2.04e Normal distribution: as model N(mu, sigma^2)

3 A bus runs from point A on the outskirts of a city, stops at point B outside the rail station, and continues to point C in the city centre.
The journey times for the sections A to B and B to C vary according to traffic conditions, and are modelled by independent Normal distributions with means and standard deviations as shown in the table.

1. Find the probability that a randomly chosen journey from A to B takes less than the scheduled time of 23 minutes. For every journey, the bus stops for 1 minute when it reaches B to drop off and pick up passengers.
2. Find the probability that a randomly chosen journey from A to C takes less than the scheduled time of 50 minutes. Mary travels on the bus from the station at B to her workplace at C every working day. You should assume that times for her bus journeys on different days are independent.
3. Find the probability that the total time taken for her five journeys on the bus in a randomly chosen week is at least \(2 \frac { 1 } { 2 }\) hours.
4. Comment on the assumption that times on different days are independent.

6 The length \(L\) of a particular type of fence panel is Normally distributed with mean 179.2 cm and standard deviation 0.8 cm . You should assume that the lengths of individual fence panels are independent of each other.

1. Find the probability that the length of a randomly chosen fence panel is at least 180 cm .
2. Find the probability that the total length of 5 randomly chosen fence panels is less than 895 cm . The width \(W\) of a fence post is Normally distributed with mean 9.8 cm and standard deviation 0.3 cm . A straight fence is constructed using 6 posts and 5 panels with no gaps between them. Fig. 6 shows a view from above of the first two posts, the first panel and the start of the second panel. You should assume that the lengths of fence panels and widths of fence posts are independent. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4caa7409-cb32-41da-ad64-012a45753296-6_213_1522_934_244} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
3. Determine the probability that the total length of the fence, including the posts, is less than 9.5 m .
4. State another assumption that is necessary for the calculation of the probability in part (c) to be valid.

5 A food company makes mini apple pies. The weight of pastry in a pie is Normally distributed with mean 75 g and standard deviation 4 g . The weight of filling in a pie is Normally distributed with mean 130 g and standard deviation 8 g . You should assume that the weights of pastry and filling in a pie are independent.

1. Find the probability that the weight of pastry in a randomly chosen pie is between 70 g and 80 g .
2. Find the probability that the mean weight of filling in 10 randomly chosen pies is at least 125 g. The pies are sold in packs of 4 . The weight of the packaging is Normally distributed with mean 165 g and standard deviation 6 g .
3. In order to find the probability that the total weight of a pack of 4 pies is less than 1 kg , you must assume that the weight of the packaging is independent of the weight of the pies.
   1. State another necessary assumption.
   2. Given that the assumptions are valid, calculate this probability.

7 Two flatmates work at the same location. One of them takes the bus to work and the other one cycles. Journey times, measured in minutes, are distributed as follows.

• By bus: Normally distributed with mean 23 and standard deviation 6
• By bicycle: Normally distributed with mean 21 and standard deviation 2

You should assume that all journey times are independent.

1. One morning the two flatmates set out at the same time. Find the probability that the person who takes the bus arrives before the cyclist.
2. Find the probability that the total time taken for 5 bus journeys is less than 2 hours.
3. Comment on the assumption that all journey times are independent. \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
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Jake works for a parcel delivery company. The masses, in kilograms, of parcels he delivers are normally distributed with mean \(2 \cdot 2\) and standard deviation \(0 \cdot 3\).

1. Calculate the probability that a randomly selected parcel will have a mass less than 1.8 kg .
   Jake delivers the lightest \(80 \%\) of parcels on his bike. The rest he puts in his car and delivers by car.
2. Find the mass of the heaviest parcel he would deliver by bike.
   
3. He randomly selects a parcel from his car. Find the probability that it has a mass less than 3 kg .
   
4. In the run-up to Christmas, Jake believes that the parcels he has to deliver are, on average, heavier. He assumes that the standard deviation is unchanged. He randomly selects 20 parcels and finds that their total mass is 46 kg . Test Jake's belief at the \(5 \%\) level of significance. Jake delivers each parcel to one of three areas, \(A , B\) or \(C\). The probabilities that a parcel has destination area \(A , B\) and \(C\) are \(\frac { 1 } { 2 } , \frac { 1 } { 6 }\) and \(\frac { 1 } { 3 }\) respectively. All parcels are considered to be independent.
5. On a particular day, Jake has three parcels to deliver. Find the probability that he will have to deliver to all three areas.
   
6. On a different day, Jake has two parcels to deliver. Find the probability that he will have to deliver to more than one area.
   

5. The masses, \(X\), in kg, of men who work for a large company are normally distributed with mean 75 and standard deviation 10.

1. Find the probability that the mean mass of a random sample of 5 men is less than 70 kg .
2. The mean mass, in kg , of a random sample of \(n\) men drawn from this distribution is \(\bar { X }\). Given that \(\mathrm { P } ( \bar { X } > 80 )\) is approximately \(0 \cdot 007\), find \(n\). The masses, in kg, of women who work for the company are normally distributed with mean 68 and standard deviation 6 . A lift in the company building will not move if the total mass in the lift is more than 500 kg .
3. A random sample of 3 men and 4 women get in the lift. Find the probability that the lift will not move.
4. State a modelling assumption you have made in calculating your answer for part (c).

1. Scaffolding poles come in two sizes, long and short. The length \(L\) of a long pole has the normal distribution \(\mathrm { N } \left( 19.6,0.6 ^ { 2 } \right)\). The length \(S\) of a short pole has the normal distribution N(4.8, 0.32). The random variables \(L\) and \(S\) are independent.

A long pole and a short pole are selected at random.

1. Find the probability that the length of the long pole is more than 4 times the length of the short pole. Show your working clearly. Four short poles are selected at random and placed end to end in a row. The random variable \(T\) represents the length of the row.
2. Find the distribution of \(T\).
3. Find \(\mathrm { P } ( | L - T | < 0.2 )\)

15 You must show detailed reasoning in this question. The screenshot in Fig. 15 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 distribution is symmetrical about the line \(x = 35\) and there is a point of inflection at \(x = 31\).
Fifty independent readings of \(X\) are made. Show that the probability that at least 45 of these readings are between 30 and 40 is less than 0.05 .

3 A discrete random variable \(X\) has the distribution \(\mathrm { U } ( 11 )\).
The mean of 50 observations of \(X\) is denoted by \(\bar { X }\).
Use an approximate method, which should be justified, to find \(\mathrm { P } ( \bar { X } \leqslant 6.10 )\).

1 The performance of a piece of music is being recorded. The piece consists of three sections, \(A , B\) and \(C\). The times, in seconds, taken to perform the three sections are normally distributed random variables with the following means and standard deviations.

1. Assume first that the times for the three sections are independent. Find the probability that the total length of the performance is greater than 720.0 seconds.
2. In fact sections \(A\) and \(C\) are musically identical, and the recording is made by using a single performance of section \(A\) twice, together with a performance of section \(B\). In this case find the probability that the total length of the performance is greater than 720.0 seconds.

8 The masses, \(X\) grams, of tomatoes are normally distributed. Half of the tomatoes have masses greater than 56.0 g and \(70 \%\) of the tomatoes have masses greater than 53.0 g .

1. Find the percentage of tomatoes with masses greater than 59.0 g .
2. Find the percentage of tomatoes with masses greater than 65.0 g .
3. Given that \(\mathrm { P } ( a < X < 50 ) = 0.1\), find \(a\).

9 The finance department of a retail firm recorded the daily income each day for 300 days. The results are summarised in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-6_689_1575_488_246}

1. Find the number of days on which the daily income was between \(\pounds 4000\) and \(\pounds 6000\).
2. Calculate an estimate of the number of days on which the daily income was between \(\pounds 2700\) and \(\pounds 3600\).
3. Use the midpoints of the classes to show that an estimate of the mean daily income is \(\pounds 3275\). An estimate of the standard deviation of the daily income is \(\pounds 1060\). The finance department uses the distribution \(\mathrm { N } \left( 3275,1060 ^ { 2 } \right)\) to model the daily income, in pounds.
4. Calculate the number of days on which, according to this model, the daily income would be between \(\pounds 4000\) and \(\pounds 6000\).
5. It is given that approximately \(95 \%\) of values of the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) lie within the range \(\mu \pm 2 \sigma\). Without further calculation, use this fact to comment briefly on whether the proposed model is a good fit to the data illustrated in the histogram.

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.

1. Find the probability that a randomly chosen employee takes more than 28.6 minutes to complete the task.
2. \(20 \%\) of employees take longer than \(t\) minutes to complete the task. Find the value of \(t\).
3. 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.

Jia writes a computer program that randomly generates values from a normal distribution. He sets the mean as 40 and the standard deviation as 2.4

1. Find the probability that a particular value generated by the computer program is less than 37 Jia changes the mean to \(m\) but leaves the standard deviation as 2.4
   The computer program then randomly generates 2 independent values from this normal distribution. The probability that both of these values are greater than 32 is 0.16
2. Find the value of \(m\), giving your answer to 2 decimal places. Jia now changes the mean to 4 and the standard deviation to 8
   The computer program then randomly generates 5 independent values from this normal distribution.
3. Find the probability that at least one of these values is negative.

At a school athletics day, the distances, in metres, achieved by students in the long jump are modelled by the normal distribution with mean 3.3 m and standard deviation 0.6 m

1. Find an estimate for the proportion of students who jump less than 2.5 m The long jump competition consists of 2 jumps. All the students can take part in the first jump and the \(40 \%\) who jump the greatest distance in their first jump qualify for the second jump.
2. Find an estimate for the minimum distance achieved in the first jump in order to qualify for the second jump.
   Give your answer correct to 4 significant figures.
3. Find an estimate for the median distance achieved in the first jump by those who qualify for the second jump. The distance of the second jump is independent of the distance of the first jump and is modelled with the same normal distribution. Students who jump a distance greater than 4.1 m in their second jump receive a certificate. At the start of the long jump competition, a student is selected at random.
4. Find the probability that this student will receive a certificate.
   

3. Hei and Tang are designing some pieces of art. They collected a large number of sticks. The random variable \(L\) represents the length of a stick in centimetres and has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). They sorted the sticks into lengths and painted them.
They found that \(60 \%\) of the sticks were longer than 45 cm and these were painted red, whilst \(15 \%\) of the sticks were shorter than 35 cm and these were painted blue. The remaining sticks were painted yellow.

1. Show that \(\mu\) and \(\sigma\) satisfy $$45 + 0.2533 \sigma = \mu$$
2. Find a second equation in \(\mu\) and \(\sigma\).
3. Hence find the value of \(\mu\) and the value of \(\sigma\).
4. Find
   1. \(\mathrm { P } ( L > 35 \mid L < 45 )\)
   2. \(\mathrm { P } ( L < 45 \mid L > 35 )\) Hei created her piece of art using a random selection of blue and yellow sticks.
      Tang created his piece of art using a random selection of red and yellow sticks.
      Hei and Tang each used the same number of sticks to create their piece of art.
      George is viewing Hei's and Tang's pieces of art. He finds a yellow stick on the floor that has fallen from one of these pieces.
5. With reference to your answers to part (d), state, giving a reason, whether the stick is more likely to have fallen from Hei's or Tang's piece of art.

Xiang is designing shelves for a bookshop. The height, \(H \mathrm {~cm}\), of books is modelled by the normal distribution with mean 25.1 cm and standard deviation 5.5 cm

1. Show that \(\mathrm { P } ( H > 30.8 ) = 0.15\) Xiang decided that the smallest \(5 \%\) of books and books taller than 30.8 cm would not be placed on the shelves. All the other books will be placed on the shelves.
2. Find the range of heights of books that will be placed on the shelves.
   (3) The books that will be placed on the shelves have heights classified as small, medium or large.
   The numbers of small, medium and large books are in the ratios \(2 : 3 : 3\)
3. The medium books have heights \(x \mathrm {~cm}\) where \(m < x < d\)
   1. Show that \(d = 25.8\) to 1 decimal place.
   2. Find the value of \(m\) Xiang wants 2 shelves for small books, 3 shelves for medium books and 3 shelves for large books.
      These shelves will be placed one above another and made of wood that is 1 cm thick.
4. Work out the minimum total height needed.

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)

8. The lifetimes of bulbs used in a lamp are normally distributed. A company \(X\) sells bulbs with a mean lifetime of 850 hours and a standard deviation of 50 hours.

1. Find the probability of a bulb, from company \(X\), having a lifetime of less than 830 hours.
2. In a box of 500 bulbs, from company \(X\), find the expected number having a lifetime of less than 830 hours. A rival company \(Y\) sells bulbs with a mean lifetime of 860 hours and \(20 \%\) of these bulbs have a lifetime of less than 818 hours.
3. Find the standard deviation of the lifetimes of bulbs from company \(Y\). Both companies sell the bulbs for the same price.
4. State which company you would recommend. Give reasons for your answer.
   \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{}
   \end{table}

2. The lifetimes of batteries used for a computer game have a mean of 12 hours and a standard deviation of 3 hours. Battery lifetimes may be assumed to be normally distributed. Find the lifetime, \(t\) hours, of a battery such that 1 battery in 5 will have a lifetime longer than \(t\).

4 Chopped lettuce is sold in bags nominally containing 100 grams.
The weight, \(X\) grams, of chopped lettuce, delivered by the machine filling the bags, may be assumed to be normally distributed with mean \(\mu\) and standard deviation 4.

1. Assuming that \(\mu = 106\), determine the probability that a randomly selected bag of chopped lettuce:
   1. weighs less than 110 grams;
   2. is underweight.
2. Determine the minimum value of \(\mu\) so that at most 2 per cent of bags of chopped lettuce are underweight. Give your answer to one decimal place.
3. Boxes each contain 10 bags of chopped lettuce. The mean weight of a bag of chopped lettuce in a box is denoted by \(\bar { X }\). Given that \(\mu = 108.5\) :
   1. write down values for the mean and variance of \(\bar { X }\);
   2. determine the probability that \(\bar { X }\) exceeds 110 .

6 When Monica walks to work from home, she uses either route A or route B.

1. Her journey time, \(X\) minutes, by route A may be assumed to be normally distributed with a mean of 37 and a standard deviation of 8 . Determine:
   1. \(\mathrm { P } ( X < 45 )\);
   2. \(\mathrm { P } ( 30 < X < 45 )\).
2. Her journey time, \(Y\) minutes, by route B may be assumed to be normally distributed with a mean of 40 and a standard deviation of \(\sigma\). Given that \(\mathrm { P } ( Y > 45 ) = 0.12\), calculate the value of \(\sigma\).
3. If Monica leaves home at 8.15 am to walk to work hoping to arrive by 9.00 am , state, with a reason, which route she should take.
4. When Monica travels to work from home by car, her journey time, \(W\) minutes, has a mean of 18 and a standard deviation of 12 . Estimate the probability that, for a random sample of 36 journeys to work from home by car, Monica's mean time is more than 20 minutes.
5. Indicate where, if anywhere, in this question you needed to make use of the Central Limit Theorem.

1 Draught excluder for doors and windows is sold in rolls of nominal length 10 metres.
The actual length, \(X\) metres, of draught excluder on a roll may be modelled by a normal distribution with mean 10.2 and standard deviation 0.15 .

1. Determine:
   1. \(\mathrm { P } ( X < 10.5 )\);
   2. \(\mathrm { P } ( 10.0 < X < 10.5 )\).
2. A customer randomly selects six 10 -metre rolls of the draught excluder. Calculate the probability that all six rolls selected contain more than 10 metres of draught excluder.

5 In a random sample of 12 bags of flour, the weight, in grams, of flour in each bag was recorded as follows. \(\begin{array} { l l l l l l l l l l l l } 1011 & 995 & 1018 & 1022 & 1014 & 1005 & 1017 & 1015 & 993 & 1018 & 992 & 1020 \end{array}\)

1. It may be assumed that the weight of flour in a bag is normally distributed with a standard deviation of 10.5 grams.
   1. Construct a \(98 \%\) confidence interval for the mean weight, \(\mu\) grams, of flour in a bag, giving the limits to four significant figures.
   2. State why, in constructing your confidence interval, use of the Central Limit Theorem was not necessary.
   3. If the distribution of the weight of flour in a bag was unknown, indicate a minimum number of weights that you would consider necessary for a confidence interval for \(\mu\) to be valid.
2. The statement ' 1 kg ' is printed on each bag. Comment on this statement using both the confidence interval that you constructed in part (a)(i) and the weights of the given sample of 12 bags.
3. Given that \(\mu = 1000\), state the probability that a \(98 \%\) confidence interval for \(\mu\) will not contain 1000.
   (l mark)