).
   (b) Determine the bias, if any, of each of the following estimators of \(p\). $$\begin{aligned} & \hat { p } _ { 1 } = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } } { 3 n } \\ & \hat { p } _ { 2 } = \frac { X _ { 1 } + 3 X _ { 2 } + X _ { 3 } } { 6 n } \\ & \hat { p } _ { 3 } = \frac { 2 X _ { 1 } + 3 X _ { 2 } + X _ { 3 } } { 6 n } \end{aligned}$$ (c) Find the variance of each of these estimators.
   (d) State, giving a reason, which of the three estimators for \(p\) is
3. the best estimator,
4. the worst estimator.

7. A bag contains marbles of which an unknown proportion \(p\) is red. A random sample of \(n\) marbles is drawn, with replacement, from the bag. The number \(X\) of red marbles drawn is noted. A second random sample of \(m\) marbles is drawn, with replacement. The number \(Y\) of red marbles drawn is noted. Given that \(p _ { 1 } = \frac { a X } { n } + \frac { b Y } { m }\) is an unbiased estimator of \(p\),

1. show that \(a + b = 1\). Given that \(p _ { 2 } = \frac { ( X + Y ) } { n + m }\),
2. show that \(p _ { 2 }\) is an unbiased estimator for \(p\).
3. Show that the variance of \(p _ { 1 }\) is \(p ( 1 - p ) \left( \frac { a ^ { 2 } } { n } + \frac { b ^ { 2 } } { m } \right)\).
4. Find the variance of \(p _ { 2 }\).
5. Given that \(a = 0.4 , m = 10\) and \(n = 20\), explain which estimator \(p _ { 1 }\) or \(p _ { 2 }\) you should use.

6. \begin{figure}[h] \end{figure} Figure 1 shows a square of side \(t\) and area \(t ^ { 2 }\) which lies in the first quadrant with one vertex at the origin. A point \(P\) with coordinates ( \(X , Y\) ) is selected at random inside the square and the coordinates are used to estimate \(t ^ { 2 }\). It is assumed that \(X\) and \(Y\) are independent random variables each having a continuous uniform distribution over the interval \([ 0 , t ]\).
[0pt] [You may assume that \(\mathrm { E } \left( X ^ { n } Y ^ { n } \right) = \mathrm { E } \left( X ^ { n } \right) \mathrm { E } \left( Y ^ { n } \right)\), where \(n\) is a positive integer.]

1. Use integration to show that \(\mathrm { E } \left( X ^ { n } \right) = \frac { t ^ { n } } { n + 1 }\). The random variable \(S = k X Y\), where \(k\) is a constant, is an unbiased estimator for \(t ^ { 2 }\).
2. Find the value of \(k\).
3. Show that \(\operatorname { Var } S = \frac { 7 t ^ { 4 } } { 9 }\). The random variable \(U = q \left( X ^ { 2 } + Y ^ { 2 } \right)\), where \(q\) is a constant, is also an unbiased estimator for \(t ^ { 2 }\).
4. Show that the value of \(q = \frac { 3 } { 2 }\).
5. Find Var \(U\).
6. State, giving a reason, which of \(S\) and \(U\) is the better estimator of \(t ^ { 2 }\). The point \(( 2,3 )\) is selected from inside the square.
7. Use the estimator chosen in part (f) to find an estimate for the area of the square.

2. The value of orders, in \(\pounds\), made to a firm over the internet has distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of \(n\) orders is taken and \(\bar { X }\) denotes the sample mean.

1. Write down the mean and variance of \(\bar { X }\) in terms of \(\mu\) and \(\sigma ^ { 2 }\). A second sample of \(m\) orders is taken and \(\bar { Y }\) denotes the mean of this sample.
   An estimator of the population mean is given by $$U = \frac { n \bar { X } + m \bar { Y } } { n + m }$$
2. Show that \(U\) is an unbiased estimator for \(\mu\).
3. Show that the variance of \(U\) is \(\frac { \sigma ^ { 2 } } { n + m }\).
4. State which of \(\bar { X }\) or \(U\) is a better estimator for \(\mu\). Give a reason for your answer.

A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }\) is taken from a population with mean \(\mu\) and variance \(\sigma ^ { 2 }\).

1. Determine the bias, if any, of each of the following estimators of \(\mu\). $$\begin{aligned} & \theta _ { 1 } = \frac { X _ { 3 } + X _ { 4 } + X _ { 5 } } { 3 } \\ & \theta _ { 2 } = \frac { X _ { 10 } - X _ { 1 } } { 3 } \\ & \theta _ { 3 } = \frac { 3 X _ { 1 } + 2 X _ { 2 } + X _ { 10 } } { 6 } \end{aligned}$$
2. Find the variance of each of these estimators.
3. State, giving reasons, which of these three estimators for \(\mu\) is
   1. the best estimator,
   2. the worst estimator.
      

6. A continuous uniform distribution on the interval \([ 0 , k ]\) has mean \(\frac { k } { 2 }\) and variance \(\frac { k ^ { 2 } } { 12 }\). A random sample of three independent variables \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) is taken from this distribution.

1. Show that \(\frac { 2 } { 3 } X _ { 1 } + \frac { 1 } { 2 } X _ { 2 } + \frac { 5 } { 6 } X _ { 3 }\) is an unbiased estimator for \(k\). An unbiased estimator for \(k\) is given by \(\hat { k } = a X _ { 1 } + b X _ { 2 }\) where \(a\) and \(b\) are constants.
2. Show that \(\operatorname { Var } ( \hat { k } ) = \left( a ^ { 2 } - 2 a + 2 \right) \frac { k ^ { 2 } } { 6 }\)
3. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat { k }\) has minimum variance, and calculate this minimum variance.

6. Faults occur in a roll of material at a rate of \(\lambda\) per \(\mathrm { m } ^ { 2 }\). To estimate \(\lambda\), three pieces of material of sizes \(3 \mathrm {~m} ^ { 2 } , 7 \mathrm {~m} ^ { 2 }\) and \(10 \mathrm {~m} ^ { 2 }\) are selected and the number of faults \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) respectively are recorded. The estimator \(\hat { \lambda }\), where $$\hat { \lambda } = k \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right)$$ is an unbiased estimator of \(\lambda\).

1. Write down the distributions of \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) and find the value of \(k\).
2. Find \(\operatorname { Var } ( \hat { \lambda } )\). A random sample of \(n\) pieces of this material, each of size \(4 \mathrm {~m} ^ { 2 }\), was taken. The number of faults on each piece, \(Y\), was recorded.
3. Show that \(\frac { 1 } { 4 } \bar { Y }\) is an unbiased estimator of \(\lambda\).
4. Find \(\operatorname { Var } \left( \frac { 1 } { 4 } \bar { Y } \right)\).
5. Find the minimum value of \(n\) for which \(\frac { 1 } { 4 } \bar { Y }\) becomes a better estimator of \(\lambda\) than \(\hat { \lambda }\).

1. A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a population where each of the \(X _ { i }\) have a continuous uniform distribution over the interval \([ 0 , \beta ]\).
   The random variable \(Y = \max \left\{ X _ { 1 } , X _ { 2 } , \ldots , X _ { n } \right\}\).
   The probability density function of \(Y\) is given by

$$f ( y ) = \left\{ \begin{array} { c c } \frac { n } { \beta ^ { n } } y ^ { n - 1 } & 0 \leqslant y \leqslant \beta \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that \(\mathrm { E } \left( Y ^ { m } \right) = \frac { n } { n + m } \beta ^ { m }\).
2. Write down \(\mathrm { E } ( Y )\).
3. Using your answers to parts (a) and (b), or otherwise, show that $$\operatorname { Var } ( Y ) = \frac { n } { ( n + 1 ) ^ { 2 } ( n + 2 ) } \beta ^ { 2 }$$
4. State, giving your reasons, whether or not \(Y\) is a consistent estimator of \(\beta\). The random variables \(M = 2 \bar { X }\), where \(\bar { X } = \frac { 1 } { n } \left( X _ { 1 } + X _ { 2 } + \ldots + X _ { n } \right)\), and \(S = k Y\), where \(k\) is a constant, are both unbiased estimators of \(\beta\).
5. Find the value of \(k\) in terms of \(n\).
6. State, giving your reasons, which of \(M\) and \(S\) is the better estimator of \(\beta\) in this case. Five observations of \(X\) are: \(\quad \begin{array} { l l l l l } 8.5 & 6.3 & 5.4 & 9.1 & 7.6 \end{array}\)
7. Calculate the better estimate of \(\beta\).
   

8. A random sample \(W _ { 1 } , W _ { 2 } \ldots , W _ { n }\) is taken from a distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\)

1. Write down \(\mathrm { E } \left( \sum _ { i = 1 } ^ { n } W _ { i } \right)\) and show that \(\mathrm { E } \left( \sum _ { i = 1 } ^ { n } W _ { i } ^ { 2 } \right) = n \left( \sigma ^ { 2 } + \mu ^ { 2 } \right)\) An estimator for \(\mu\) is $$\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i }$$
2. Show that \(\bar { X }\) is a consistent estimator for \(\mu\). An estimator of \(\sigma ^ { 2 }\) is $$U = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i } ^ { 2 } - \left( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i } \right) ^ { 2 }$$
3. Find the bias of \(U\).
4. Write down an unbiased estimator of \(\sigma ^ { 2 }\) in the form \(k U\), where \(k\) is in terms of \(n\).

A random sample of size \(2 , X _ { 1 }\) and \(X _ { 2 }\), is taken from the random variable \(X\) which has a continuous uniform distribution over the interval \([ - a , 2 a ] , a > 0\)

1. Show that \(\bar { X } = \frac { X _ { 1 } + X _ { 2 } } { 2 }\) is a biased estimator of \(a\) and find the bias. The random variable \(Y = k \bar { X }\) is an unbiased estimator of \(a\).
2. Write down the value of the constant \(k\).
3. Find \(\operatorname { Var } ( Y )\). The random variable \(M\) is the maximum of \(X _ { 1 }\) and \(X _ { 2 }\) The probability density function, \(m ( x )\), of \(M\) is given by $$m ( x ) = \left\{ \begin{array} { c l } \frac { 2 ( x + a ) } { 9 a ^ { 2 } } & - a \leqslant x \leqslant 2 a \\ 0 & \text { otherwise } \end{array} \right.$$
4. Show that \(M\) is an unbiased estimator of \(a\). Given that \(\mathrm { E } \left( M ^ { 2 } \right) = \frac { 3 } { 2 } a ^ { 2 }\)
5. find \(\operatorname { Var } ( M )\).
6. State, giving a reason, whether you would use \(Y\) or \(M\) as an estimator of \(a\). A random sample of two values of \(X\) are 5 and - 1
7. Use your answer to part (f) to estimate \(a\).
   

6. A random sample of size \(n\) is taken from the random variable \(X\), which has a continuous uniform distribution over the interval [ \(0 , a\) ], \(a > 0\) The sample mean is denoted by \(\bar { X }\)

1. Show that \(Y = 2 \bar { X }\) is an unbiased estimator of \(a\) The maximum value, \(M\), in the sample has probability density function $$f ( m ) = \left\{ \begin{array} { c c } \frac { n m ^ { n - 1 } } { a ^ { n } } & 0 \leqslant m \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$
2. Find E(M)
3. Show that \(\operatorname { Var } ( M ) = \frac { n a ^ { 2 } } { ( n + 2 ) ( n + 1 ) ^ { 2 } }\) The estimator \(S\) is defined by \(S = \frac { n + 1 } { n } M\) Given that \(n > 1\)
4. state which of \(Y\) or \(S\) is the better estimator for \(a\). Give a reason for your answer.

6. The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) are each distributed \(\mathrm { B } ( n , p )\), where \(n > 1\) An unbiased estimator for \(p\) is given by $$\hat { p } = \frac { a X _ { 1 } + b X _ { 2 } } { n }$$ where \(a\) and \(b\) are constants.
[0pt] [You may assume that if \(X _ { 1 }\) and \(X _ { 2 }\) are independent then \(\mathrm { E } \left( X _ { 1 } X _ { 2 } \right) = \mathrm { E } \left( X _ { 1 } \right) \mathrm { E } \left( X _ { 2 } \right)\) ]

1. Show that \(a + b = 1\)
2. Show that \(\operatorname { Var } ( \hat { p } ) = \frac { \left( 2 a ^ { 2 } - 2 a + 1 \right) p ( 1 - p ) } { n }\)
3. Hence, justifying your answer, determine the value of \(a\) and the value of \(b\) for which \(\hat { p }\) has minimum variance.
4. 1. Show that \(\hat { p } ^ { 2 }\) is a biased estimator for \(p ^ { 2 }\)
   2. Show that the bias \(\rightarrow 0\) as \(n \rightarrow \infty\)
5. By considering \(\mathrm { E } \left[ X _ { 1 } \left( X _ { 1 } - 1 \right) \right]\) find an unbiased estimator for \(p ^ { 2 }\)

1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\)

$$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 2 \theta ^ { 2 } } & 0 \leqslant x \leqslant 2 \theta \\ 0 & \text { otherwise } \end{array} \right.$$ where \(\theta\) is a constant.

1. Use integration to show that \(\mathrm { E } \left( X ^ { N } \right) = \frac { 2 ^ { N + 1 } } { N + 2 } \theta ^ { N }\)
2. Hence
   1. write down an expression for \(\mathrm { E } ( X )\) in terms of \(\theta\)
   2. find \(\operatorname { Var } ( X )\) in terms of \(\theta\) A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) where \(n \geqslant 2\) is taken to estimate the value of \(\theta\) The random variable \(S _ { 1 } = q \bar { X }\) is an unbiased estimator of \(\theta\)
3. Write down the value of \(q\) and show that \(S _ { 1 }\) is a consistent estimator of \(\theta\) The continuous random variable \(Y\) is independent of \(X\) and is uniformly distributed over the interval \(\left[ 0 , \frac { 2 \theta } { 3 } \right]\), where \(\theta\) is the same unknown constant as in \(\mathrm { f } ( x )\). The random variable \(S _ { 2 } = a X + b Y\) is an unbiased estimator of \(\theta\) and is based on one observation of \(X\) and one observation of \(Y\).
4. Find the value of \(a\) and the value of \(b\) for which \(S _ { 2 }\) has minimum variance.
5. Show that the minimum variance of \(S _ { 2 }\) is \(\frac { \theta ^ { 2 } } { 11 }\)
6. Explain which of \(S _ { 1 }\) or \(S _ { 2 }\) is the better estimator for \(\theta\)

6. A statistics student is trying to estimate the probability, \(p\), of rolling a 6 with a particular die. The die is rolled 10 times and the random variable \(X _ { 1 }\) represents the number of sixes obtained. The random variable \(R _ { 1 } = \frac { X _ { 1 } } { 10 }\) is proposed as an estimator of \(p\).

1. Show that \(R _ { 1 }\) is an unbiased estimator of \(p\). The student decided to roll the die again \(n\) times ( \(n > 10\) ) and the random variable \(X _ { 2 }\) represents the number of sixes in these \(n\) rolls. The random variable \(R _ { 2 } = \frac { X _ { 2 } } { n }\) and the random variable \(Y = \frac { 1 } { 2 } \left( R _ { 1 } + R _ { 2 } \right)\).
2. Show that both \(R _ { 2 }\) and \(Y\) are unbiased estimators of \(p\).
3. Find \(\operatorname { Var } \left( R _ { 2 } \right)\) and \(\operatorname { Var } ( Y )\).
4. State giving a reason which of the 3 estimators \(R _ { 1 } , R _ { 2 }\) and \(Y\) are consistent estimators of \(p\).
5. For the case \(n = 20\) state, giving a reason, which of the 3 estimators \(R _ { 1 } , R _ { 2 }\) and \(Y\) you would recommend. The student's teacher pointed out that a better estimator could be found based on the random variable \(X _ { 1 } + X _ { 2 }\).
6. Find a suitable estimator and explain why it is better than \(R _ { 1 } , R _ { 2 }\) and \(Y\). END

1 It is known that the red blood cell count of adults in a particular country, measured in suitable units, has mean 4.96 and variance 0.15.

1. Find the probability that the mean red blood cell count of a random sample of 50 adults from this country is at least 5.00.
2. Explain how you can find the probability in part (a) despite the fact that you do not know the distribution of red blood cell counts.

1 When babies are born, their head circumferences are measured. A random sample of 50 newborn female babies is selected. The sample mean head circumference is 34.711 cm . The sample standard deviation head circumference is 1.530 cm .

1. Determine a 95\% confidence interval for the population mean head circumference of newborn female babies.
2. Explain why you can calculate this interval even though the distribution of the population of head circumferences of newborn female babies is unknown.

1. The average time it takes for a new kettle to boil, when full of water, is 305 seconds. An old kettle will take longer, on average, to boil. Alun suspects that a particular kettle is an old kettle. He boils the full kettle on 9 occasions and the times taken, in seconds, are shown below.
   305
   295
   310
   310
   315
   307
   300
   311
   306

You may assume the times taken to boil the full kettle are normally distributed.

1. Calculate unbiased estimates for the mean and variance of the times taken to boil the full kettle.
2. Test, at the \(5 \%\) level of significance, whether there is evidence to suggest that this is an old kettle.
3. State a factor that Alun should control when carrying out this investigation.
   

2. The random variables \(X\) and \(Y\) are independent, with \(X\) having mean \(\mu\) and variance \(\sigma ^ { 2 }\), and \(Y\) having mean \(\mu\) and variance \(k \sigma ^ { 2 }\), where \(k\) is a positive constant. Let \(\bar { X }\) denote the mean of a random sample of 20 observations of \(X\), and let \(\bar { Y }\) denote the mean of a random sample of 25 observations of \(Y\).

1. Given that \(T _ { 1 } = \frac { 3 \bar { X } + 7 \bar { Y } } { 10 }\), show that \(T _ { 1 }\) is an unbiased estimator for \(\mu\).
2. Given that \(T _ { 2 } = \frac { \bar { X } + a ^ { 2 } \bar { Y } } { 1 + a } , a > 0\), and \(T _ { 2 }\) is an unbiased estimator for \(\mu\), prove that \(a = 1\).
3. Find and simplify expressions for the variances of \(T _ { 1 }\) and \(T _ { 2 }\).
4. Show that the value of \(k\) for which \(T _ { 1 }\) and \(T _ { 2 }\) are equally good estimators is \(\frac { 5 } { 6 }\).
5. Given that \(T _ { 3 } = ( 1 - \lambda ) \bar { X } + \lambda \bar { Y }\), find an expression for \(\lambda\), in terms of \(k\), for which \(T _ { 3 }\) has the smallest possible variance.

Some of the components produced by a factory are defective. The management requires that no more than \(3 \%\) of the components produced are defective.
Niluki monitors the production process and takes a random sample of \(n\) components.

1. Write down the hypotheses Niluki should use in a test to assess whether or not the proportion of defective components is greater than 0.03 Niluki defines the random variable \(D _ { n }\) to represent the number of defective components in a sample of size \(n\). She considers two tests \(\mathbf { A }\) and \(\mathbf { B }\) In test \(\mathbf { A }\), Niluki uses \(n = 100\) and if \(D _ { 100 } \geqslant 5\) she rejects \(H _ { 0 }\)
2. Find the size of test \(\mathbf { A }\) In test B, Niluki uses \(n = 80\) and
   • if \(D _ { 80 } \geqslant 5\) she rejects \(\mathrm { H } _ { 0 }\)
   • if \(D _ { 80 } \leqslant 3\) she does not reject \(\mathrm { H } _ { 0 }\)
   • if \(D _ { 80 } = 4\) she takes a second random sample of size 80 and if \(D _ { 80 } \geqslant 1\) in this second sample then she rejects \(\mathrm { H } _ { 0 }\) otherwise she does not reject \(\mathrm { H } _ { 0 }\)
   • Find the size of test \(\mathbf { B }\)
   Given that the actual proportion of defective components is 0.06
3. 1. find the power of test \(\mathbf { A }\)
   2. find the expected number of components sampled using test \(\mathbf { B }\) Given also that, when the actual proportion of defective components is 0.06 , the power of test \(\mathbf { B }\) is 0.713
4. suggest, giving your reasons, which test Niluki should use.

4 A biased coin has a probability \(p\) of landing on heads, where \(0 < p < 1\) Simon spins the coin \(n\) times and the random variable \(X\) represents the number of heads. Taruni spins the coin \(m\) times, \(m \neq n\), and the random variable \(Y\) represents the number of heads. Simon and Taruni want to combine their results to find unbiased estimators of \(p\).
Simon proposes the estimator \(S = \frac { X + Y } { m + n }\) and Taruni proposes \(T = \frac { 1 } { 2 } \left[ \frac { X } { n } + \frac { Y } { m } \right]\)

1. Show that both \(S\) and \(T\) are unbiased estimators of \(p\).
2. Prove that, for all values of \(m\) and \(n , S\) is the better estimator.

1. The continuous random variable \(X\) is uniformly distributed over the interval \([ 0,4 \beta ]\), where \(\beta\) is an unknown constant.

Three independent observations, \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), are taken of \(X\) and the following estimators for \(\beta\) are proposed $$\begin{aligned} & A = \frac { X _ { 1 } + X _ { 2 } } { 2 } \\ & B = \frac { X _ { 1 } + 2 X _ { 2 } + 3 X _ { 3 } } { 8 } \\ & C = \frac { X _ { 1 } + 2 X _ { 2 } - X _ { 3 } } { 8 } \end{aligned}$$

1. Calculate the bias of \(A\), the bias of \(B\) and the bias of \(C\)
2. By calculating the variances, explain which of \(B\) or \(C\) is the better estimator for \(\beta\)
3. Find an unbiased estimator for \(\beta\)