5.05b Unbiased estimates: of population mean and variance

A computer company repairs large numbers of PCs and wants to estimate the mean time to repair a particular fault. Five repairs are chosen at random from the company's records and the times taken, in seconds, are 205 \quad 310 \quad 405 \quad 195 \quad 320.

1. Calculate unbiased estimates of the mean and the variance of the population of repair times from which this sample has been taken. [4]

It is known from previous results that the standard deviation of the repair time for this fault is 100 seconds. The company manager wants to ensure that there is a probability of at least 0.95 that the estimate of the population mean lies within 20 seconds of its true value.

1. Find the minimum sample size required. [6]

(Total 10 marks)

A machine produces metal containers. The weights of the containers are normally distributed. A random sample of 10 containers from the production line was weighed, to the nearest 0.1 kg, and gave the following results 49.7, 50.3, 51.0, 49.5, 49.9 50.1, 50.2, 50.0, 49.6, 49.7.

1. Find unbiased estimates of the mean and variance of the weights of the population of metal containers. [5]

The machine is set to produce metal containers whose weights have a population standard deviation of 0.5 kg.

1. Estimate the limits between which 95\% of the weights of metal containers lie. [4]
2. Determine the 99\% confidence interval for the mean weight of metal containers. [5]

A company produces climbing ropes. The lengths of the climbing ropes are normally distributed. A random sample of 5 ropes is taken and the length, in metres, of each rope is measured. The results are given below. 120.3 \quad 120.1 \quad 120.4 \quad 120.2 \quad 119.9

1. Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company. [5]

The lengths of climbing rope are known to have a standard deviation of 0.2 m. The company wants to make sure that there is a probability of at least 0.90 that the estimate of the population mean, based on a random sample size of \(n\), lies within 0.05 m of its true value.

1. Find the minimum sample size required. [6]

Observations have been made over many years of \(T\), the noon temperature in °C, on 21st March at Sunnymere. The records for a random sample of 12 years are given below. 5.2, 3.1, 10.6, 12.4, 4.6, 8.7, 2.5, 15.3, \(-1.5\), 1.8, 13.2, 9.3.

1. Find unbiased estimates of the mean and variance of \(T\). [5]

Over the years, the standard deviation of \(T\) has been found to be 5.1.

1. Assuming a normal distribution find a 90\% confidence interval for the mean of \(T\). [5]

A meteorologist claims that the mean temperature at noon in Sunnymere on 21st March is 4 °C.

1. Use your interval to comment on the meteorologist's claim. [2]

A company sells sugar in bags which are labelled as containing 450 grams. Although the mean weight of sugar in a bag is more than 450 grams, there is concern that too many bags are underweight. The company can adjust the mean or the standard deviation of the weight of sugar in a bag.

1. State two adjustments the company could make. [2]

The weights, \(x\) grams, of a random sample of 25 bags are now recorded.

1. Given that \(\sum x = 11409\) and \(\sum x^2 = 5206937\), calculate the sample mean and sample standard deviation of these weights. [3]

A supplier of widgets claims that only 10\% of his widgets have faults.

1. In a consignment of 50 widgets, 9 are faulty. Test, at the 5\% significance level, whether this suggests that the supplier's claim is false. [6 marks]
2. Find how many faulty widgets would be needed to provide evidence against the claim at the 1\% significance level. [3 marks]

When a park is redeveloped, it is claimed that 70\% of the local population approve of the new design. Assuming this to be true, find the probability that, in a group of 10 residents selected at random,

1. 6 or more approve, [3 marks]
2. exactly 7 approve. [3 marks]

A conservation group, however, carries out a survey of 20 people, and finds that only 9 approve.

1. Use this information to carry out a hypothesis test on the original claim, working at the 5\% significance level. State your conclusion clearly. [5 marks]

If the conservationists are right, and only 45\% approve of the new park,

1. use a suitable approximation to the binomial distribution to estimate the probability that in a larger survey, of 500 people, less than half will approve. [7 marks]

A pharmaceutical company produces an ointment for earache that, in 80\% of cases, relieves pain within 6 hours. A new drug is tried out on a sample of 25 people with earache, and 24 of them get better within 6 hours.

1. Test, at the 5\% significance level, the claim that the new treatment is better than the old one. State your hypotheses carefully. [6 marks] A rival company suggests that the sample does not give a conclusive result;
2. Might they be right, and how could a more conclusive statement be achieved? [3 marks]

The values of 5 independent observations from a population can be summarised by $$\sum x = 75.8, \quad \sum x^2 = 1154.58.$$ Find unbiased estimates of the population mean and variance. [4]

A random sample of 50 observations of the random variable \(X\) is summarised by $$n = 50, \Sigma x = 182.5, \Sigma x^2 = 739.625.$$ Calculate unbiased estimates of the expectation and variance of \(X\). [4]

The results of 14 observations of a random variable \(V\) are summarised by $$n = 14, \quad \sum v = 3752, \quad \sum v^2 = 1007448.$$ Calculate unbiased estimates of E\((V)\) and Var\((V)\). [4]

It is known that the lifetime of a certain species of animal in the wild has mean 13.3 years. A zoologist reads a study of 50 randomly chosen animals of this species that have been kept in zoos. According to the study, for these 50 animals the sample mean lifetime is 12.48 years and the population variance is 12.25 years\(^2\).

1. Test at the 5% significance level whether these results provide evidence that animals of this species that have been kept in zoos have a shorter expected lifetime than those in the wild. [7]
2. Subsequently the zoologist discovered that there had been a mistake in the study. The quoted variance of 12.25 years\(^2\) was in fact the sample variance. Determine whether this makes a difference to the conclusion of the test. [5]
3. Explain whether the Central Limit Theorem is needed in these tests. [1]

An advert for Tatty's Crisps claims that 1 in 10 bags contain a free scratchcard game. Tatty's Crisps can be bought in a Family Pack containing 10 bags. Find the probability that the bags in one of these Family Packs contain

1. no scratchcards, [2]
2. more than 2 scratchcards. [2]

Tatty's Crisps can also be bought wholesale in boxes containing 50 bags. A pub Landlord notices that her customers only found 2 scratchcards in the crisps from one of these boxes.

1. Stating your hypotheses clearly, test at the 10\% level of significance whether or not this gives evidence of there being fewer free scratchcards than is claimed by the advert. [4]

A student collects data on the number of bicycles passing outside his house in 5-minute intervals during one morning.

1. Suggest, with reasons, a suitable distribution for modelling this situation. [3]

The student's data is shown in the table below.

1. Show that the mean and variance of these data are 1.5 and 1.58 (to 3 significant figures) respectively and explain how these values support your answer to part (a). [6]

An environmental organisation declares a "car free day" encouraging the public to leave their cars at home. The student wishes to test whether or not there are more bicycles passing along his road on this day and records 16 bicycles in a half-hour period during the morning.

1. Stating your hypotheses clearly, test at the 5\% level of significance whether or not there are more than 1.5 bicycles passing along his road per 5-minute interval that morning. [5]

Car parking in a market town's high street was, until 31 May 2014, limited to one hour free of charge between 8 am and 6 pm. Records show that, during a period of 30 days prior to this date, a total of 315 penalty tickets were issued. Car parking in the high street later became limited to thirty minutes free of charge between 8 am and 6 pm. A subsequent investigation revealed that, during a period of 60 days from 1 October 2014, a total of 747 penalty tickets were issued. The daily numbers of penalty tickets issued may be modelled by independent Poisson distributions with means \(\lambda_A\) until 31 May 2014 and \(\lambda_B\) from 1 October 2014. Investigate, at the 1\% level of significance, a claim by traders on the high street that \(\lambda_B > \lambda_A\). [7 marks]

An investigation in 2007 into the incidence of tuberculosis (TB) in badgers in a certain area found that 42 out of a random sample of 190 badgers tested positive for TB. In 2010, 48 out of a random sample of 150 badgers tested positive for TB.

1. Assuming that the population proportions of badgers with TB are the same in 2007 and 2010, obtain the best estimate of this proportion. [1]
2. Carry out a test at the \(2\frac{1}{2}\%\) significance level of whether the population proportion of badgers with TB increased from 2007 to 2010. [6]

A student collected data on the number of text messages, \(t\), sent by 30 students in her year group in the previous week. Her results are summarised as follows: \(\Sigma t = 1039\), \(\Sigma t^2 = 65393\).

1. Calculate unbiased estimates of the mean and variance of the number of text messages sent by these students per week. [4]

Another student collected similar data for 20 different students and calculated unbiased estimates of the mean and variance of 32.0 and 963.4 respectively.

1. Calculate unbiased estimates of the mean and variance for the combined sample of 50 students. [6]

A random sample of three independent variables \(X_1, X_2\) and \(X_3\) is taken from a distribution with mean \(\mu\) and variance \(\sigma^2\).

1. Show that \(\frac{1}{3}X_1 + \frac{1}{3}X_2 + \frac{1}{3}X_3\) is an unbiased estimator for \(\mu\). [3]

An unbiased estimator for \(\mu\) is given by \(\hat{\mu} = aX_1 + bX_2\) where \(a\) and \(b\) are constants.

1. [(b)] Show that Var(\(\hat{\mu}\)) = \((2a^2 - 2a + 1)\sigma^2\). [6]
2. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat{\mu}\) has minimum variance. [5]

1. Explain briefly what you understand by
   1. an unbiased estimator,
   2. a consistent estimator.

of an unknown population parameter \(\theta\) [3] 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 \(p\).

1. [(b)] Determine the bias, if any, of each of the following estimators of \(p\). \(\hat{p}_1 = \frac{X_1 + X_2 + X_3}{3n}\), \(\hat{p}_2 = \frac{X_1 + 3X_2 + X_3}{6n}\), \(\hat{p}_3 = \frac{2X_1 + 3X_2 + X_3}{6n}\) [4]
2. Find the variance of each of these estimators. [4]
3. State, giving a reason, which of the three estimators for \(p\) is
   1. the best estimator,
   2. the worst estimator. [4]

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 \(p_1 = \frac{aX}{n} + \frac{bY}{m}\) is an unbiased estimator of \(p_1\),

1. show that \(a + b = 1\). [4]

Given that \(p_2 = \frac{(X + Y)}{n + m}\)

1. [(b)] show that \(p_2\) is an unbiased estimator for \(p\). [3]
2. Show that the variance of \(p_1\) is p(1 - \(p\))\(\left(\frac{a^2}{n} + \frac{b^2}{m}\right)\). [3]
3. Find the variance of \(p_2\). [3]
4. Given that \(a = 0.4\), \(m = 10\) and \(n = 20\), explain which estimator \(p_1\) or \(p_2\) you should use. [4]

(Total 17 marks)

\includegraphics{figure_6} Figure 1 shows a square of side 1 and area \(l^2\) 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 \(l^2\). It is assumed that \(X\) and \(Y\) are independent random variables each having a continuous uniform distribution over the interval \([0, l]\). [You may assume that E\((X^n Y^m) = \) E\((X^n)\)E\((Y^m)\), where \(n\) is a positive integer.]

1. Use integration to show that E\((X^n) = \frac{l^{n+1}}{n+1}\). [3]

The random variable \(S = kXY\), where \(k\) is a constant, is an unbiased estimator for \(l^2\).

1. [(b)] Find the value of \(k\). [3]
2. Show that Var \(S = \frac{7l^4}{9}\). [3]

The random variable \(U = q(X^2 + Y^2)\), where \(q\) is a constant, is also an unbiased estimator for \(l^2\).

1. [(d)] Show that the value of \(q = \frac{3}{2}\). [3]
2. Find Var \(U\). [3]
3. State, giving a reason, which of \(S\) and \(U\) is the better estimator of \(l^2\). [1]

The point (2, 3) is selected from inside the square.

1. [(g)] Use the estimator chosen in part (f) to find an estimate for the area of the square. [1]

TOTAL FOR PAPER: 75 MARKS

The value of orders, in £, made to a firm over the internet has distribution N(\(\mu, \sigma^2\)). 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 \(\sigma^2\). [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 $$U = \frac{n\bar{X} + m\bar{Y}}{n + m}$$

1. [(b)] Show that \(U\) is an unbiased estimator for \(\mu\). [3]
2. Show that the variance of \(U\) is \(\frac{\sigma^2}{n + m}\). [4]
3. State which of \(\bar{X}\) or \(U\) is a better estimator for \(\mu\). Give a reason for your answer. [2]

A random sample \(X_1, X_2, ..., X_{10}\) 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 \(\mu\). $$\theta_1 = \frac{X_1 + X_4 + X_5}{3}$$ $$\theta_2 = \frac{X_{10} - X_1}{3}$$ $$\theta_3 = \frac{3X_1 + 2X_5 + X_{10}}{6}$$ [4]
2. Find the variance of each of these estimators. [5]
3. State, giving reasons, which of these three estimators for \(\mu\) is
   1. the best estimator,
   2. the worst estimator.
   [4]

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 \(k\). [3]

An unbiased estimator for \(k\) is given by \(\hat{k} = aX_1 + bX_2\) where \(a\) and \(b\) are constants.

1. Show that Var(\(\hat{k}\)) = \((a^2 - 2a + 2) \frac{k^2}{6}\) [6]
2. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat{k}\) has minimum variance, and calculate this minimum variance. [6]

Two methods of extracting juice from an orange are to be compared. Eight oranges are halved. One half of each orange is chosen at random and allocated to Method \(A\) and the other half is allocated to Method \(B\). The amounts of juice extracted, in ml, are given in the table. One statistician suggests performing a two-sample \(t\)-test to investigate whether or not there is a difference between the mean amounts of juice extracted by the two methods.

1. Stating your hypotheses clearly and using a 5\% significance level, carry out this test. (You may assume \(\bar{x}_A = 26.125\), \(s_A^2 = 7.84\), \(\bar{x}_B = 25\), \(s_B^2 = 4\) and \(\sigma_A^2 = \sigma_B^2\)) [7]

Another statistician suggests analysing these data using a paired \(t\)-test.

1. Using a 5\% significance level, carry out this test. [9]
2. State which of these two tests you consider to be more appropriate. Give a reason for your choice. [1]