Normal Distribution

3 Jack's journey time, in minutes, to work each morning is modelled by the normal distribution \(\mathrm { N } \left( 43.2,6.3 ^ { 2 } \right)\).

1. If Jack leaves home at 0810 , find the probability that he arrives at work by 0900 .
2. Find the time by which Jack should leave home in order to be at least \(95 \%\) certain that he arrives at work by 0900 .

5 The time in hours that Davin plays on his games machine each day is normally distributed with mean 3.5 and standard deviation 0.9.

1. Find the probability that on a randomly chosen day Davin plays on his games machine for more than 4.2 hours.
2. On 90\% of days Davin plays on his games machine for more than \(t\) hours. Find the value of \(t\).
3. Calculate an estimate for the number of days in a year ( 365 days) on which Davin plays on his games machine for between 2.8 and 4.2 hours.

The heights of the students at a university are assumed to follow a normal distribution. 1% of the students are over 200 cm tall and 76% are between 165 cm and 200 cm tall. Find

1. the mean and the variance of the distribution, [9 marks]
2. the percentage of the students who are under 158 cm tall. [3 marks]
3. Comment briefly on the suitability of a normal distribution to model such a population. [2 marks]

6. The random variable \(X\) has a normal distribution with mean 20 and standard deviation 4 .

1. Find \(\mathrm { P } ( X > 25 )\).
2. Find the value of \(d\) such that \(\mathrm { P } ( 20 < X < d ) = 0.4641\)

3 In a certain town, the time, \(X\) hours, for which people watch television in a week has a normal distribution with mean 15.8 hours and standard deviation 4.2 hours.

1. Find the probability that a randomly chosen person from this town watches television for less than 21 hours in a week.
2. Find the value of \(k\) such that \(\mathrm { P } ( X < k ) = 0.75\).

4 In a large population, the systolic blood pressure (SBP) of adults is normally distributed with mean 125.4 and standard deviation 18.6.

1. Find the probability that the SBP of a randomly chosen adult is less than 132.
   The SBP of 12-year-old children in the same population is normally distributed with mean 117. Of these children 88\% have SBP more than 108.
2. Find the standard deviation of this distribution.
   Three adults are chosen at random from this population.
3. Find the probability that each of these three adults has SBP within 1.5 standard deviations of the mean.

3 In a normal distribution, 69\% of the distribution is less than 28 and 90\% is less than 35. Find the mean and standard deviation of the distribution.

The continuous random variable \(X\) has the distribution N(\(\mu\), \(\sigma^2\)).

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 \(\sigma\).)
   1. P(\(X > 50\)) = 0.7 and P(\(X < 50\)) = 0.2 [1]
   2. P(\(X > 50\)) = 0.7 and P(\(X > 70\)) = 0.8 [1]
   3. P(\(X > 50\)) = 0.3 and P(\(X < 70\)) = 0.3 [1]
2. Given that P(\(X > 50\)) = 0.7 and P(\(X < 70\)) = 0.7, find the values of \(\mu\) and \(\sigma\). [4]

6 The volume of shampoo, \(V\) millilitres, delivered by a machine into bottles may be modelled by a normal random variable with mean \(\mu\) and standard deviation \(\sigma\).

1. Given that \(\mu = 412\) and \(\sigma = 8\), determine:
   1. \(\mathrm { P } ( V < 400 )\);
   2. \(\mathrm { P } ( V > 420 )\);
   3. \(\mathrm { P } ( V = 410 )\).
2. A new quality control specification requires that the values of \(\mu\) and \(\sigma\) are changed so that $$\mathrm { P } ( V < 400 ) = 0.05 \quad \text { and } \quad \mathrm { P } ( V > 420 ) = 0.01$$
   1. Show, with the aid of a suitable sketch, or otherwise, that $$400 - \mu = - 1.6449 \sigma \quad \text { and } \quad 420 - \mu = 2.3263 \sigma$$
   2. Hence calculate values for \(\mu\) and \(\sigma\).

2 A factory produces flower pots. The base diameters have a normal distribution with mean 14 cm and standard deviation 0.52 cm . Find the probability that the base diameters of exactly 8 out of 10 randomly chosen flower pots are between 13.6 cm and 14.8 cm .

1. The continuous random variable \(W\) has the normal distribution \(\mathrm { N } \left( 32,4 { } ^ { 2 } \right)\)
   1. Write down the value of \(\mathrm { P } ( W = 36 )\)
   The discrete random variable \(X\) has the binomial distribution \(\mathrm { B } ( 20,0.45 )\)
2. Find \(\mathrm { P } ( X = 8 )\)
3. Find the probability that \(X\) lies within one standard deviation of its mean.

5 When Moses makes a phone call, the amount of time that the call takes has a normal distribution with mean 6.5 minutes and standard deviation 1.76 minutes.

1. \(90 \%\) of Moses's phone calls take longer than \(t\) minutes. Find the value of \(t\).
2. Find the probability that, in a random sample of 9 phone calls made by Moses, more than 7 take a time which is within 1 standard deviation of the mean.

1 The random variable \(Y\) is normally distributed with mean equal to five times the standard deviation. It is given that \(\mathrm { P } ( Y > 20 ) = 0.0732\). Find the mean.

1. The time, \(X\) hours, for which people sleep in one night has a normal distribution with mean 7.15 hours and standard deviation 0.88 hours.
   1. Find the probability that a randomly chosen person sleeps for less than 8 hours in a night. [2]
   2. Find the value of \(q\) such that P\((X < q) = 0.75\). [3]
2. The random variable \(Y\) has the distribution N\((\mu, \sigma^2)\), where \(2\sigma = 3\mu\) and \(\mu \neq 0\). Find P\((Y > 4\mu)\). [3]

4 The mean of a certain normally distributed variable is four times the standard deviation. The probability that a randomly chosen value is greater than 5 is 0.15 .

1. Find the mean and standard deviation.
2. 200 values of the variable are chosen at random. Find the probability that at least 160 of these values are less than 5 .

1 In a normal distribution with mean 9.3, the probability of a randomly chosen value being greater than 5.6 is 0.85 . Find the standard deviation.

1 Packets of tea are labelled as containing 250 g . The actual weight of tea in a packet has a normal distribution with mean 260 g and standard deviation \(\sigma \mathrm { g }\). Any packet with a weight less than 250 g is classed as 'underweight'. Given that \(1 \%\) of packets of tea are underweight, find the value of \(\sigma\). [3]

The rules for the weight of a cricket ball state: ``The ball, when new, shall weigh not less than 155.9 g, nor more than 163 g.'' A company produces cricket balls whose weights are normally distributed. It wants 99\% of the balls it produces to be an acceptable weight.

1. What is the largest acceptable standard deviation? [3]

The weights of the cricket balls are in fact normally distributed with mean 159.5 grams and standard deviation 1.2 grams. The company also produces tennis balls. The weights of the tennis balls are normally distributed with mean 58.5 grams and standard deviation 1.3 grams.

1. Find the probability that the weight of a randomly chosen cricket ball is more than three times the weight of a randomly chosen tennis ball. [6]

3 Lengths of rolls of parcel tape have a normal distribution with mean 75 m , and 15\% of the rolls have lengths less than 73 m .

1. Find the standard deviation of the lengths. Alison buys 8 rolls of parcel tape.
2. Find the probability that fewer than 3 of these rolls have lengths more than 77 m .

7 The times, in minutes, that Karli spends each day on social media are normally distributed with mean 125 and standard deviation 24.

1. 1. On how many days of the year ( 365 days) would you expect Karli to spend more than 142 minutes on social media?
   2. Find the probability that Karli spends more than 142 minutes on social media on fewer than 2 of 10 randomly chosen days.
2. On \(90 \%\) of days, Karli spends more than \(t\) minutes on social media. Find the value of \(t\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\frac { 1 } { 4 } \mu\). It is given that \(\mathrm { P } ( X > 20 ) = 0.04\).

1. Find \(\mu\).
2. Find \(\mathrm { P } ( 10 < X < 20 )\).
3. 250 independent observations of \(X\) are taken. Find the probability that at least 235 of them are less than 20.

The random variable \(X\) is normally distributed with mean 17. The probability that \(X\) is less than 16 is 0.3707.

1. Calculate the standard deviation of \(X\). [4 marks]
2. In 75 independent observations of \(X\), how many would you expect to be greater than 20? [6 marks]

4 The heights of students at the Mainland college are normally distributed with mean 148 cm and standard deviation 8 cm .

1. The probability that a Mainland student chosen at random has a height less than \(h \mathrm {~cm}\) is 0.67 . Find the value of \(h\).
   120 Mainland students are chosen at random.
2. Find the number of these students that would be expected to have a height within half a standard deviation of the mean.

4 A mathematical puzzle is given to a large number of students. The times taken to complete the puzzle are normally distributed with mean 14.6 minutes and standard deviation 5.2 minutes.

1. In a random sample of 250 of the students, how many would you expect to have taken more than 20 minutes to complete the puzzle?
   All the students are given a second puzzle to complete. Their times, in minutes, are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is found that \(20 \%\) of the students have times less than 14.5 minutes and \(67 \%\) of the students have times greater than 18.5 minutes.
2. Find the value of \(\mu\) and the value of \(\sigma\).

A company makes two cars, model \(A\) and model \(B\). The distance that model \(A\) travels on 10 litres of petrol is normally distributed with mean 109 km and variance 72.25 km\(^2\). The distance that model \(B\) travels on 10 litres of petrol is normally distributed with mean 108.5 km and variance 169 km\(^2\). In a trial, one of each model is filled with 10 litres of petrol and sent on a journey of 110 km. Find which model has the greater probability of completing this journey, and state the value of this probability. [8 marks]

2

1. Tim rings the church bell in his village every Sunday morning. The time that he spends ringing the bell may be modelled by a normal distribution with mean 7.5 minutes and standard deviation 1.6 minutes. Determine the probability that, on a particular Sunday morning, the time that Tim spends ringing the bell is:
   1. at most 10 minutes;
   2. more than 6 minutes;
   3. between 5 minutes and 10 minutes.
2. June rings the same church bell for weekday weddings. The time that she spends, in minutes, ringing the bell may be modelled by the distribution \(\mathrm { N } \left( \mu , 2.4 ^ { 2 } \right)\). Given that 80 per cent of the times that she spends ringing the bell are less than 15 minutes, find the value of \(\mu\).
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-04_1477_1707_1226_153}

2 In a certain country, the heights of the adult population are normally distributed with mean 1.64 m and standard deviation 0.25 m .

1. Find the probability that an adult chosen at random from this country will have height greater than 1.93 m . \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-04_2716_35_143_2012} \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-05_2724_35_136_20} In another country, the heights of the adult population are also normally distributed. \(33 \%\) of the adult population have height less than \(1.56 \mathrm {~m} .25 \%\) of the adult population have height greater than 1.86 m .
2. Find the mean and the standard deviation of this distribution.

4. The random variable \(Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( Y < 17 ) = 0.6\) find

1. \(\mathrm { P } ( Y > 17 )\)
2. \(\mathrm { P } ( \mu < Y < 17 )\)
3. \(\mathrm { P } ( Y < \mu \mid Y < 17 )\)

4. The random variable \(Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( Y < 17 ) = 0.6\) find

1. \(\mathrm { P } ( Y > 17 )\)
2. \(\mathrm { P } ( \mu < Y < 17 )\)
3. \(\mathrm { P } ( Y < \mu \mid Y < 17 )\)

1. The lifetime, \(L\) hours, of a battery has a normal distribution with mean 18 hours and standard deviation 4 hours.

Alice's calculator requires 4 batteries and will stop working when any one battery reaches the end of its lifetime.

1. Find the probability that a randomly selected battery will last for longer than 16 hours. At the start of her exams Alice put 4 new batteries in her calculator. She has used her calculator for 16 hours, but has another 4 hours of exams to sit.
2. Find the probability that her calculator will not stop working for Alice's remaining exams. Alice only has 2 new batteries so, after the first 16 hours of her exams, although her calculator is still working, she randomly selects 2 of the batteries from her calculator and replaces these with the 2 new batteries.
3. Show that the probability that her calculator will not stop working for the remainder of her exams is 0.199 to 3 significant figures. After her exams, Alice believed that the lifetime of the batteries was more than 18 hours. She took a random sample of 20 of these batteries and found that their mean lifetime was 19.2 hours.
4. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test Alice's belief.

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.

9 The continuous random variable \(X\) has distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 2 , \\ \frac { 1 } { 8 } x - \frac { 1 } { 4 } & 2 \leqslant x \leqslant 10 , \\ 1 & x > 10 . \end{cases}$$ Find the value of \(k\) for which \(\mathrm { P } ( X \geqslant k ) = 0.6\). The random variable \(Y\) is defined by \(Y = 2 \ln X\). Find the distribution function of \(Y\). Find the probability density function of \(Y\) and sketch its graph.

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.

1 It is given that \(X \sim \mathrm {~N} \left( 1.5,3.2 ^ { 2 } \right)\). Find the probability that a randomly chosen value of \(X\) is less than - 2.4 .

\section*{INSTRUCTIONS}

• Answer all questions.
• Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
• Write your name, centre number and candidate number in the boxes at the top of the page.
• Write your answer to each question in the space provided.
• Do not use an erasable pen or correction fluid.
• Do not write on any bar codes.
• If additional space is needed, you should use the lined page at the end of this booklet; the question number or numbers must be clearly shown.
• You should use a calculator where appropriate.
• You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
• Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

\section*{INFORMATION}

• The total mark for this paper is 50.
• The number of marks for each question or part question is shown in brackets [ ].

1 Each of the 180 students at a college plays exactly one of the piano, the guitar and the drums. The numbers of male and female students who play the piano, the guitar and the drums are given in the following table. A student at the college is chosen at random.

1. Find the probability that the student plays the guitar.
2. Find the probability that the student is male given that the student plays the drums.
3. Determine whether the events 'the student plays the guitar' and 'the student is female' are independent, justifying your answer.
   2 A group of 6 people is to be chosen from 4 men and 11 women.
   1. In how many different ways can a group of 6 be chosen if it must contain exactly 1 man?
      Two of the 11 women are sisters Jane and Kate.
   2. In how many different ways can a group of 6 be chosen if Jane and Kate cannot both be in the group?
      3 A bag contains 5 yellow and 4 green marbles. Three marbles are selected at random from the bag, without replacement.
   3. Show that the probability that exactly one of the marbles is yellow is \(\frac { 5 } { 14 }\).
      The random variable \(X\) is the number of yellow marbles selected.
   4. Draw up the probability distribution table for \(X\).
   5. Find \(\mathrm { E } ( X )\).
      4
   6. In how many different ways can the 9 letters of the word TELESCOPE be arranged?
   7. In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C ?
      5 In a certain region, the probability that any given day in October is wet is 0.16 , independently of other days.
   8. Find the probability that, in a 10-day period in October, fewer than 3 days will be wet.
   9. Find the probability that the first wet day in October is 8 October.
   10. For 4 randomly chosen years, find the probability that in exactly 1 of these years the first wet day in October is 8 October.
       6 The times taken, in minutes, to complete a particular task by employees at a large company are normally distributed with mean 32.2 and standard deviation 9.6.
   11. Find the probability that a randomly chosen employee takes more than 28.6 minutes to complete the task.
   12. \(20 \%\) of employees take longer than \(t\) minutes to complete the task. Find the value of \(t\).
   13. Find the probability that the time taken to complete the task by a randomly chosen employee differs from the mean by less than 15.0 minutes.
       7 The distances, \(x \mathrm {~m}\), travelled to school by 140 children were recorded. The results are summarised in the table below.

       Distance, \(x \mathrm {~m}\)	\(x \leqslant 200\)	\(x \leqslant 300\)	\(x \leqslant 500\)	\(x \leqslant 900\)	\(x \leqslant 1200\)	\(x \leqslant 1600\)
       Cumulative frequency	16	46	88	122	134	140

   14. On the grid, draw a cumulative frequency graph to represent these results. \includegraphics[max width=\textwidth, alt={}, center]{93ff111b-0267-4b4b-a41c-64c3307115af-10_1593_1593_701_306}
   15. Use your graph to estimate the interquartile range of the distances.
   16. Calculate estimates of the mean and standard deviation of the distances.
       If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 Alan's journey time to work can be modelled by a normal distribution with standard deviation 6 minutes. Alan measures the journey time to work for a random sample of 5 journeys. The mean of the 5 journey times is 36 minutes. 3

1. Construct a 95\% confidence interval for Alan's mean journey time to work, giving your values to one decimal place.
   3
2. Alan claims that his mean journey time to work is 30 minutes.
   State, with a reason, whether or not the confidence interval found in part (a) supports Alan's claim.
   3
3. Suppose that the standard deviation is not known but a sample standard deviation is found from Alan's sample and calculated to be 6 Explain how the working in part (a) would change.

The random variable \(X\) has the normal distribution \(N(2, 1.7^2)\).

1. State the standard deviation of \(X\). [1 mark]
2. Find \(P(X < 0)\). [2 marks]
3. Find \(P(0.6 < X < 3.4)\). [4 marks]

\(X\) is a continuous random variable such that \(X \sim N(\mu, \sigma^2)\). On the sketch of this Normal distribution in the Printed Answer Booklet, shade the area bounded by the curve, the \(x\)-axis and the lines \(x = \mu \pm \sigma\). [2]

2 The volume of ink in a certain type of ink cartridge has a normal distribution with mean 30 ml and standard deviation 1.5 ml . People in an office use a total of 8 cartridges of this ink per month. Find the expected number of cartridges per month that contain less than 28.9 ml of this ink.

6. Skilled operators make a particular component for an engine. The company believes that the time taken to make this component may be modelled by the normal distribution. They timed one of their operators, Sheila, over a long period. They find that when she makes a component, she takes over 90 minutes to make one \(10 \%\) of the time, and that \(20 \%\) of the time, a component was less than 70 minutes to make. Estimate the mean and standard deviation of the time Sheila takes to make a component.

2 A random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 10 observations of \(C\) is obtained, and the results are summarised as $$n = 10 , \Sigma c = 380 , \Sigma c ^ { 2 } = 14602 .$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Hence calculate an estimate of the probability that \(C > 40\).

2 The lengths of the rods produced by a company are normally distributed with mean 55.6 mm and standard deviation 1.2 mm .

1. In a random sample of 400 of these rods, how many would you expect to have length less than 54.8 mm ?
2. Find the probability that a randomly chosen rod produced by this company has a length that is within half a standard deviation of the mean.

\(X \sim \text{N}(14, 0.35)\) Find the standard deviation of \(X\), correct to two decimal places. Circle your answer. [1 mark] 0.12 \quad\quad 0.35 \quad\quad 0.59 \quad\quad 1.78

1 \includegraphics[max width=\textwidth, alt={}, center]{6f677fc6-3ca2-4a0d-82a2-69a7cbb8574d-2_211_1169_267_488} Measurements of wind speed on a certain island were taken over a period of one year. A box-andwhisker plot of the data obtained is displayed above, and the values of the quartiles are as shown. It is suggested that wind speed can be modelled approximately by a normal distribution with mean \(\mu \mathrm { km } \mathrm { h } ^ { - 1 }\) and standard deviation \(\sigma \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Estimate the value of \(\mu\).
2. Estimate the value of \(\sigma\).

4 It is known that 20\% of male giant pandas in a certain area weigh more than 121 kg and \(71.9 \%\) weigh more than 102 kg . Weights of male giant pandas in this area have a normal distribution. Find the mean and standard deviation of the weights of male giant pandas in this area.

5 The random variable \(X\) is such that \(X \sim \mathrm {~N} ( 82,126 )\).

1. A value of \(X\) is chosen at random and rounded to the nearest whole number. Find the probability that this whole number is 84 .
2. Five independent observations of \(X\) are taken. Find the probability that at most one of them is greater than 87.
3. Find the value of \(k\) such that \(\mathrm { P } ( 87 < X < k ) = 0.3\).

1 The random variable \(X\) is such that \(X \sim \mathrm {~N} ( 20,49 )\). Given that \(\mathrm { P } ( X > k ) = 0.25\), find the value of \(k\).

4 In a certain country the time taken for a common infection to clear up is normally distributed with mean \(\mu\) days and standard deviation 2.6 days. \(25 \%\) of these infections clear up in less than 7 days.

1. Find the value of \(\mu\). In another country the standard deviation of the time taken for the infection to clear up is the same as in part (i), but the mean is 6.5 days. The time taken is normally distributed.
2. Find the probability that, in a randomly chosen case from this country, the infection takes longer than 6.2 days to clear up.

5

1. The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(3 \mu = 7 \sigma ^ { 2 }\) and that \(\mathrm { P } ( X > 2 \mu ) = 0.1016\). Find \(\mu\) and \(\sigma\).
2. It is given that \(Y \sim \mathrm {~N} ( 33,21 )\). Find the value of \(a\) given that \(\mathrm { P } ( 33 - a < Y < 33 + a ) = 0.5\).

1 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 22 ) = 0.242\). Find the value of \(\mu\).

The random variable \(G\) has a normal distribution. It is known that $$\text{P}(G < 56.2) = \text{P}(G > 63.8) = 0.1.$$ Find P(\(G > 65\)). [6]

8 A market gardener records the masses of a random sample of 100 of this year's crop of plums. The table shows his results.

1. Explain why the normal distribution might be a reasonable model for this distribution. The market gardener models the distribution of masses by \(\mathrm { N } \left( 47.5,10 ^ { 2 } \right)\).
2. Find the number of plums in the sample that this model would predict to have masses in the range:
   1. \(35 \leq m < 45\)
   2. \(m < 25\).
3. Use your answers to parts (b)(i) and (b)(ii) to comment on the suitability of this model. The market gardener plans to use this model to predict the distribution of the masses of next year's crop of plums.
4. Comment on this plan.

3 Buildings in a certain city centre are classified by height as tall, medium or short. The heights can be modelled by a normal distribution with mean 50 metres and standard deviation 16 metres. Buildings with a height of more than 70 metres are classified as tall.

1. Find the probability that a building chosen at random is classified as tall.
2. The rest of the buildings are classified as medium and short in such a way that there are twice as many medium buildings as there are short ones. Find the height below which buildings are classified as short.

1 It is given that \(X \sim \mathrm {~N} ( 28.3,4.5 )\). Find the probability that a randomly chosen value of \(X\) lies between 25 and 30 .

7 Josie aims to catch a bus which departs at a fixed time every day. Josie arrives at the bus stop \(T\) minutes before the bus departs, where \(T \sim \mathrm {~N} \left( 5.3,2.1 ^ { 2 } \right)\).

1. Find the probability that Josie has to wait longer than 6 minutes at the bus stop.
   On \(5 \%\) of days Josie has to wait longer than \(x\) minutes at the bus stop.
2. Find the value of \(x\).
3. Find the probability that Josie waits longer than \(x\) minutes on fewer than 3 days in 10 days.
4. Find the probability that Josie misses the bus.

4 Trees in the Redian forest are classified as tall, medium or short, according to their height. The heights can be modelled by a normal distribution with mean 40 m and standard deviation 12 m . Trees with a height of less than 25 m are classified as short.

1. Find the probability that a randomly chosen tree is classified as short.
   Of the trees that are classified as tall or medium, one third are tall and two thirds are medium.
2. Show that the probability that a randomly chosen tree is classified as tall is 0.298 , correct to 3 decimal places.
3. Find the height above which trees are classified as tall.

10 The screenshot in Fig. 10 shows the probability distribution for the continuous random variable \(X\), where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). \begin{figure}[h] \end{figure} The area of each of the unshaded regions under the curve is 0.025 . The lower boundary of the shaded region is at 16.452 and the upper boundary of the shaded region is at 21.548 .

1. Calculate the value of \(\mu\).
2. Calculate the value of \(\sigma ^ { 2 }\).
3. \(Y\) is the random variable given by \(Y = 4 X + 5\).
   (A) Write down the distribution of \(Y\).
   (B) Find \(\mathrm { P } ( \mathrm { Y } > 90 )\).

A call-centre dealing with complaints collected data on how long customers had to wait before an operator was free to take their call. The lower quartile of the data was 12.7 minutes and the interquartile range was 5.8 minutes.

1. Find the value of the upper quartile of the data. [1 mark]

It is suggested that a normal distribution could be used to model the waiting time.

1. Calculate correct to 3 significant figures the mean and variance of this normal distribution based on the values of the quartiles. [8 marks]

The actual mean and variance of the data were 15.3 minutes and 20.1 minutes\(^2\) respectively.

1. Comment on the suitability of the model. [2 marks]

6 The diameters of apples in an orchard have a normal distribution with mean 5.7 cm and standard deviation 0.8 cm . Apples with diameters between 4.1 cm and 5 cm can be used as toffee apples.

1. Find the probability that an apple selected at random can be used as a toffee apple.
2. 250 apples are chosen at random. Use a suitable approximation to find the probability that fewer than 50 can be used as toffee apples.