5.05b Unbiased estimates: of population mean and variance

1. Korhan and Louise challenge each other to find an estimator for the mean, \(\mu\), of the continuous random variable \(X\) which has variance \(\sigma ^ { 2 }\) \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n }\) are \(n\) independent observations taken from \(X\) Korhan's estimator is given by

$$K = \frac { 2 } { n ( n + 1 ) } \sum _ { r = 1 } ^ { n } r X _ { r }$$ Louise's estimator is given by $$L = \frac { X _ { 1 } + X _ { 2 } } { 3 } + \frac { X _ { 3 } + X _ { 4 } + \ldots + X _ { n } } { 3 ( n - 2 ) }$$

1. Show that \(K\) and \(L\) are both unbiased estimators of \(\mu\)
2. 1. Find \(\operatorname { Var } ( K )\)
   2. Find \(\operatorname { Var } ( L )\) The winner of the challenge is the person who finds the better estimator.
3. Determine the winner of the challenge for large values of \(n\). Give reasons for your answer.

1. A bag contains a large number of marbles of which an unknown proportion, \(p\), is yellow.

Three random samples of size \(n\) are taken, and the number of yellow marbles in each sample, \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\), is recorded. Two estimators \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\) are proposed to estimate the value of \(p\) $$\begin{aligned} & \hat { p } _ { 1 } = \frac { Y _ { 1 } + 3 Y _ { 2 } - 2 Y _ { 3 } } { 2 n } \\ & \hat { p } _ { 2 } = \frac { 2 Y _ { 1 } + 3 Y _ { 2 } + Y _ { 3 } } { 6 n } \end{aligned}$$

1. Show that \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\) are both unbiased estimators of \(p\)
2. Find the variance of \(\hat { p } _ { 1 }\) The variance of \(\hat { \mathrm { p } } _ { 2 }\) is \(\frac { 7 p ( 1 - p ) } { 18 n }\)
3. State, giving a reason, which is the better estimator. The estimator \(\hat { p } _ { 3 } = \frac { Y _ { 1 } + a Y _ { 2 } + 3 Y _ { 3 } } { b n }\) where \(a\) and \(b\) are positive integers.
4. Find the pair of values of \(a\) and \(b\) such that \(\hat { \mathrm { p } } _ { 3 }\) is a better unbiased estimator of \(p\) than both \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\) You must show all stages of your working.

1. Two organisations are each asked to carry out a survey to find out the proportion, \(p\), of the population that would vote for a particular political party.

The first organisation finds that out of \(m\) people, \(X\) would vote for this particular political party. The second organisation finds that out of \(n\) people, \(Y\) would vote for this particular political party. An unbiased estimator, \(Q\), of \(p\) is proposed where $$Q = k \left( \frac { X } { m } + \frac { Y } { n } \right)$$

1. Show that \(k = \frac { 1 } { 2 }\) A second unbiased estimator, \(R\), of \(p\) is proposed where $$R = \frac { a X } { m } + \frac { b Y } { n }$$
2. Show that \(a + b = 1\) Given that \(m = 100\) and \(n = 200\) and that \(R\) is a better estimator of \(p\) than \(Q\)
3. calculate the range of possible values of \(a\) Show your working clearly.

9 The continuous random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The sum of a random sample of 16 observations of \(C\) is 224.0 .

1. Find an unbiased estimate of \(\mu\).
2. It is given that an unbiased estimate of \(\sigma ^ { 2 }\) is 0.24. Find the value of \(\Sigma c ^ { 2 }\). \(D\) is the sum of 10 independent observations of \(C\).
3. Explain whether \(D\) has a normal distribution. The continuous random variable \(F\) is normally distributed with mean 15.0, and it is known that \(\mathrm { P } ( F < 13.2 ) = 0.115\).
4. Use the unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) to find \(\mathrm { P } ( D + F > 157.0 )\). \section*{OCR} \section*{Oxford Cambridge and RSA}

7 Over a period of time it has been shown that the mean number of vehicles passing a service station on a motorway is 50 per minute. After a new motorway junction was built nearby, Xander observed that 30 vehicles passed the service station in one minute. 7

1. Xander claims that the construction of the new motorway junction has reduced the mean number of vehicles passing the service station per minute. Investigate Xander's claim, using a suitable test at the \(1 \%\) level of significance.
   7
2. For your test carried out in part (a) state, in context, the meaning of a Type 1 error. 7
3. Explain why the model used in part (a) might be invalid.

2 A binomial hypothesis test was carried out at the \(5 \%\) level of significance with the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : p = 0.6 \\ & \mathrm { H } _ { 1 } : p > 0.6 \end{aligned}$$ A sample of size 30 was used to carry out the test.
Find the probability that a Type I error was made.
Circle your answer.
[0pt] [1 mark] \(4.4 \%\) 4.8\% 5.0\% 9.4\%

10 The label on a particular size of milk carton states that it contains 1.5 litres of milk. In an investigation at the packaging plant the contents, \(x\) litres, of each of 60 randomly selected cartons are measured. The data are summarised as follows. $$\Sigma x = 89.758 \quad \Sigma x ^ { 2 } = 134.280$$

1. Estimate the variance of the underlying population.
2. Find a 95\% confidence interval for the mean of the underlying population.
3. What does the confidence interval which you have calculated suggest about the statement on the carton? Each day for 300 days a random sample of 60 cartons is selected and for each sample a \(95 \%\) confidence interval is constructed.
4. Explain why the confidence intervals will not be identical.
5. What is the expected number of confidence intervals to contain the population mean?

7. \includegraphics[max width=\textwidth, alt={}, center]{65369843-222f-48b2-b8cd-a1c304eac3d9-6_707_718_347_660} The diagram above shows a cyclic quadrilateral \(A B C D\), where \(\widehat { B A D } = \alpha , \widehat { B C D } = \beta\) and \(\alpha + \beta = 180 ^ { \circ }\). These angles are measured.
The random variables \(X\) and \(Y\) denote the measured values, in degrees, of \(\widehat { B A D }\) and \(\widehat { B C D }\) respectively. You are given that \(X\) and \(Y\) are independently normally distributed with standard deviation \(\sigma\) and means \(\alpha\) and \(\beta\) respectively.

1. Calculate, correct to two decimal places, the probability that \(X + Y\) will differ from \(180 ^ { \circ }\) by less than \(\sigma\).
2. Show that \(T _ { 1 } = 45 ^ { \circ } + \frac { 1 } { 4 } ( 3 X - Y )\) is an unbiased estimator for \(\alpha\) and verify that it is a better estimator than \(X\) for \(\alpha\).
3. Now consider \(T _ { 2 } = \lambda X + ( 1 - \lambda ) \left( 180 ^ { \circ } - Y \right)\).
   1. Show that \(T _ { 2 }\) is an unbiased estimator for \(\alpha\) for all values of \(\lambda\).
   2. Find \(\operatorname { Var } \left( T _ { 2 } \right)\) in terms of \(\lambda\) and \(\sigma\).
   3. Hence determine the value of \(\lambda\) which gives the best unbiased estimator for \(\alpha\).

2 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

2

1. The statistic \(T\) is derived from a random sample taken from a population which has an unknown parameter \(\theta\). \(T\) is an unbiased estimator of \(\theta\). What does the statement ' \(T\) is an unbiased estimator of \(\theta ^ { \prime }\) imply?
2. A random sample of size \(n\) is taken from each of two independent populations. The first population has a non-zero mean \(\mu\) and variance \(\sigma ^ { 2 }\) and \(\bar { X } _ { 1 }\) denotes the sample mean. The second population has mean \(\frac { 1 } { 2 } \mu\) and variance \(b \sigma ^ { 2 }\), where \(b\) is a positive constant, and \(\bar { X } _ { 2 }\) denotes the sample mean. Two unbiased estimators for \(\mu\) are defined by $$T _ { 1 } = 3 \bar { X } _ { 1 } - a \bar { X } _ { 2 } \quad \text { and } \quad T _ { 2 } = \frac { 1 } { 5 } \left( 4 \bar { X } _ { 1 } + 2 \bar { X } _ { 2 } \right) .$$
   1. Determine the value of \(a\).
   2. Show that \(\operatorname { Var } \left( T _ { 1 } \right) = \frac { \sigma ^ { 2 } } { n } ( 9 + 16 b )\) and find a similar expression for \(\operatorname { Var } \left( T _ { 2 } \right)\).
   3. The estimator with the smaller variance is preferred. State which of \(T _ { 1 }\) and \(T _ { 2 }\) is the preferred estimator of \(\mu\).

7 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 x ^ { 2 } } { k ^ { 3 } } & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a parameter.

1. Find \(\mathrm { E } ( X )\). Hence show that \(\frac { 4 } { 3 } X\) is an unbiased estimator of \(k\). Three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), and the largest value of \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) is denoted by \(M\).
2. Write down an expression for \(\mathrm { P } ( M \leqslant x )\), and hence show that the probability density function of \(M\) is $$f _ { M } ( x ) = \begin{cases} \frac { 9 x ^ { 8 } } { k ^ { 9 } } & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } . \end{cases}$$
3. Find \(\mathrm { E } ( M )\) and use your answer to construct an unbiased estimator of \(k\) based on \(M\).

4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

6 In a certain city there are \(N\) taxis. Each taxi displays a different licensing number which is an integer in the range 1 to \(N\). A visitor to the city attempts to estimate the value of \(N\), assuming that the licensing number of each taxi observed is equally likely to be any integer from 1 to \(N\) inclusive.

1. The visitor observes one randomly chosen licensing number, \(X\). Using standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\mathrm { E } ( X ) = \frac { 1 } { 2 } ( N + 1 )\) and \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } \left( N ^ { 2 } - 1 \right)\). The mean of 40 independent observations of \(X\) is denoted by \(A\).
2. Find an unbiased estimator \(E _ { 1 }\) of \(N\) based on \(A\), and state the approximate distribution of \(E _ { 1 }\), giving the value(s) of any parameter(s). \(B\) is another random variable based on a random sample of 40 independent observations of \(X\). It is given that \(\mathrm { E } ( B ) = \frac { 40 } { 27 } N\) and that \(\operatorname { Var } ( B ) = \alpha N ^ { 2 }\) where \(\alpha\) is a constant.
3. Find an unbiased estimator \(E _ { 2 }\) of \(N\) based on \(B\), and determine the set of values of \(\alpha\) for which \(\operatorname { Var } \left( E _ { 2 } \right) > \operatorname { Var } \left( E _ { 1 } \right)\) for all values of \(N\).

4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( \overline { a X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

100 randomly chosen adults each throw a ball once. The length, \(l\) metres, of each throw is recorded. The results are summarised below. $$n = 100 \qquad \sum l = 3820 \qquad \sum l^2 = 182200$$ Calculate a 94% confidence interval for the population mean length of throws by adults. [6]

Each of a random sample of 80 adults gave an estimate, \(h\) metres, of the height of a particular building. The results were summarised as follows. $$n = 80 \quad \sum h = 2048 \quad \sum h^2 = 52760$$

1. Calculate unbiased estimates of the population mean and variance. [3]
2. Using this sample, the upper boundary of an \(\alpha\%\) confidence interval for the population mean is 26.0. Find the value of \(\alpha\). [4]

The times, \(T\) minutes, taken by a random sample of \(75\) students to complete a test were noted. The results were summarised by \(\sum t = 230\) and \(\sum t^2 = 930\).

1. Calculate unbiased estimates of the population mean and variance of \(T\). [3]

You should now assume that your estimates from part (a) are the true values of the population mean and variance of \(T\).

1. The times taken by another random sample of \(75\) students were noted, and the sample mean, \(\overline{T}\), was found. Find the value of \(a\) such that \(P(\overline{T} > a) = 0.234\). [3]

The number of accidents per year on a certain road has the distribution \(\text{Po}(\lambda)\). In the past the value of \(\lambda\) was \(3.3\). Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5\%\) significance level.

1. Calculate the probability of a Type I error. [4]
2. Given that \(X = 2\), carry out the test. [3]
3. The council decides to carry out another similar test at the \(5\%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is \(0.6\), calculate the probability of a Type II error. [3]
4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than \(10\) accidents in \(30\) years. [4]

The number of adult customers arriving in a shop during a 5-minute period is modelled by a random variable with distribution \(\text{Po}(6)\). The number of child customers arriving in the same shop during a 10-minute period is modelled by an independent random variable with distribution \(\text{Po}(4.5)\).

1. Find the probability that during a randomly chosen 2-minute period, the total number of adult and child customers who arrive in the shop is less than 3. [3]
2. During a sale, the manager claims that more adult customers are arriving than usual. In a randomly selected 30-minute period during the sale, 49 adult customers arrive. Test the manager's claim at the 2.5\% significance level. [6]

The number of sightings of a golden eagle at a certain location has a Poisson distribution with mean 2.5 per week. Drilling for oil is started nearby. A naturalist wishes to test at the 5\% significance level whether there are fewer sightings since the drilling began. He notes that during the following 3 weeks there are 2 sightings.

1. Find the critical region for the test and carry out the test. [5]
2. State the probability of a Type I error. [1]
3. State why the naturalist could not have made a Type II error. [1]

Jack has to choose a random sample of 8 people from the 750 members of a sports club.

1. Explain fully how he can use random numbers to choose the sample. [3]

Jack asks each person in the sample how much they spent last week in the club café. The results, in dollars, were as follows. 15 \quad 25 \quad 30 \quad 8 \quad 12 \quad 18 \quad 27 \quad 25

1. Find unbiased estimates of the population mean and variance. [3]
2. Explain briefly what is meant by 'population' in this question. [1]

As part of an investigation, a random sample was taken of 50 footballers who had completed an obstacle course in the early morning. The time taken by each of these footballers to complete the obstacle course, \(x\) minutes, was recorded and the results are summarised by $$\sum x = 1570 \quad \text{and} \quad \sum x^2 = 49467.58$$

1. Find unbiased estimates for the mean and variance of the time taken by footballers to complete the obstacle course in the early morning. [4]

An independent random sample was taken of 50 footballers who had completed the same obstacle course in the late afternoon. The time taken by each of these footballers to complete the obstacle course, \(y\) minutes, was recorded and the results are summarised as $$\bar{y} = 30.9 \quad \text{and} \quad s_y^2 = 3.03$$

1. Test, at the 5\% level of significance, whether or not the mean time taken by footballers to complete the obstacle course in the early morning, is greater than the mean time taken by footballers to complete the obstacle course in the late afternoon. State your hypotheses clearly. [7]
2. Explain the relevance of the Central Limit Theorem to the test in part (b). [1]
3. State an assumption you have made in carrying out the test in part (b). [1]

The weights of tubs of margarine are known to be normally distributed. A random sample of 10 tubs of margarine were weighed, to the nearest gram, and the results were as follows. $$498 \quad 502 \quad 500 \quad 496 \quad 509 \quad 504 \quad 511 \quad 497 \quad 506 \quad 499$$

1. Find unbiased estimates of the mean and the variance of the population from which this sample was taken. [5]

Given that the population standard deviation is 5.0 g,

1. estimate limits, to 2 decimal places, between which 90\% of the weights of the tubs lie, [2]
2. find a 95\% confidence interval for the mean weight of the tubs. [5]

A second random sample of 15 tubs was found to have a mean weight of 501.9 g.

1. Stating your hypotheses clearly and using a 1\% level of significance, test whether or not the mean weight of these tubs is greater than 500 g. [5]

The weights of tubs of margarine are known to be normally distributed. A random sample of 10 tubs of margarine were weighed, to the nearest gram, and the results were as follows. 498 502 500 496 509 504 511 497 506 499

1. Find unbiased estimates of the mean and the variance of the population from which this sample was taken. [5]

Given that the population standard deviation is 5.0 g,

1. estimate limits, to 2 decimal places, between which 90\% of the weights of the tubs lie, [2]
2. find a 95\% confidence interval for the mean weight of the tubs. [5]

A second random sample of 15 tubs was found to have a mean weight of 501.9 g.

1. Stating your hypotheses clearly and using a 1\% level of significance, test whether or not the mean weight of these tubs is greater than 500 g. [5]