2.04f Find normal probabilities: Z transformation

1. A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, \(D \mathrm { ml }\), follows a normal distribution with mean 25 ml

Given that 15\% of bottles contain less than 24.63 ml

1. find, to 2 decimal places, the value of \(k\) such that \(\mathrm { P } ( 24.63 < D < k ) = 0.45\) A random sample of 200 bottles is taken.
2. Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and \(k \mathrm { ml }\) The machine is adjusted so that the standard deviation of the liquid put in the bottles is now 0.16 ml Following the adjustments, Hannah believes that the mean amount of liquid put in each bottle is less than 25 ml She takes a random sample of 20 bottles and finds the mean amount of liquid to be 24.94 ml
3. Test Hannah's belief at the \(5 \%\) level of significance. You should state your hypotheses clearly.

1. A manufacturer uses a machine to make metal rods.

The length of a metal rod, \(L \mathrm {~cm}\), is normally distributed with

• a mean of 8 cm
• a standard deviation of \(x \mathrm {~cm}\)

Given that the proportion of metal rods less than 7.902 cm in length is \(2.5 \%\)

1. show that \(x = 0.05\) to 2 decimal places.
2. Calculate the proportion of metal rods that are between 7.94 cm and 8.09 cm in length. The cost of producing a single metal rod is 20p
   A metal rod
   • where \(L < 7.94\) is sold for scrap for 5 p
   • where \(7.94 \leqslant L \leqslant 8.09\) is sold for 50 p
   • where \(L > 8.09\) is shortened for an extra cost of 10 p and then sold for 50 p
   • Calculate the expected profit per 500 of the metal rods.
   Give your answer to the nearest pound. The same manufacturer makes metal hinges in large batches.
   The hinges each have a probability of 0.015 of having a fault.
   A random sample of 200 hinges is taken from each batch and the batch is accepted if fewer than 6 hinges are faulty. The manufacturer's aim is for 95\% of batches to be accepted.
3. Explain whether the manufacturer is likely to achieve its aim.

A study was made of adult men from region \(A\) of a country. It was found that their heights were normally distributed with a mean of 175.4 cm and standard deviation 6.8 cm .

1. Find the proportion of these men that are taller than 180 cm . A student claimed that the mean height of adult men from region \(B\) of this country was different from the mean height of adult men from region \(A\). A random sample of 52 adult men from region \(B\) had a mean height of 177.2 cm
   The student assumed that the standard deviation of heights of adult men was 6.8 cm both for region \(A\) and region \(B\).
2. Use a suitable test to assess the student's claim. You should
   • state your hypotheses clearly
   • use a \(5 \%\) level of significance
   • Find the \(p\)-value for the test in part (b)

1. A medical researcher is studying the number of hours, \(T\), a patient stays in hospital following a particular operation.

The histogram on the page opposite summarises the results for a random sample of 90 patients.

1. Use the histogram to estimate \(\mathrm { P } ( 10 < T < 30 )\) For these 90 patients the time spent in hospital following the operation had
   • a mean of 14.9 hours
   • a standard deviation of 9.3 hours
   Tomas suggests that \(T\) can be modelled by \(\mathrm { N } \left( 14.9,9.3 ^ { 2 } \right)\)
2. With reference to the histogram, state, giving a reason, whether or not Tomas' model could be suitable. Xiang suggests that the frequency polygon based on this histogram could be modelled by a curve with equation $$y = k x \mathrm { e } ^ { - x } \quad 0 \leqslant x \leqslant 4$$ where
   • \(x\) is measured in tens of hours
   • \(k\) is a constant
   • Use algebraic integration to show that
   $$\int _ { 0 } ^ { n } x \mathrm { e } ^ { - x } \mathrm {~d} x = 1 - ( n + 1 ) \mathrm { e } ^ { - n }$$
3. Show that, for Xiang's model, \(k = 99\) to the nearest integer.
4. Estimate \(\mathrm { P } ( 10 < T < 30 )\) using
   1. Tomas' model of \(T \sim \mathrm {~N} \left( 14.9,9.3 ^ { 2 } \right)\)
   2. Xiang's curve with equation \(y = 99 x \mathrm { e } ^ { - x }\) and the answer to part (c) The researcher decides to use Xiang's curve to model \(\mathrm { P } ( a < T < b )\)
5. State one limitation of Xiang's model. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Question 6 continued} \includegraphics[alt={},max width=\textwidth]{a067577e-e2a6-440b-9d22-d558fade15f0-17_1164_1778_294_146}
   \end{figure} Time in hours

The records for a school athletics club show that the height, \(H\) metres, achieved by students in the high jump is normally distributed with mean 1.4 metres and standard deviation 0.15 metres.

1. Find the proportion of these students achieving a height of more than 1.6 metres. The records also show that the time, \(T\) seconds, to run 1500 metres is normally distributed with mean 330 seconds and standard deviation 26 seconds. The school's Head would like to use these distributions to estimate the proportion of students from the school athletics club who can jump higher than 1.6 metres and can run 1500 metres in less than 5 minutes.
2. State a necessary assumption about \(H\) and \(T\) for the Head to calculate an estimate of this proportion.
3. Find the Head's estimate of this proportion. Students in the school athletics club also throw the discus.
   The random variable \(D \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) represents the distance, in metres, that a student can throw the discus. Given that \(\mathrm { P } ( D < 16.3 ) = 0.30\) and \(\mathrm { P } ( D > 29.0 ) = 0.10\)
4. calculate the value of \(\mu\) and the value of \(\sigma\)

A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes.

1. Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes. The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes.
2. Stating your hypotheses clearly and using a \(5 \%\) significance level, test whether or not there is evidence to support the patients' complaint. The health centre also claims that the time a dentist spends with a patient during a routine appointment, \(T\) minutes, can be modelled by the normal distribution where \(T \sim \mathrm {~N} \left( 5,3.5 ^ { 2 } \right)\)
3. Using this model,
   1. find the probability that a routine appointment with the dentist takes less than 2 minutes
   2. find \(\mathrm { P } ( T < 2 \mid T > 0 )\)
   3. hence explain why this normal distribution may not be a good model for \(T\). The dentist believes that she cannot complete a routine appointment in less than 2 minutes.
      She suggests that the health centre should use a refined model only including values of \(T > 2\)
4. Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place.

1. The heights of females from a country are normally distributed with

• a mean of 166.5 cm
• a standard deviation of 6.1 cm

Given that \(1 \%\) of females from this country are shorter than \(k \mathrm {~cm}\),

1. find the value of \(k\)
2. Find the proportion of females from this country with heights between 150 cm and 175 cm A female, from this country, is chosen at random from those with heights between 150 cm and 175 cm
3. Find the probability that her height is more than 160 cm The heights of females from a different country are normally distributed with a standard deviation of 7.4 cm Mia believes that the mean height of females from this country is less than 166.5 cm
   Mia takes a random sample of 50 females from this country and finds the mean of her sample is 164.6 cm
4. Carry out a suitable test to assess Mia's belief. You should
   \section*{Question 5 continued.} \section*{Question 5 continued.} \section*{Question 5 continued.}

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 \(\mathrm { P } ( \mathrm { Y } > 90 )\).

10 Club 65-80 Holidays fly jets between Liverpool and Magaluf. Over a long period of time records show that half of the flights from Liverpool to Magaluf take less than 153 minutes and \(5 \%\) of the flights take more than 183 minutes. An operations manager believes that flight times from Liverpool to Magaluf may be modelled by the Normal distribution.

1. Use the information above to write down the mean time the operations manager will use in his Normal model for flight times from Liverpool to Magaluf.
2. Use the information above to find the standard deviation the operations manager will use in his Normal model for flight times from Liverpool to Magaluf, giving your answer correct to 1 decimal place.
3. Data is available for 452 flights. A flight time of under 2 hours was recorded in 16 of these flights. Use your answers to parts (a) and (b) to determine whether the model is consistent with this data. The operations manager suspects that the mean time for the journey from Magaluf to Liverpool is less than from Liverpool to Magaluf. He collects a random sample of 24 flight times from Magaluf to Liverpool. He finds that the mean flight time is 143.6 minutes.
4. Use the Normal model used in part (c) to conduct a hypothesis test to determine whether there is evidence at the \(1 \%\) level to suggest that the mean flight time from Magaluf to Liverpool is less than the mean flight time from Liverpool to Magaluf.
   [0pt]
5. Identify two ways in which the Normal model for flight times from Liverpool to Magaluf might be adapted to provide a better model for the flight times from Magaluf to Liverpool. [2]

18 Riley is investigating the daily water consumption, in litres, of his household.
He records the amount used for a random sample of 120 days from the previous twelve-month period. The daily water consumption, in litres, is denoted by \(x\). Summary statistics for Riley's sample are given below. \(\sum \mathrm { x } = 31164.7 \sum \mathrm { x } ^ { 2 } = 8101050.91 \mathrm { n } = 120\)

1. Calculate the sample mean giving your answer correct to \(\mathbf { 3 }\) significant figures. Riley displays the data in a histogram. \includegraphics[max width=\textwidth, alt={}, center]{11788aaf-98fb-4a78-8a40-a40743b1fe15-13_832_1383_934_242}
2. Find the number of days on which between 255 and 260 litres were used.
3. Give two reasons why a Normal distribution may be an appropriate model for the daily consumption of water. Riley uses the sample mean and the sample variance, both correct to \(\mathbf { 3 }\) significant figures, as parameters of a Normal distribution to model the daily consumption of water.
4. Use Riley's model to calculate the probability that on a randomly chosen day the household uses less than 255 litres of water.
5. Calculate the probability that the household uses less than 255 litres of water on at least 5 days out of a random sample of 28 days. The company which supplies the water makes charges relating to water consumption which are shown in the table below.

   Standing charge per day in pence	7.8
   Charge per litre in pence	0.18

6. Adapt Riley's model for daily water consumption to model the daily charges for water consumption. \section*{END OF QUESTION PAPER}

15 Bottles of Fizzipop nominally contain 330 ml of drink. A consumer affairs researcher collects a random sample of 55 bottles of Fizzipop and records the volume of drink in each bottle. Summary statistics for the researcher's sample are shown in the table.

1. 1. Calculate the mean volume of drink in a bottle of Fizzipop.
   2. Show that the standard deviation of the volume of drink in a bottle of Fizzipop is 3.78 ml . The researcher uses software to produce a histogram with equal class intervals, which is shown below. \includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-10_533_759_1181_251}
2. Explain why the researcher decides that the Normal distribution is a suitable model for the volume of drink in a bottle of Fizzipop.
3. Use your answers to parts (a) and (b) to determine the expected number of bottles which contain less than 330 ml in a random sample of 100 bottles. In order to comply with new regulations, no more than 1\% of bottles of Fizzipop should contain less than 330 ml . The manufacturer decides to meet the new regulations by adjusting the manufacturing process so that the mean volume of drink in a bottle of Fizzipop is increased. The standard deviation is unaltered.
4. Determine the minimum mean volume of drink in a bottle of Fizzipop which should ensure that the new regulations are met. Give your answer to \(\mathbf { 3 }\) significant figures. The mean volume of drink in a bottle of Fizzipop is set to 340 ml . After several weeks the quality control manager suspects the mean volume may have reduced. She collects a random sample of 100 bottles of Fizzipop. The mean volume of drink in a bottle in the sample is found to be 339.37 ml .
5. Assuming the standard deviation is unaltered, conduct a hypothesis test at the \(5 \%\) level to determine whether there is any evidence to suggest that the mean volume of drink in a bottle of Fizzipop is less than 340 ml .

7. A machine fills bottles with water. The volume of water delivered by the machine to a bottle is \(X \mathrm { ml }\) where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) One of these bottles of water is selected at random. Given that \(\mu = 503\) and \(\sigma = 1.6\)

1. find
   1. \(\mathrm { P } ( X > 505 )\)
   2. \(\mathrm { P } ( 501 < X < 505 )\)
2. Find \(w\) such that \(\mathrm { P } ( 1006 - w < X < w ) = 0.9426\) Following adjustments to the machine, the volume of water delivered by the machine to a bottle is such that \(\mu = 503\) and \(\sigma = q\) Given that \(\mathrm { P } ( X < r ) = 0.01\) and \(\mathrm { P } ( X > r + 6 ) = 0.05\)
3. find the value of \(r\) and the value of \(q\)

6. The waiting time, \(L\) minutes, to see a doctor at a health centre is normally distributed with \(L \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). Given that \(\mathrm { P } ( L < 15 ) = 0.9\) and \(\mathrm { P } ( L < 5 ) = 0.05\)

1. find the value of \(\mu\) and the value of \(\sigma\). There are 23 people waiting to see a doctor at the health centre.
2. Determine the expected number of these people who will have a waiting time of more than 12 minutes.

The weights of packages delivered to Susie are normally distributed with a mean of 510 grams and a standard deviation of 45 grams.

1. Find the probability that a randomly selected package delivered to Susie weighs less than 450 grams. The heaviest 5\% of packages delivered to Susie are delivered by Rav in his van, the others are delivered by Taruni on foot.
2. Find the weight of the lightest package that Rav would deliver to Susie. Susie randomly selects a package from those delivered by Taruni.
3. Find the probability that this package weighs more than 450 grams. On Tuesday there are 5 packages delivered to Susie.
4. Find the probability that 4 are delivered by Taruni and 1 is delivered by Rav.

3. The distance achieved in a long jump competition by students at a school is normally Students who achieve a distance greater than 4.3 metres receive a medal.

1. Find the proportion of students who receive medals. The school wishes to give a certificate of achievement or a medal to the \(80 \%\) of students who achieve a distance of at least \(d\) metres.
2. Find the value of \(d\). Of those who received medals, the \(\frac { 1 } { 3 }\) who jump the furthest will receive gold medals.
3. Find the shortest distance, \(g\) metres, that must be achieved to receive a gold medal. A journalist from the local newspaper interviews a randomly selected group of 3 medal winners.
4. Find the exact probability that there is at least one gold medal winner in the group. \includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-08_79_1153_233_251} Students who achieve a distance greater than 4.3 metres receive a medal.
   1. Find the proportion of students who receive medals.

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6. The random variable \(A\) represents the score when a spinner is spun. The probability distribution for \(A\) is given in the following table.

1. Show that \(\mathrm { E } ( A ) = 3.5\)
2. Find \(\operatorname { Var } ( A )\) The random variable \(B\) represents the score on a 4 -sided die. The probability distribution for \(B\) is given in the following table where \(k\) is a positive integer.

   \(b\)	1	3	4	\(k\)
   \(\mathrm { P } ( B = b )\)	0.25	0.25	0.25	0.25

3. Write down the name of the probability distribution of \(B\).
4. Given that \(\mathrm { E } ( B ) = \mathrm { E } ( A )\) state, giving a reason, the value of \(k\). The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Sam and Tim are playing a game with the spinner and the die. They each spin the spinner once to obtain their value of \(A\) and each roll the die once to obtain their value of \(B\).
   Their value of \(A\) is taken as their value of \(\mu\) and their value of \(B\) is taken as their value of \(\sigma\). The person with the larger value of \(\mathrm { P } ( X > 3.5 )\) is the winner.
5. Given that Sam obtained values of \(a = 4\) and \(b = 3\) and Tim obtained \(b = 4\) find, giving a reason, the probability that Tim wins.
6. Find the largest value of \(\mathrm { P } ( X > 3.5 )\) achievable in this game.
7. Find the probability of achieving this value. \includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-21_2255_50_314_34}

Kris works in the mailroom of a large company and is responsible for all the letters sent by the company. The weights of letters sent by the company, \(W\) grams, have a normal distribution with mean 165 g and standard deviation 35 g .

1. Estimate the proportion of letters sent by the company that weigh less than 120 g . Kris splits the letters to be sent into 3 categories: heavy, medium and light, with \(\frac { 1 } { 3 }\) of the letters in each category.
2. Find the weight limits that determine medium letters. A heavy letter is chosen at random.
3. Find the probability that this letter weighs less than 200 g . Kris chooses a random sample of 3 letters from those in the mailroom one day.
4. Find the probability that there is one letter in each of the 3 categories.
   

A manufacturer fills bottles with oil. The volume of oil in a bottle, \(V \mathrm { ml }\), is normally distributed with \(V \sim \mathrm {~N} \left( 100,2.5 ^ { 2 } \right)\)

1. Find \(\mathrm { P } ( V > 104.9 )\)
2. In a pack of 150 bottles, find the expected number of bottles containing more than 104.9 ml
3. Find the value of \(v\), to 2 decimal places, such that \(\mathrm { P } ( V > v \mid V < 104.9 ) = 0.2801\)

1. A competition consists of two rounds.

The time, in minutes, taken by adults to complete round one is modelled by a normal distribution with mean 15 minutes and standard deviation 2 minutes.

1. Use standardisation to find the proportion of adults that take less than 18 minutes to complete round one. Only the fastest \(60 \%\) of adults from round one take part in round two.
2. Use standardisation to find the longest time that an adult can take to complete round one if they are to take part in round two. The time, \(T\) minutes, taken by adults to complete round two is modelled by a normal distribution with mean \(\mu\) Given that \(\mathrm { P } ( \mu - 10 < T < \mu + 10 ) = 0.95\)
3. find \(\mathrm { P } ( T > \mu - 5 \mid T > \mu - 10 )\)

1. The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\)

Given that \(\mathrm { P } ( X > \mu + a ) = 0.35\) where \(a\) is a constant, find

1. \(\mathrm { P } ( X > \mu - a )\)
2. \(\mathrm { P } ( \mu - a < X < \mu + a )\)
3. \(\mathrm { P } ( X < \mu + a \mid X > \mu - a )\)

1. The label on a jar of Amy's jam states that the jar contains about 400 grams of jam. For each jar that contains less than 388 grams of jam, Amy will be fined \(\pounds 100\). If a jar contains more than 410 grams of jam then Amy makes a loss of \(\pounds 0.30\) on that jar.

The weight of jam, \(A\) grams, in a jar of Amy's jam has a normal distribution with mean \(\mu\) grams and standard deviation \(\sigma\) grams. Amy chooses \(\mu\) and \(\sigma\) so that \(\mathrm { P } ( A < 388 ) = 0.001\) and \(\mathrm { P } ( A > 410 ) = 0.0197\)

1. Find the value of \(\mu\) and the value of \(\sigma\). Amy can sell jars of jam containing between 388 grams and 410 grams for a profit of \(\pounds 0.25\)
2. Calculate the expected amount, in £s, that Amy receives for each jar of jam.
   

1. A machine makes bolts such that the length, \(L \mathrm {~cm}\), of a bolt has distribution \(L \sim \mathrm {~N} \left( 4.1,0.125 ^ { 2 } \right)\)

A bolt is selected at random.

1. Find the probability that the length of this bolt is more than 4.3 cm .
2. Show that \(\mathrm { P } ( 3.9 < L < 4.3 )\) is 0.890 correct to 3 decimal places. The machine makes 500 bolts.
   The cost to make each bolt is 5 pence.
   Only bolts with length between 3.9 cm and 4.3 cm can be used. These are sold for 9 pence each. All the bolts that cannot be used are recycled with a scrap value of 1 pence each.
3. Calculate an estimate for the profit made on these 500 bolts. Following adjustments to the machine, the length of a bolt, \(B \mathrm {~cm}\), made by the machine is such that \(B \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( B > 4.198 ) = 0.025\) and \(\mathrm { P } ( B < 4.065 ) = 0.242\)
4. find the value of \(\mu\) and the value of \(\sigma\)
5. State, giving a reason, whether the adjustments to the machine will result in a decrease or an increase in the profit made on 500 bolts.

1. The weights, \(W\) grams, of kiwi fruit grown on a farm are normally distributed with mean 80 grams and standard deviation 8 grams.

The table shows the classifications of the kiwi fruit by their weight, where \(k\) is a positive constant. One kiwi fruit is selected at random from those grown on the farm.

1. Find the probability that this kiwi fruit is Large. 35\% of the kiwi fruit are Jumbo.
2. Find the value of \(k\) to one decimal place. 75\% of Tiny kiwi fruit weigh more than \(y\) grams.
3. Find the value of \(y\) giving your answer to one decimal place.

1. The weights, \(X\) grams, of a particular variety of fruit are normally distributed with

$$X \sim \mathrm {~N} \left( 210,25 ^ { 2 } \right)$$ A fruit of this variety is selected at random.

1. Show that the probability that the weight of this fruit is less than 240 grams is 0.8849
2. Find the probability that the weight of this fruit is between 190 grams and 240 grams.
3. Find the value of \(k\) such that \(\mathrm { P } ( 210 - k < X < 210 + k ) = 0.95\) A wholesaler buys large numbers of this variety of fruit and classifies the lightest \(15 \%\) as small.
4. Find the maximum weight of a fruit that is classified as small. You must show your working clearly. The weights, \(Y\) grams, of a second variety of this fruit are normally distributed with $$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$ Given that \(5 \%\) of these fruit weigh less than 152 grams and \(40 \%\) weigh more than 180 grams,
5. calculate the mean and standard deviation of the weights of this variety of fruit.

1. A machine fills bottles with water. The volume of water delivered by the machine to a bottle is \(X \mathrm { ml }\) where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\)

One of these bottles of water is selected at random.
Given that \(\mu = 503\) and \(\sigma = 1.6\)

1. find
   1. \(\mathrm { P } ( X > 505 )\)
   2. \(\mathrm { P } ( 501 < X < 505 )\)
2. Find \(w\) such that \(\mathrm { P } ( 1006 - w < X < w ) = 0.9426\) Following adjustments to the machine, the volume of water delivered by the machine to a bottle is such that \(\mu = 503\) and \(\sigma = q\) Given that \(\mathrm { P } ( X < r ) = 0.01\) and \(\mathrm { P } ( X > r + 6 ) = 0.05\)
3. find the value of \(r\) and the value of \(q\) \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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