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1 It is given that \(X \sim \mathrm {~N} ( 30,49 ) , Y \sim \mathrm {~N} ( 30,16 )\) and \(Z \sim \mathrm {~N} ( 50,16 )\). On a single diagram, with the horizontal axis going from 0 to 70 , sketch three curves to represent the distributions of \(X , Y\) and \(Z\).

2 Lengths of a certain species of lizard are known to be normally distributed with standard deviation 3.2 cm . A naturalist measures the lengths of a random sample of 100 lizards of this species and obtains an \(\alpha \%\) confidence interval for the population mean. He finds that the total width of this interval is 1.25 cm . Find \(\alpha\).

3 A random sample of 500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, \(p\), of all households in that town that owned two or more cars was found to be \(0.355 < p < 0.445\). Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.

1 Name the distribution and suggest suitable numerical parameters that you could use to model the weights in kilograms of female 18-year-old students.

2 In a random sample of 200 shareholders of a company, 103 said that they wanted a change in the management.

1. Find an approximate \(92 \%\) confidence interval for the proportion, \(p\), of all shareholders who want a change in the management.
2. State the probability that a \(92 \%\) confidence interval does not contain \(p\).

2 In a survey of 300 randomly chosen adults in Rickton, 134 said that they exercised regularly. This information was used to calculate an \(\alpha \%\) confidence interval for the proportion of adults in Rickton who exercise regularly. The upper bound of the confidence interval was found to be 0.487 , correct to 3 significant figures. Find the value of \(\alpha\) correct to the nearest integer.

1 Leat s frm a certain tro tree \(\mathbf { h }\) leg \(\mathbf { b }\) th t are \(\dot { \mathbf { d } }\) strib ed with stad \(\operatorname { rd } \mathbf { d } \dot { \mathbf { v } }\) atio \(\mathbf { Z }\) cm. A rach sample 6 6 th se lead s is tak n ad the mean leg \(\mathrm { h } \mathbf { 6 }\) th s samp e is fo d to b © cm .

1. Calch ate a 90 cf id \(n\) e in erd \(l\) fo th \(p p\) atim earl eg \(h\)
2. Write d n th p b b lity th t th wh e 6 a \(9 \%\) co id n e in ery l will lie b low th p atim ean

3 A die is thrown 100 times and shows an odd number on 56 throws. Calculate an approximate \(97 \%\) confidence interval for the probability that the die shows an odd number on one throw.

1 The time taken, \(T\) minutes, for a special anti-rust paint to dry was measured for a random sample of 120 painted pieces of metal. The sample mean was 51.2 minutes and an unbiased estimate of the population variance was 37.4 minutes \(^ { 2 }\). Determine a \(99 \%\) confidence interval for the mean drying time.
\(21.5 \%\) of the population of the UK can be classified as 'very tall'.

1. The random variable \(X\) denotes the number of people in a sample of \(n\) people who are classified as very tall. Given that \(\mathrm { E } ( X ) = 2.55\), find \(n\).
2. By using the Poisson distribution as an approximation to a binomial distribution, calculate an approximate value for the probability that a sample of size 210 will contain fewer than 3 people who are classified as very tall.

2. The random variable \(X\) is normally distributed with mean 177.0 and standard deviation 6.4.

1. Find \(\mathrm { P } ( 166 < X < 185 )\). It is suggested that \(X\) might be a suitable random variable to model the height, in cm , of adult males.
2. Give two reasons why this is a sensible suggestion.
3. Explain briefly why mathematical models can help to improve our understanding of real-world problems.

2. The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{0e73f1d0-5532-4995-b39e-759d82c2bd92-04_860_1684_367_130} One of the 150 plants is chosen at random, and its height, \(X \mathrm {~cm}\), is noted.

1. Show that \(\mathrm { P } ( 20 < X < 30 ) = 0.147\), correct to 3 significant figures. Sam suggests that the distribution of \(X\) can be well modelled by the distribution \(\mathrm { N } ( 40,100 )\).
2. 1. Give a brief justification for the use of the normal distribution in this context.
   2. Give a brief justification for the choice of the parameter values 40 and 100 .
3. Use Sam's model to find \(\mathrm { P } ( 20 < X < 30 )\). Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, \(m\) and \(s\), for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution \(\mathrm { N } \left( m , s ^ { 2 } \right)\) as her model.
4. Use Nina's model to find \(\mathrm { P } ( 20 < X < 30 )\).
5. 1. Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model.
   2. By considering the different ranges of values of \(X\) given in the table, discuss how well the two models fit the original distribution. Table for (e)(i):

      \(x\)	Below 20	20 to 30	30 to 35	35 to 40	40 to 45	45 to 50	50 to 60	Above 60
      Probability obtained from histogram	0.027	0.147	0.153	0.187	0.193	0.147	0.133	0.013
      Probability obtained from Sam's model, N(40, 100)	0.023		0.150	0.191			0.136	0.023
      Probability obtained from Nina's model, \(\mathrm { N } \left( m , s ^ { 2 } \right)\)	0.030		0.153	0.188			0.130	0.023

2. The random variable \(X\) is normally distributed with mean 177.0 and standard deviation 6.4.

1. Find \(\mathrm { P } ( 166 < X < 185 )\).
   (4 marks)
   It is suggested that \(X\) might be a suitable random variable to model the height, in cm , of adult males.
2. Give two reasons why this is a sensible suggestion.
   (2 marks)
3. Explain briefly why mathematical models can help to improve our understanding of real-world problems.
   (2 marks)

6 The continuous random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\).

1. Each of the three following sets of probabilities is impossible. Give a reason in each case why the probabilities cannot both be correct. (You should not attempt to find \(\mu\) or \(\sigma\).)
   (a) \(\mathrm { P } ( X > 50 ) = 0.7\) and \(\mathrm { P } ( X < 50 ) = 0.2\)
   (b) \(\mathrm { P } ( X > 50 ) = 0.7\) and \(\mathrm { P } ( X > 70 ) = 0.8\)
   (c) \(\quad \mathrm { P } ( X > 50 ) = 0.3\) and \(\mathrm { P } ( X < 70 ) = 0.3\)
2. Given that \(\mathrm { P } ( X > 50 ) = 0.7\) and \(\mathrm { P } ( X < 70 ) = 0.7\), find the values of \(\mu\) and \(\sigma\).

9 The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-08_842_1651_495_207} One of the 150 plants is chosen at random, and its height, \(X \mathrm {~cm}\), is noted.

1. Show that \(\mathrm { P } ( 20 < X < 30 ) = 0.147\), correct to 3 significant figures. Sam suggests that the distribution of \(X\) can be well modelled by the distribution \(\mathrm { N } ( 40,100 )\).
2. 1. Give a brief justification for the use of the normal distribution in this context.
   2. Give a brief justification for the choice of the parameter values 40 and 100 .
3. Use Sam's model to find \(\mathrm { P } ( 20 < X < 30 )\). Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, \(m\) and \(s\), for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution \(\mathrm { N } \left( m , s ^ { 2 } \right)\) as her model.
4. Use Nina's model to find \(\mathrm { P } ( 20 < X < 30 )\).
5. 1. Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model.
   2. By considering the different ranges of values of \(X\) given in the table, discuss how well the two models fit the original distribution.

16 A survey of 120 adults found that the volume, \(X\) litres per person, of carbonated drinks they consumed in a week had the following results: $$\sum x = 165.6 \quad \sum x ^ { 2 } = 261.8$$ 16

1. 1. Calculate the mean of \(X\).
      16
2. (ii) Calculate the standard deviation of \(X\).
   16
3. Assuming that \(X\) can be modelled by a normal distribution find
   16
4. 1. \(\mathrm { P } ( 0.5 < X < 1.5 )\)
      16
5. (ii) \(\mathrm { P } ( X = 1 )\) 16
6. Determine with a reason, whether a normal distribution is suitable to model this data. [2 marks]
   16
7. It is known that the volume, \(Y\) litres per person, of energy drinks consumed in a week may be modelled by a normal distribution with standard deviation 0.21 Given that \(\mathrm { P } ( Y > 0.75 ) = 0.10\), find the value of \(\mu\), correct to three significant figures. [4 marks]

18 In a particular year, the height of a male athlete at the Summer Olympics has a mean 1.78 metres and standard deviation 0.23 metres. The heights of \(95 \%\) of male athletes are between 1.33 metres and 2.22 metres.
18

1. Comment on whether a normal distribution may be suitable to model the height of a male athlete at the Summer Olympics in this particular year.
   18
2. You may assume that the height of a male athlete at the Summer Olympics may be modelled by a normal distribution with mean 1.78 metres and standard deviation 0.23 metres. 18
3. 1. Find the probability that the height of a randomly selected male athlete is 1.82 metres.
      18
4. (ii) Find the probability that the height of a randomly selected male athlete is between 1.70 metres and 1.90 metres.
   18
5. (iii) Two male athletes are chosen at random. Calculate the probability that both of their heights are between 1.70 metres and 1.90 metres. 18
6. The summarised data for the heights, \(h\) metres, of a random sample of 40 male athletes at the Winter Olympics is given below. $$\sum h = 69.2 \quad \sum ( h - \bar { h } ) ^ { 2 } = 2.81$$ Use this data to calculate estimates of the mean and standard deviation of the heights of male athletes at the Winter Olympics.
   [0pt] [3 marks]
   18
7. Using your answers from part (c), compare the heights of male athletes at the Summer Olympics and male athletes at the Winter Olympics.
   [0pt] [2 marks]
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4. Automatic coin counting machines sort, count and batch coins. A particular brand of these machines rejects \(2 p\) coins that are less than 6.12 grams or greater than 8.12 grams.

1. The histogram represents the distribution of the weight of UK 2p coins supplied by the Royal Mint. This distribution has mean 7.12 grams and standard deviation 0.357 grams. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Weight of UK two pence coins} \includegraphics[alt={},max width=\textwidth]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-3_602_969_664_589}
   \end{figure} Explain why the weight of 2 p coins can be modelled using a normal distribution.
2. Assume the distribution of the weight of \(2 p\) coins is normally distributed. Calculate the proportion of \(2 p\) coins that are rejected by this brand of coin counting machine.
3. A manager suspects that a large batch of \(2 p\) coins is counterfeit. A random sample of 30 of the suspect coins is selected. Each of the coins in the sample is weighed. The results are shown in the summary statistics table.

   Summary statistics
   Mean	Standard deviation	Minimum	Lower quartile	Median	Upper quartile	Maximum
   6.89	0.296	6.45	6.63	6.88	7.08	7.48

   i) What assumption must be made about the weights of coins in this batch in order to conduct a test of significance on the sample mean? State, with a reason, whether you think this assumption is reasonable.
   ii) Assuming the population standard deviation is 0.357 grams, test at the \(1 \%\) significance level whether the mean weight of the \(2 p\) coins in this batch is less than 7.12 grams.