2.04f Find normal probabilities: Z transformation

5 Still mineral water is supplied in 1.5-litre bottles. The actual volume, \(X\) millilitres, in a bottle may be modelled by a normal distribution with mean \(\mu = 1525\) and standard deviation \(\sigma = 9.6\).

1. Determine the probability that the volume of water in a randomly selected bottle is:
   1. less than 1540 ml ;
   2. more than 1535 ml ;
   3. between 1515 ml and 1540 ml ;
   4. not 1500 ml .
2. The supplier requires that only 10 per cent of bottles should contain more than 1535 ml of water. Assuming that there has been no change in the value of \(\sigma\), calculate the reduction in the value of \(\mu\) in order to satisfy this requirement. Give your answer to one decimal place.
3. Sparkling spring water is supplied in packs of six 0.5 -litre bottles. The actual volume in a bottle may be modelled by a normal distribution with mean 508.5 ml and standard deviation 3.5 ml . Stating a necessary assumption, determine the probability that:
   1. the volume of water in each of the 6 bottles from a randomly selected pack is more than 505 ml ;
   2. the mean volume of water in the 6 bottles from a randomly selected pack is more than 505 ml .
      [0pt] [7 marks]

7. The times taken by a large number of people to read a certain book can be modelled by a normal distribution with mean \(5 \cdot 2\) hours. It is found that \(62 \cdot 5 \%\) of the people took more than \(4 \cdot 5\) hours to read the book.

1. Show that the standard deviation of the times is approximately \(2 \cdot 2\) hours.
2. Calculate the percentage of the people who took between 4 and 7 hours to read the book.
3. Calculate the probability that two of the people chosen at random both took less than 5 hours to read the book, stating any assumption that you make.
4. If a number of extra people were taken into account, all of whom took exactly \(5 \cdot 2\) hours to read the book, state with reasons what would happen to (i) the mean, (ii) the variance and explain briefly why the distribution would no longer be normal.

4. A botanist believes that the lengths of the branches on trees of a certain species can be modelled by a normal distribution.
When he measures the lengths of 500 branches, he finds 55 which are less than 30 cm long and 200 which are more than 90 cm long.

1. Find the mean and the standard deviation of the lengths.
2. In a sample of 1000 branches, how many would he expect to find with lengths greater than 1 metre? \section*{STATISTICS 1 (A) TEST PAPER 7 Page 2}

7. The volume of liquid in bottles of sparkling water from one producer is believed to be normally distributed with a mean of 704 ml and a variance of \(3.2 \mathrm { ml } ^ { 2 }\). Calculate the probability that a randomly chosen bottle from this producer contains

1. more than 706 ml ,
2. between 703 and 708 ml . The bottles are labelled as containing 700 ml .
3. In a delivery of 1200 bottles, how many could be expected to contain less than the stated 700 ml ? The bottling process can be adjusted so that the mean changes but the variance is unchanged.
4. What should the mean be changed to in order to have only a \(0.1 \%\) chance of a bottle having less than 700 ml of sparkling water? Give your answer correct to 1 decimal place.

5. The time taken in minutes, \(T\), for a mechanic to service a bicycle follows a normal distribution with a mean of 25 minutes and a variance of 16 minutes \(^ { 2 }\). Find

1. \(\mathrm { P } ( T < 28 )\),
2. \(\quad \mathrm { P } ( | T - 25 | < 5 )\). One afternoon the mechanic has 3 bicycles to service.
3. Find the probability that he will take less than 23 minutes on each of the three bicycles.
   (4 marks)

3. The random variable \(X\) is normally distributed with a mean of 42 and a variance of 18 . Find

1. \(\mathrm { P } ( X \leq 45 )\),
2. \(\mathrm { P } ( 32 \leq X \leq 38 )\),
3. the value of \(x\) such that \(\mathrm { P } ( X \leq x ) = 0.95\)

3. The time it takes girls aged 15 to complete an obstacle course is found to be normally distributed with a mean of 21.5 minutes and a standard deviation of 2.2 minutes.

1. Find the probability that a randomly chosen 15 year-old girl completes the course in less than 25 minutes. A 13 year-old girl completes the course in exactly 19 minutes.
2. What percentage of 15 year-old girls would she beat over the course? Anyone completing the course in less than 20 minutes is presented with a certificate of achievement. Three friends all complete the course one afternoon.
3. What is the probability that exactly two of them get certificates?

4. The random variable \(A\) is normally distributed with a mean of 32.5 and a variance of 18.6 Find

1. \(\mathrm { P } ( A < 38.2 )\),
2. \(\mathrm { P } ( 31 \leq A \leq 35 )\), The random variable \(B\) is normally distributed with a standard deviation of 7.2
   Given also that \(\mathrm { P } ( B > 110 ) = 0.138\),
3. find the mean of \(B\).

4. A company produces jars of English Honey. The weight of the glass jars used is normally distributed with a mean of 122.3 g and a standard deviation of 2.6 g . Calculate the probability that a randomly chosen jar will weigh

1. less than 127 g ,
2. less than 121.5 g . The weight of honey put into each jar by a machine is normally distributed with a standard deviation of 1.6 g . The machine operator can adjust the mean weight of the honey put into each jar without changing the standard deviation.
3. Find, correct to 4 significant figures, the minimum that the mean weight can be set to such that at most 1 in 20 of the jars will contain less than 454 g .
   (4 marks)

3. A study was made of the heights of boys of different ages in Lancashire. The study concluded that the heights of 13 year-old boys are normally distributed with a mean of 156 cm and a variance of \(73 \mathrm {~cm} ^ { 2 }\). Find the probability that a 13 year-old boy chosen at random will be

1. more than 165 cm tall,
2. between 156 and 165 cm tall. The study also concluded that the heights of 14 year-old boys are normally distributed with a mean of 160 cm and a variance of \(79 \mathrm {~cm} ^ { 2 }\). One 13 year-old and one 14 year-old boy are chosen at random.
3. Find the probability that both boys are more than 165 cm tall.
4. State, with a reason, whether the probability that the combined height of the two boys is more than 330 cm is more or less than your answer to part (c).
   (2 marks)

4. Alan runs on a treadmill each day for as long as he can at 7 miles per hour. The length of time for which he runs is normally distributed with a mean of 21.6 minutes and a standard deviation of 1.8 minutes.

1. Calculate the probability that on any one day Alan will run for less than 20 minutes.
2. Estimate the number of times in a ninety-day period that Alan will run for more than 24 minutes.
3. On a particular day Alan is still running after 22 minutes. Find the probability that he will stop running in the next 2 minutes.

7. A cyber-cafe recorded how long each user stayed during one day giving the following results.

1. Use linear interpolation to estimate the median and quartiles of these data. The results of a previous study had led to the suggestion that the length of time each user stays can be modelled by a normal distribution with a mean of 72 minutes and a standard deviation of 48 minutes.
2. Find the median and quartiles that this model would predict.
3. Comment on the suitability of the suggested model in the light of the new results.

3. The time that a school pupil spends on French homework each week is normally distributed with a mean of 55 minutes and a standard deviation of 10 minutes. The time that this pupil spends on English homework each week is normally distributed with a mean of 1 hour 30 minutes and a standard deviation of 18 minutes. Find the probability that in a randomly chosen week

1. the pupil spends more than 2 hours in total doing French and English homework,
2. the pupil spends more than twice as long doing English homework as he spends doing French homework.
   (6 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)

2 A supermarket sells oranges. Their weights are modelled by the random variable \(X\) which has a Normal distribution with mean 345 grams and standard deviation 15 grams. When the oranges have been peeled, their weights in grams, \(Y\), are modelled by \(Y = 0.7 X\).

1. Find the probability that a randomly chosen peeled orange weighs less than 250 grams. I randomly choose 5 oranges to buy.
2. Find the probability that the total weight of the 5 unpeeled oranges is at least 1800 grams.
3. I peel three of the oranges and leave the remaining two unpeeled. Find the probability that the total weight of the two unpeeled oranges is greater than the total weight of the three peeled ones.

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).

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 .

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.