Mixed calculations with boundaries

3 The weight, \(X\) grams, of talcum powder in a tin may be modelled by a normal distribution with mean 253 and standard deviation \(\sigma\).

1. Given that \(\sigma = 5\), determine:
   1. \(\mathrm { P } ( X < 250 )\);
   2. \(\mathrm { P } ( 245 < X < 250 )\);
   3. \(\mathrm { P } ( X = 245 )\).
2. Assuming that the value of the mean remains unchanged, determine the value of \(\sigma\) necessary to ensure that \(98 \%\) of tins contain more than 245 grams of talcum powder.
   (4 marks) \includegraphics[max width=\textwidth, alt={}, center]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-07_38_118_440_159} \includegraphics[max width=\textwidth, alt={}, center]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-07_40_118_529_159}

2 A garden centre sells bamboo canes of nominal length 1.8 metres. The length, \(X\) metres, of the canes can be modelled by a normal distribution with mean 1.86 and standard deviation \(\sigma\).

1. Assuming that \(\sigma = 0.04\), determine:
   1. \(\mathrm { P } ( X < 1.90 )\);
   2. \(\mathrm { P } ( X > 1.80 )\);
   3. \(\mathrm { P } ( 1.80 < X < 1.90 )\);
   4. \(\mathrm { P } ( X \neq 1.86 )\).
2. It is subsequently found that \(\mathrm { P } ( X > 1.80 ) = 0.98\). Determine the value of \(\sigma\).
   [0pt] [3 marks]
   \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-06_1529_1717_1178_150}

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]

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?

2 The weight, \(X\) grams, of a particular variety of orange is normally distributed with mean 205 and standard deviation 25.

1. Determine the probability that the weight of an orange is:
   1. less than 250 grams;
   2. between 200 grams and 250 grams.
2. A wholesaler decides to grade such oranges by weight. He decides that the smallest 30 per cent should be graded as small, the largest 20 per cent graded as large, and the remainder graded as medium. Determine, to one decimal place, the maximum weight of an orange graded as:
   1. small;
   2. medium.
3. The weight, \(Y\) grams, of a second variety of orange is normally distributed with mean 175. Given that 90 per cent of these oranges weigh less than 200 grams, calculate the standard deviation of their weights.
   (4 marks)

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The distances, \(m\) miles, a motorbike travels on a full tank of petrol can be modelled by a normal distribution with mean 170 miles and standard deviation 16 miles.

1. Find the probability that, on a randomly selected journey, the motorbike could travel at least 190 miles on a full tank of petrol. The probability that, on a randomly selected journey, the motorbike could travel at least \(d\) miles on a full tank of petrol is 0.9
2. Find the value of \(d\)

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 .

14 The maximum pressure exerted by the blood on the arteries in a population of elderly male patients may be modelled by a random variable having a normal distribution with a mean of 150 and standard deviation 15, measured in suitable units.

1. Find the probability that the maximum pressure for a randomly chosen patient is more than 160.
2. If the maximum pressure is found to be \(t\) or more, the patient must be referred to a consultant. If \(5 \%\) of the patients are referred to a consultant, find the value of \(t\).
3. Find the percentage of patients whose maximum pressure is between 130 and 160 . The probability that a randomly chosen patient attending a doctor's surgery has their blood pressure measured is 0.4 .
4. Find the probability that of 18 people attending a doctor's surgery more than 8 have their blood pressure measured, assuming that each measurement is random and independent of any other.
5. If 450 patients visited the surgery in a week, find the expected number of patients whose blood pressure would be measured.

10 Cheeky Cola is sold in bottles of two sizes, small and large. For each size, the content of a randomly chosen bottle is normally distributed with mean and standard deviation, in litres, as given in the table.

1. Find the probability that a randomly chosen small bottle contains more than 0.51 litres.
2. Find \(x\) if the probability that a randomly chosen large bottle contains less than 1.45 litres is 0.1 . The manufacturer introduces a new size of bottle of Cheeky Cola, called the mega bottle. It is found that the probabilities that a randomly chosen mega bottle contains less than 2.97 litres or more than 3.05 litres are both 0.05 .
3. Assuming that the contents of the mega bottle are normally distributed, find the mean and variance of the distribution.

In a cycling event the times taken to complete a course are modelled by a normal distribution with mean 62.3 minutes and standard deviation 8.4 minutes.

1. Find the probability that a randomly chosen cyclist has a time less than 74 minutes. [2]
2. Find the probability that 4 randomly chosen cyclists all have times between 50 and 74 minutes. [4]

In a different cycling event, the times can also be modelled by a normal distribution. 23\% of the cyclists have times less than 36 minutes and 10\% of the cyclists have times greater than 54 minutes.

1. Find estimates for the mean and standard deviation of this distribution. [5]

The heights of a population of women are normally distributed with mean \(\mu\) cm and standard deviation \(\sigma\) cm. It is known that 30% of the women are taller than 172 cm and 5% are shorter than 154 cm.

1. Sketch a diagram to show the distribution of heights represented by this information. [3]
2. Show that \(\mu = 154 + 1.6449\sigma\). [3]
3. Obtain a second equation and hence find the value of \(\mu\) and the value of \(\sigma\). [4]

A woman is chosen at random from the population.

1. Find the probability that she is taller than 160 cm. [3]

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]

The times taken by a group of people to complete a task are modelled by a normal distribution with mean 8 hours and standard deviation 2 hours. Use this model to calculate

1. the probability that a person chosen at random took between 5 and 9 hours to complete the task, [4 marks]
2. the range, symmetrical about the mean, within which 80% of the people's times lie. [5 marks]

It is found that, in fact, 80% of the people take more than 5 hours. The model is modified so that the mean is still 8 hours but the standard deviation is no longer 2 hours.

1. Find the standard deviation of the times in the modified model. [3 marks]

The ages of the residents of a retirement community are assumed to be normally distributed. 15% of the residents are under 60 years old and 5% are over 90 years old.

1. Using this information, find the mean and the standard deviation of the ages. [7 marks]
2. If there are 200 residents, find how many are over 80 years old. [3 marks]

A geologist is analysing the size of quartz crystals in a sample of granite. She estimates that the longest diameter of 75% of the crystals is greater than 2 mm, but only 10% of the crystals have a longest diameter of more than 6 mm. The geologist believes that the distribution of the longest diameters of the quartz crystals can be modelled by a normal distribution.

1. Find the mean and variance of this normal distribution. [9 marks]

The geologist also estimated that only 2% of the longest diameters were smaller than 1 mm.

1. Calculate the corresponding percentage that would be predicted by a normal distribution with the parameters you calculated in part \((a)\). [3 marks]
2. Hence, comment on the suitability of the normal distribution as a model in this situation. [2 marks]

As part of a research project, the masses, \(m\) grams, of a random sample of 1000 pebbles from a certain beach were recorded. The results are summarised in the table.

Mass (g)	\(50 \leq m < 150\)	\(150 \leq m < 200\)	\(200 \leq m < 250\)	\(250 \leq m < 350\)
Frequency	162	318	355	165

1. Calculate estimates of the mean and standard deviation of these masses. [2]

The masses, \(x\) grams, of a random sample of 1000 pebbles on a different beach were also found. It was proposed that the distribution of these masses should be modelled by the random variable \(X \sim N(200, 3600)\).

1. Use the model to find \(P(150 < X < 210)\). [1]
2. Use the model to determine \(x_1\) such that \(P(160 < X < x_1) = 0.6\), giving your answer correct to five significant figures. [3]

It was found that the smallest and largest masses of the pebbles in this second sample were 112 g and 288 g respectively.

1. Use these results to show that the model may not be appropriate. [1]
2. Suggest a different value of a parameter of the model in the light of these results. [2]

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

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

A farm supplies apples to a supermarket. The diameters of the apples, \(D\) centimetres, are normally distributed with mean 6.5 and standard deviation 0.73

1. 1. Find \(P(D < 5.2)\) [1 mark]
   2. Find \(P(D > 7)\) [1 mark]
   3. The supermarket only accepts apples with diameters between 5 cm and 8 cm. Find the proportion of apples that the supermarket accepts. [1 mark]
2. The farm also supplies plums to the supermarket. These plums have diameters that are normally distributed. It is found that 60% of these plums have a diameter less than 5.9 cm. It is found that 20% of these plums have a diameter greater than 6.1 cm. Find the mean and standard deviation of the diameter, in centimetres, of the plums supplied by the farm. [6 marks]

The heights of a population of men are normally distributed with mean \(\mu\) cm and standard deviation \(\sigma\) cm. It is known that 20% of the men are taller than 180 cm and 5% are shorter than 170 cm.

1. Sketch a diagram to show the distribution of heights represented by this information. [2 marks]
2. Find the value of \(\mu\) and \(\sigma\). [5 marks]
3. Three men are selected at random, find the probability that they are all taller than 175 cm. [2 marks]

The random variable \(X\) represents the weight in kg of a randomly selected male dog of a particular breed. \(X\) is Normally distributed with mean 30.7 and standard deviation 3.5.

1. Find the 90th percentile for the weights of these dogs. [2]
2. Five of these dogs are chosen at random. Find the probability that exactly four of them weighs at least 30 kg. [3]

The weights of females of the same breed of dog are Normally distributed with mean 26.8 kg.

1. Given that 5% of female dogs of this breed weigh more than 30 kg, find the standard deviation of their weights. [3]
2. Sketch the distributions of the weights of male and female dogs of this breed on a single diagram. [3]

A manufacturer produces components designed with length \(L\) mm such that \(12 < L < 15\). The Quality Control department finds that 15% of the components sampled are longer than 15 mm while 8% are shorter than 12 mm. Assume that \(L\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\).

1. Calculate \(\mu\) and \(\sigma\). [6]
2. The shortest 5% of components are rejected. Find the minimum length which a component may have before it is rejected. [3]
3. It was found in a random sample that 10% of components were longer than 16 mm. Determine whether this finding is consistent with the assumption that \(L\) is normally distributed with the \(\mu\) and \(\sigma\) found in part (i). [3]

A company supplies tubs of coleslaw to a large supermarket chain. According to the labels on the tubs, each tub contains 300 grams of coleslaw. In practice the weights of coleslaw in the tubs are normally distributed with mean 305 grams and standard deviation 6 grams.

1. Find the proportion of tubs that are underweight, according to the label. [3]

The supermarket chain requires that the proportion of underweight tubs should be reduced to 5%.

1. If the standard deviation is kept at 6 grams, find the new mean weight needed to achieve the required reduction. [3]
2. If the mean weight is kept at 305 grams, find the new standard deviation needed to achieve the required reduction. Explain why the company might prefer to adjust the standard deviation rather than the mean. [3]