5.02c Linear coding: effects on mean and variance

A netball team are in a league with three other teams from which one team will progress to the next stage of the competition. The team's coach estimates their chances of winning each of their three matches in the league to be 0.6, 0.5 and 0.3 respectively, and believes these probabilities to be independent of each other.

1. Show that the probability of the team winning exactly two of their three matches is 0.36 [4 marks]

Let the random variable \(W\) be the number of matches that the team win in the league.

1. Find the probability distribution of \(W\). [4 marks]
2. Find E\((W)\) and Var\((W)\). [6 marks]
3. Comment on the coach's assumption that the probabilities of success in each of the three matches are independent. [2 marks]

Members of a library may borrow up to 6 books. Past experience has shown that the number of books borrowed, \(X\), follows the distribution shown in the table.

\(x\)	0	1	2	3	4	5	6
P(X = x)	0	0.19	0.26	0.20	0.13	0.07	0.15

1. Find the probability that a member borrows more than 3 books. [1 mark]
2. Assume that the numbers of books borrowed by two particular members are independent. Find the probability that one of these members borrows more than 3 books and the other borrows fewer than 3 books. [3 marks]
3. Show that the mean of \(X\) is 3.08, and calculate the variance of \(X\). [4 marks]
4. One of the library staff notices that the values of the mean and the variance of \(X\) are similar and suggests that a Poisson distribution could be used to model \(X\). Without further calculations, give two reasons why a Poisson distribution would not be suitable to model \(X\). [2 marks]
5. The library introduces a fee of 10 pence for each book borrowed. Assuming that the probabilities do not change, calculate:
   1. the mean amount that will be paid by a member;
   2. the standard deviation of the amount that will be paid by a member.
   [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]

In an orchard, all the trees are either apple or pear trees. There are four times as many apple trees as pear trees. Find the probability that, in a random sample of 10 trees, there are

1. equal numbers of apple and pear trees, [3 marks]
2. more than 7 apple trees. [3 marks]

In a sample of 60 trees in the orchard,

1. find the expected number of pear trees. [1 mark]
2. Calculate the standard deviation of the number of pear trees and compare this result with the standard deviation of the number of apple trees. [2 marks]
3. Find the probability that exactly 35 in the sample of 60 trees are pear trees. [4 marks]
4. Find an approximate value for the probability that more than 15 of the 60 trees are pear trees. [5 marks]

1. Briefly describe the main features of a binomial distribution. [2 marks]

I conduct an experiment by randomly selecting 10 cards, without replacement, from a normal pack of 52.

1. Explain why the distribution of \(X\), the number of hearts obtained, is not \(\text{B}(10, \frac{1}{4})\). [2 marks]

After making the appropriate adjustment to the experiment, which should be stated, so that the distribution is \(\text{B}(10, \frac{1}{4})\), find

1. the probability of getting no hearts, [3 marks]
2. the probability of getting 4 or more hearts. [2 marks]
3. If the modified experiment is repeated 50 times, find the total number of hearts that you would you expect to have drawn. [2 marks]

A shop receives weekly deliveries of 120 eggs from a local farm. The proportion of eggs received from the farm that are broken is 0.008

1. Explain why it is reasonable to use the binomial distribution to model the number of eggs that are broken in each delivery. [3 marks]
2. Use the binomial distribution to calculate the probability that at most one egg in a delivery will be broken. [4 marks]
3. State the conditions under which the binomial distribution can be approximated by the Poisson distribution. [1 mark]
4. Using the Poisson approximation to the binomial, find the probability that at most one egg in a delivery will be broken. Comment on your answer. [5 marks]

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 bag contains 40 beads of the same shape and size. The ratio of red to green to blue beads is \(1 : 3 : 4\) and there are no beads of any other colour. In an experiment, a bead is picked at random, its colour noted and the bead replaced in the bag. This is done ten times.

1. Suggest a suitable distribution for modelling the number of times a blue bead is picked out and give the value of any parameters needed. [2]
2. Explain why this distribution would not be suitable if the beads were not replaced in the bag. [1]
3. Find the probability that of the ten beads picked out
   1. five are blue,
   2. at least one is red. [6]

The experiment is repeated, but this time a bead is picked out and replaced \(n\) times.

1. Find in the form \(a^n < b\), where \(a\) and \(b\) are exact fractions, the condition which \(n\) must satisfy in order to have at least a 99\% chance of picking out at least one red bead. [3]

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]

Faults occur in a roll of material at a rate of \(\lambda\) per m\(^2\). To estimate \(\lambda\), three pieces of material of sizes 3 m\(^2\), 7 m\(^2\) and 10 m\(^2\) are selected and the number of faults \(X_1\), \(X_2\) and \(X_3\) respectively are recorded. The estimator \(\hat{\lambda}\), where $$\hat{\lambda} = k(X_1 + X_2 + X_3)$$ is an unbiased estimator of \(\lambda\).

1. Write down the distributions of \(X_1\), \(X_2\) and \(X_3\) and find the value of \(k\). [4]
2. Find Var(\(\hat{\lambda}\)). [3]

A random sample of \(n\) pieces of this material, each of size 4 m\(^2\), was taken. The number of faults on each piece, \(Y\), was recorded.

1. Show that \(\frac{1}{4}\bar{Y}\) is an unbiased estimator of \(\lambda\). [2]
2. Find Var(\(\frac{1}{4}\bar{Y}\)). [3]
3. Find the minimum value of \(n\) for which \(\frac{1}{4}\bar{Y}\) becomes a better estimator of \(\lambda\) than \(\hat{\lambda}\). [2]

A technician is trying to estimate the area \(\mu^2\) of a metal square. The independent random variables \(X_1\) and \(X_2\) are each distributed \(\text{N}(\mu, \sigma^2)\) and represent two measurements of the sides of the square. Two estimators of the area, \(A_1\) and \(A_2\), are proposed where $$A_1 = X_1X_2 \text{ and } A_2 = \left(\frac{X_1 + X_2}{2}\right)^2.$$ [You may assume that if \(X_1\) and \(X_2\) are independent random variables then $$\text{E}(X_1X_2) = \text{E}(X_1)\text{E}(X_2)$$]

1. Find \(\text{E}(A_1)\) and show that \(\text{E}(A_2) = \mu^2 + \frac{\sigma^2}{2}\). [4]
2. Find the bias of each of these estimators. [2]

The technician is told that \(\text{Var}(A_1) = \sigma^4 + 2\mu^2\sigma^2\) and \(\text{Var}(A_2) = \frac{1}{2}\sigma^4 + 2\mu^2\sigma^2\). The technician decided to use \(A_1\) as the estimator for \(\mu^2\).

1. Suggest a possible reason for this decision. [1]

A statistician suggests taking a random sample of \(n\) measurements of sides of the square and finding the mean \(\overline{X}\). He knows that \(\text{E}(\overline{X}^2) = \mu^2 + \frac{\sigma^2}{n}\) and $$\text{Var}(\overline{X}^2) = \frac{2\sigma^4}{n^2} + \frac{4\sigma^2\mu^2}{n}.$$

1. Explain whether or not \(\overline{X}^2\) is a consistent estimator of \(\mu^2\). [3]

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 \(\frac{2}{5}X_1 - \frac{1}{5}X_2 + \frac{4}{5}X_3\) 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. 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]

When a tree seed is planted the probability of it germinating is \(p\). A random sample of size \(n\) is taken and the number of tree seeds, \(X\), which germinate is recorded.

1. 1. Show that \(\hat{p}_1 = \frac{X}{n}\) is an unbiased estimator of \(p\).
   2. Find the variance of \(\hat{p}_1\). [4]
   A second sample of size \(m\) is taken and the number of tree seeds, \(Y\), which germinate is recorded. Given that \(\hat{p}_2 = \frac{Y}{m}\) and that \(\hat{p}_3 = a(3\hat{p}_1 + 2\hat{p}_2)\) is an unbiased estimator of \(p\),
2. show that
   1. \(a = \frac{1}{5}\),
   2. \(\text{Var}(\hat{p}_3) = \frac{p(1-p)}{25}\left(\frac{9}{n} + \frac{4}{m}\right)\). [6]
3. Find the range of values of \(\frac{n}{m}\) for which $$\text{Var}(\hat{p}_3) < \text{Var}(\hat{p}_1) \text{ and } \text{Var}(\hat{p}_3) < \text{Var}(\hat{p}_2)$$ [3]
4. Given that \(n = 20\) and \(m = 60\), explain which of \(\hat{p}_1\), \(\hat{p}_2\) or \(\hat{p}_3\) is the best estimator. [3]

Each day, for many years, the maximum temperature in degrees Celsius at a particular location is recorded. The maximum temperatures for days in October can be modelled by a Normal distribution. The appropriate Normal curve is shown in Fig. 6. \includegraphics{figure_6}

1. 1. Use the model to write down the mean of the maximum temperatures. [1]
   2. Explain why the curve indicates that the standard deviation is approximately 3 degrees Celsius. [1]

Temperatures can be converted from Celsius to Fahrenheit using the formula \(F = 1.8C + 32\), where \(F\) is the temperature in degrees Fahrenheit and \(C\) is the temperature in degrees Celsius.

1. For maximum temperature in October in degrees Fahrenheit, estimate
   [2]

The discrete random variable \(X\) has the following probability distribution.

1. Find P\((X \leq 6)\) [1 mark]
2. Let \(Y = 3X + 2\) Show that Var\((Y) = 32.49\) [5 marks]
3. The continuous random variable \(T\) is independent of \(Y\). Given that Var\((T) = 5\), find Var\((T + Y)\) [1 mark]

The random variable \(X\) is equally likely to take any of the \(n\) integer values from \(m + 1\) to \(m + n\) inclusive. It is given that \(\text{E}(3X) = 30\) and \(\text{Var}(3X) = 36\). Determine the value of \(m\) and the value of \(n\). [7]

The number of televisions of a particular model sold per week at a retail store can be modelled by a random variable \(X\) with the probability function shown in the table.

\(x\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(P(X = x)\)	\(0.05\)	\(0.2\)	\(0.5\)	\(0.2\)	\(0.05\)

1. 1. Explain why \(\text{E}(X) = 2\). [1]
   2. Find \(\text{Var}(X)\). [3]
2. The profit, measured in pounds made in a week, on the sales of this model of television is given by \(Y\), where \(Y = 250X - 80\). Find

The remote controls for the televisions are quality tested by the manufacturer to see how long they last before they fail.

1. Explain why it would be inappropriate to test all the remote controls in this way. [1]
2. State an advantage of using random sampling in this context. [1]

The discrete random variable \(X\) has the following probability distribution, where \(\theta\) is an unknown parameter belonging to the interval \(\left(0, \frac{1}{3}\right)\).

1. Obtain an expression for \(E(X)\) in terms of \(\theta\) and show that $$\text{Var}(X) = 4\theta(3 - \theta).$$ [4] In order to estimate the value of \(\theta\), a random sample of \(n\) observations on \(X\) was obtained and \(\bar{X}\) denotes the sample mean.
2. 1. Show that $$V = \frac{\bar{X} - 3}{2}$$ is an unbiased estimator for \(\theta\).
   2. Find an expression for the variance of \(V\). [4]
3. Let \(Y\) denote the number of observations in the random sample that are equal to 1. Show that $$W = \frac{Y}{n}$$ is an unbiased estimator for \(\theta\) and find an expression for \(\text{Var}(W)\). [5]
4. Determine which of \(V\) and \(W\) is the better estimator, explaining your method clearly. [4]

A spinner has edges numbered 1, 2, 3, 4 and 5. When the spinner is spun, the number of the edge on which it lands is the score. The probability distribution of the score, \(N\), is given in the table. It is known that E\((N) = 2.55\).

1. Find Var\((N)\). [7]
2. Find E\((3N + 2)\). [1]
3. Find Var\((3N + 2)\). [1]

A researcher has collected data on the heights of a sample of adults but has encoded the actual values using a linear transformation of the form \(aX + b\), where \(X\) represents the original height in centimetres. Given the following information about the encoded data: The mean of the encoded heights is 5.4 cm The standard deviation of the encoded heights is 2.0 cm The researcher knows that the transformation used was \(0.2X - 30\)

1. Find the mean of the original heights in the sample. [2]
2. Find the standard deviation of the original heights in the sample. [2]
3. If an encoded height value is 6.8, what was the original height in centimetres? [1]

At a factory that makes crockery the quality control department has found that 10\% of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by \(X\).

1. State an appropriate distribution with which to model \(X\). Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. [4]

Assume now that your model is valid.

1. Find
   1. P\((X = 3)\), [2]
2. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3. [4]

The probability distribution for the discrete random variable \(W\) is given in the table.

\(w\)	1	2	3	4
\(P(W = w)\)	0.25	0.36	\(x\)	\(x^2\)

1. Show that \(\text{Var}(W) = 0.8571\). [7]
2. Find \(\text{Var}(3W + 6)\). [1]

The probability distribution of a discrete random variable \(W\) is given in the table. Given that \(\mathrm{E}(W) = 1.61\), find the value of \(\text{Var}(3W + 2)\). [7]

Chris plays for his local hockey club. In his first 20 games for the club, the mean number of goals per game he has scored is \(0.7\), with a standard deviation of \(0.9\). In the next 5 games he scores \(0, 1, 0, 2, 1\) goals.

1. Find the mean and standard deviation for the number of goals per game Chris has scored in all 25 games. [7]
2. A sponsor pays Chris £65 each time he plays for the club and a further £25 for each goal he scores. Find the mean and standard deviation of the amount per game he earns from the sponsor for all 25 games. [4]