5.05a Sample mean distribution: central limit theorem

1. The continuous random variable \(D\) is uniformly distributed over the interval \([ x - 1 , x + 5 ]\) where \(x\) is a constant.

A random sample of \(n\) observations of \(D\) is taken, where \(n\) is large.

1. Use the Central Limit Theorem to find an approximate distribution for \(\bar { D }\) Give your answer in terms of \(n\) and \(x\) where appropriate. The \(n\) observations of \(D\) have a sample mean of 24.6
   Given that the lower bound of the \(99 \%\) confidence interval for \(x\) is 22.101 to 3 decimal places,
2. find the value of \(n\) Show your working clearly.

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 \(\lambda\)

7. A bag contains a large number of coins of which \(30 \%\) are 50 p coins, \(20 \%\) are 10 p coins and the rest are 2 p coins.

1. Find the mean \(\mu\) and the variance \(\sigma ^ { 2 }\) of this population of coins. A random sample of 2 coins is drawn from the bag one after the other.
2. List all possible samples that could be drawn.
3. Find the sampling distribution of \(\bar { X }\), the mean of the coins drawn.
4. Find \(\mathrm { P } ( 2 \leq \bar { X } < 7 )\).
5. Use the sampling distribution of \(\bar { X }\) to verify \(\mathrm { E } ( \bar { X } ) = \mu\) and \(\operatorname { Var } ( \bar { X } ) = \frac { 1 } { 2 } \sigma ^ { 2 }\). END

3. A woodwork teacher measures the width, \(w \mathrm {~mm}\), of a board. The measured width, \(X \mathrm {~mm}\), is normally distributed with mean \(w \mathrm {~mm}\) and standard deviation 0.5 mm .

1. Find the probability that \(X\) is within 0.6 mm of \(w\). The same board is measured 16 times and the results are recorded.
2. Find the probability that the mean of these results is within 0.3 mm of \(w\). Given that the mean of these 16 measurements is 35.6 mm ,
3. find a \(98 \%\) confidence interval for \(w\).

7. A farmer monitored the amount of lead in soil in a field next to a factory. He took 100 samples of soil, randomly selected from different parts of the field, and found the mean weight of lead to be \(67 \mathrm { mg } / \mathrm { kg }\) with standard deviation \(25 \mathrm { mg } / \mathrm { kg }\).
After the factory closed, the farmer took 150 samples of soil, randomly selected from different parts of the field, and found the mean weight of lead to be \(60 \mathrm { mg } / \mathrm { kg }\) with standard deviation \(10 \mathrm { mg } / \mathrm { kg }\).

1. Test at the \(5 \%\) level of significance whether or not the mean weight of lead in the soil decreased after the factory closed. State your hypotheses clearly.
2. Explain the significance of the Central Limit Theorem to the test in part(a).
3. State an assumption you have made to carry out this test.

6. Fruit-n-Veg4U Market Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is \(0.5 \mathrm {~kg} ^ { 2 }\) and the variance of the yield for the new variety of plant is \(0.75 \mathrm {~kg} ^ { 2 }\). A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg .

1. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
2. Explain the relevance of the Central Limit Theorem to the test in part (a).

5. A student believes that there is a difference in the mean lengths of English and French films. He goes to the university video library and randomly selects a sample of 120 English films and a sample of 70 French films. He notes the length, \(x\) minutes, of each of the films in his samples. His data are summarised in the table below.

1. Verify that the unbiased estimate of the variance, \(s ^ { 2 }\), of the lengths of English films is 98.5 minutes \({ } ^ { 2 }\)
2. Stating your hypotheses clearly, test, at the 1\% level of significance, whether or not the mean lengths of English and French films are different.
3. Explain the significance of the Central Limit Theorem to the test in part (b).
4. The university video library contained 724 English films and 473 French films. Explain how the student could have taken a stratified sample of 190 of these films.

2. The weights of pears in an orchard are assumed to have unknown mean \(\mu\) and unknown standard deviation \(\sigma\). A random sample of 20 pears is taken and their weights recorded.
The sample is represented by \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 20 }\). State whether or not the following are statistics. Give reasons for your answers.

1. 1. \(\frac { X _ { 1 } + 3 X _ { 20 } } { 2 }\)
   2. \(\sum _ { i = 1 } ^ { 20 } \left( X _ { i } - \mu \right)\)
   3. \(\sum _ { i = 1 } ^ { 20 } \left( \frac { X _ { i } - \mu } { \sigma } \right)\)
2. Find the mean and variance of \(\frac { 3 X _ { 1 } - X _ { 20 } } { 2 }\)

2. A researcher believes that the mean weight loss of those people using a slimming plan as part of a group is more than 1.5 kg a year greater than the mean weight loss of those using the plan on their own. The mean weight loss of a random sample of 80 people using the plan as part of a group is 8.7 kg with a standard deviation of 2.1 kg . The mean weight loss of a random sample of 65 people using the plan on their own is 6.6 kg with a standard deviation of 1.4 kg .

1. Stating your hypotheses clearly, test the researcher's claim. Use a \(1 \%\) level of significance.
2. For the test in part (a), state whether or not it is necessary to assume that the weight loss of a person using this plan has a normal distribution. Give a reason for your answer.

5 The times taken by new recruits to complete an assault course may be modelled by a normal distribution with a standard deviation of 8 minutes. A group of 30 new recruits takes a total time of 1620 minutes to complete the course.

1. Calculate the mean time taken by these 30 new recruits.
2. Assuming that the 30 recruits may be considered to be a random sample, construct a \(98 \%\) confidence interval for the mean time taken by new recruits to complete the course.
3. Construct an interval within which approximately \(98 \%\) of the times taken by individual new recruits to complete the course will lie.
4. State where, if at all, in this question you made use of the Central Limit Theorem.

7

1. Electra is employed by E \& G Ltd to install electricity meters in new houses on an estate. Her time, \(X\) minutes, to install a meter may be assumed to be normally distributed with a mean of 48 and a standard deviation of 20 . Determine:
   1. \(\mathrm { P } ( X < 60 )\);
   2. \(\mathrm { P } ( 30 < X < 60 )\);
   3. the time, \(k\) minutes, such that \(\mathrm { P } ( X < k ) = 0.9\).
2. Gazali is employed by E \& G Ltd to install gas meters in the same new houses. His time, \(Y\) minutes, to install a meter has a mean of 37 and a standard deviation of 25 .
   1. Explain why \(Y\) is unlikely to be normally distributed.
   2. State why \(\bar { Y }\), the mean of a random sample of 35 gas meter installations, is likely to be approximately normally distributed.
   3. Determine \(\mathrm { P } ( \bar { Y } > 40 )\).

6

1. The time taken, in minutes, by Domesat to install a domestic satellite system may be modelled by a normal distribution with unknown mean, \(\mu\), and standard deviation 15 . The times taken, in minutes, for a random sample of 10 installations are as follows. \(\begin{array} { l l l l l l l l l l } 47 & 39 & 25 & 51 & 47 & 36 & 63 & 41 & 78 & 43 \end{array}\) Construct a \(98 \%\) confidence interval for \(\mu\).
2. The time taken, \(Y\) minutes, by Teleair to erect a TV aerial and then connect it to a TV is known to have a mean of 108 and a standard deviation of 28. Estimate the probability that the mean of a random sample of 40 observations of \(Y\) is more than 120 .
3. Indicate, with a reason, where, if at all, in this question you made use of the Central Limit Theorem.
   (2 marks)
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7 An ambulance control centre responds to emergency calls in a rural area. The response time, \(T\) minutes, is defined as the time between the answering of an emergency call at the centre and the arrival of an ambulance at the given location of the emergency. Response times have an unknown mean \(\mu _ { T }\) and an unknown variance.
Anita, the centre's manager, asked Peng, a student on supervised work experience, to record and summarise the values of \(T\) obtained from a random sample of 80 emergency calls. Peng's summarised results were $$\text { Mean, } \bar { t } = 6.31 \quad \text { Variance (unbiased estimate), } s ^ { 2 } = 19.3$$ Only 1 of the 80 values of \(T\) exceeded 20

1. Anita then asked Peng to determine a confidence interval for \(\mu _ { T }\). Peng replied that, from his summarised results, \(T\) was not normally distributed and so a valid confidence interval for \(\mu _ { T }\) could not be constructed.
   1. Explain, using the value of \(\bar { t } - 2 s\), why Peng's conclusion that \(T\) was not normally distributed was likely to be correct.
   2. Explain why Peng's conclusion that a valid confidence interval for \(\mu _ { T }\) could not be constructed was incorrect.
2. Construct a \(98 \%\) confidence interval for \(\mu _ { T }\).
3. Anita had two targets for \(T\). These were that \(\mu _ { T } < 8\) and that \(\mathrm { P } ( T \leqslant 20 ) > 95 \%\). Indicate, with justification, whether each of these two targets was likely to have been met.
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7 The volume of water, \(V\), used by a guest in an en suite shower room at a small guest house may be modelled by a random variable with mean \(\mu\) litres and standard deviation 65 litres. A random sample of 80 guests using this shower room showed a mean usage of 118 litres of water.

1. 1. Give a numerical justification as to why \(V\) is unlikely to be normally distributed.
   2. Explain why \(\bar { V }\), the mean of a random sample of 80 observations of \(V\), may be assumed to be approximately normally distributed.
2. 1. Construct a \(98 \%\) confidence interval for \(\mu\).
   2. Hence comment on a claim that \(\mu\) is 140 .
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1 The number of telephone calls per hour to an out-of-hours doctors' service may be modelled by a Poisson distribution. The total number of telephone calls received during a random sample of 12 weekday night shifts, all of the same duration, was 392.

1. Calculate an approximate \(98 \%\) confidence interval for the mean number of calls received per weekday night shift.
2. The mean number of calls received during weekend shifts of 48 hours' total duration is 136.8 . Comment on a claim that the mean number of calls per hour during weekend shifts is greater than that during weekday night shifts, which are each of \(\mathbf { 1 4 }\) hours' duration.
   (3 marks)

4 An analysis of a sample of 250 patients visiting a medical centre showed that 38 per cent were aged over 65 years. An analysis of a sample of 100 patients visiting a dental practice showed that 21 per cent were aged over 65 years. Assume that each of these two samples has been randomly selected.
Investigate, at the \(5 \%\) level of significance, the hypothesis that the percentage of patients visiting the medical centre, who are aged over 65 years, exceeds that of patients visiting the dental practice, who are aged over 65 years, by more than 10 per cent.

7 It is claimed that the proportion, \(P\), of people who prefer cooked fresh garden peas to cooked frozen garden peas is greater than 0.50 .

1. In an attempt to investigate this claim, a sample of 50 people were each given an unlabelled portion of cooked fresh garden peas and an unlabelled portion of cooked frozen garden peas to taste. After tasting each portion, the people were each asked to state which of the two portions they preferred. Of the 50 people sampled, 29 preferred the cooked fresh garden peas. Assuming that the 50 people may be considered to constitute a random sample, use a binomial distribution and the \(10 \%\) level of significance to investigate the claim.
   (6 marks)
2. It was then decided to repeat the tasting in part (a) but to involve a sample of 500 , rather than 50, people. Of the 500 people sampled, 271 preferred the cooked fresh garden peas.
   1. Assuming that the 500 people may be considered to constitute a random sample, use an approximation to the distribution of the sample proportion, \(\widehat { P }\), and the \(10 \%\) level of significance to again investigate the claim.
   2. The critical value of \(\widehat { P }\) for the test in part (b)(i) is 0.529 , correct to three significant figures. It is also given that, in fact, 55 per cent of people prefer cooked fresh garden peas. Estimate the power for a test of the claim that \(P > 0.50\) based on a random sample of 500 people and using the \(10 \%\) level of significance.
      (5 marks)

1 A hotel's management is concerned about the quality of the free pens that it provides in its meeting rooms. The hotel's assistant manager tests a random sample of 200 such pens and finds that 23 of them fail to write immediately.

1. Calculate an approximate \(\mathbf { 9 6 \% }\) confidence interval for the proportion of pens that fail to write immediately.
2. The supplier of the pens to the hotel claims that at most 2 in 50 pens fail to write immediately. Comment, with numerical justification, on the supplier's claim.
   [0pt] [2 marks] QUESTION
   PART Answer space for question 1

7

1. The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
   2. Given that \(\mathrm { E } \left( X ^ { 2 } - X \right) = \lambda ^ { 2 }\), deduce that \(\operatorname { Var } ( X ) = \lambda\).
2. The number of faults in a 100-metre ball of nylon string may be modelled by a Poisson distribution with parameter \(\lambda\).
   1. An analysis of one ball of string, selected at random, showed 15 faults. Using an exact test, investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
   2. A subsequent analysis of a random sample of 20 balls of string showed a total of 241 faults.
      (A) Using an approximate test, re-investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
      (B) Determine the critical value of the total number of faults for the test in part (b)(ii)(A).
      (C) Given that, in fact, \(\lambda = 12\), estimate the probability of a Type II error for a test of the claim that \(\lambda > 10\) based upon a random sample of 20 balls of string and using the \(5 \%\) level of significance.
      [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-22_2490_1728_219_141} \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-23_2490_1719_217_150} \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-24_2489_1728_221_141}

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]

6. The weight of a particular electrical component is normally distributed with a mean of 46.7 grams and a variance of 1.8 grams \(^ { 2 }\). The component is sold in boxes of 12 .

1. State the distribution of the mean weight of the components in one box.
2. Find the probability that the mean weight of the components in a randomly chosen box is more than 47 grams.
   (3 marks)
   After a break in production the component manufacturer wishes to find out if the mean weight of the components has changed. A random sample of 30 components is found to have a mean weight of 46.5 grams.
3. Assuming that the variance of the weight of the components is unchanged, test at the \(5 \%\) level of significance if there has been any change in the mean weight of the components.
   (7 marks)

7. (a) Briefly state the central limit theorem. A student throws ten dice and records the number of sixes showing. The dice are fair, numbered 1 to 6 on the faces.
(b) Write down the distribution of the number of sixes obtained when the ten dice are thrown.
(c) Find the mean and variance of this distribution. The student throws the ten dice 100 times, recording the number of sixes showing each time.
(d) Find the probability that the mean number of sixes obtained is more than 1.8