Probability calculation plus find unknown boundary

6. The random variable \(X\) has a normal distribution with mean 30 and standard deviation 5 .

1. Find \(\mathrm { P } ( X < 39 )\).
2. Find the value of \(d\) such that \(\mathrm { P } ( X < d ) = 0.1151\)
3. Find the value of \(e\) such that \(\mathrm { P } ( X > e ) = 0.1151\)
4. Find \(\mathrm { P } ( d < X < e )\).

3. The continuous random variable \(Y\) is normally distributed with mean 100 and variance 256 .

1. Find \(\mathrm { P } ( Y < 80 )\).
2. Find \(k\) such that \(\mathrm { P } ( 100 - k \leq Y \leq 100 + k ) = 0.516\).

1. A scientist found that the time taken, \(M\) minutes, to carry out an experiment can be modelled by a normal random variable with mean 155 minutes and standard deviation 3.5 minutes.

Find

1. \(\mathrm { P } ( M > 160 )\).
2. \(\mathrm { P } ( 150 \leqslant M \leqslant 157 )\).
3. the value of \(m\), to 1 decimal place, such that \(\mathrm { P } ( M \leqslant m ) = 0.30\).

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 \(\mathrm { P } ( 20 < X < d ) = 0.4641\)

7

1. The weight, \(X\) grams, of soup in a carton may be modelled by a normal random variable with mean 406 and standard deviation 4.2. Find the probability that the weight of soup in a carton:
   1. is less than 400 grams;
   2. is between 402.5 grams and 407.5 grams.
2. The weight, \(Y\) grams, of chopped tomatoes in a tin is a normal random variable with mean \(\mu\) and standard deviation \(\sigma\).
   1. Given that \(\mathrm { P } ( Y < 310 ) = 0.975\), explain why: $$310 - \mu = 1.96 \sigma$$
   2. Given that \(\mathrm { P } ( Y < 307.5 ) = 0.86\), find, to two decimal places, values for \(\mu\) and \(\sigma\).
      (4 marks)

1 In large-scale tree-felling operations, a machine cuts down trees, strips off the branches and then cuts the trunks into logs of length \(X\) metres for transporting to a sawmill. It may be assumed that values of \(X\) are normally distributed with mean \(\mu\) and standard deviation 0.16 , where \(\mu\) can be set to a specific value.

1. Given that \(\mu\) is set to 3.3 , determine:
   1. \(\mathrm { P } ( X < 3.5 )\);
   2. \(\mathrm { P } ( X > 3.0 )\);
   3. \(\mathrm { P } ( 3.0 < X < 3.5 )\).
2. The sawmill now requires a batch of logs such that there is a probability of 0.025 that any given log will have a length less than 3.1 metres. Determine, to two decimal places, the new value of \(\mu\).

3 UPVC facia board is supplied in lengths labelled as 5 metres. The actual length, \(X\) metres, of a board may be modelled by a normal distribution with a mean of 5.08 and a standard deviation of 0.05 .

1. Determine:
   1. \(\mathrm { P } ( X < 5 )\);
   2. \(\mathrm { P } ( 5 < X < 5.10 )\).
2. Determine the probability that the mean length of a random sample of 4 boards:
   1. exceeds 5.05 metres;
   2. is exactly 5 metres.
3. Assuming that the value of the standard deviation remains unchanged, determine the mean length necessary to ensure that only 1 per cent of boards have lengths less than 5 metres.

2 The diameter, \(D\) millimetres, of an American pool ball may be modelled by a normal random variable with mean 57.15 and standard deviation 0.04 .

1. Determine:
   1. \(\mathrm { P } ( D < 57.2 )\);
   2. \(\mathrm { P } ( 57.1 < D < 57.2 )\).
2. A box contains 16 of these pool balls. Given that the balls may be regarded as a random sample, determine the probability that:
   1. all 16 balls have diameters less than 57.2 mm ;
   2. the mean diameter of the 16 balls is greater than 57.16 mm .

2 The weight, \(X\) grams, of the contents of a tin of baked beans can be modelled by a normal random variable with a mean of 421 and a standard deviation of 2.5.

1. Find:
   1. \(\mathrm { P } ( X = 421 )\);
   2. \(\mathrm { P } ( X < 425 )\);
   3. \(\mathrm { P } ( 418 < X < 424 )\).
2. Determine the value of \(x\) such that \(\mathrm { P } ( X < x ) = 0.98\).
3. The weight, \(Y\) grams, of the contents of a tin of ravioli can be modelled by a normal random variable with a mean of \(\mu\) and a standard deviation of 3.0 . Find the value of \(\mu\) such that \(\mathrm { P } ( Y < 410 ) = 0.01\).

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 \(\mathrm { P } ( X \leq x ) = 0.95\)

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 \(B\).

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.

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 .

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.

A machine squeezes apples to extract their juice. The volume of juice, \(J\) ml, extracted from 1 kg of apples is modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) Given that \(\mu = 500\) and \(\sigma = 25\) use standardisation to

1. 1. show that P\((J > 510) = 0.3446\) [2]
   2. calculate the value of \(d\) such that P\((J > d) = 0.9192\) [3]

Zen randomly selects 5 bags each containing 1 kg of apples and records the volume of juice extracted from each bag of apples.

1. Calculate the probability that each of the 5 bags of apples produce less than 510ml of juice. [2]

Following adjustments to the machine, the volume of juice, \(R\) ml, extracted from 1 kg of apples is such that \(\mu = 520\) and \(\sigma = k\) Given that P\((R < r) = 0.15\) and P\((R > 3r - 800) = 0.005\)

1. find the value of \(r\) and the value of \(k\) [7]

Past records show that the times, in seconds, taken to run 100 m by children at a school can be modelled by a normal distribution with a mean of 16.12 and a standard deviation of 1.60 A child from the school is selected at random.

1. Find the probability that this child runs 100 m in less than 15 s. [3]

On sports day the school awards certificates to the fastest 30\% of the children in the 100 m race.

1. Estimate, to 2 decimal places, the slowest time taken to run 100 m for which a child will be awarded a certificate. [4]

Strips of metal are cut to length \(L\) cm, where \(L \sim N(\mu, 0.5^2)\).

1. Given that 2.5\% of the cut lengths exceed 50.98 cm, show that \(\mu = 50\). [5]
2. Find \(P(49.25 < L < 50.75)\). [4]

Those strips with length either less than 49.25 cm or greater than 50.75 cm cannot be used. Two strips of metal are selected at random.

1. Find the probability that both strips cannot be used. [2]

The entrance to a car park is \(1.9\) m wide. It is found that this is too narrow for \(2\%\) of the vehicles which need to use the car park. The widths of these vehicles are modelled by a normal distribution with mean \(1.6\) m.

1. Find the standard deviation of the distribution. [4 marks]

It is decided to widen the entrance so that \(99.5\%\) of vehicles will be able to use it.

1. Find the minimum width needed to achieve this. [4 marks]

The rainfall at a weather station was recorded every day of the twentieth century. One year is selected at random from the records and the total rainfall, in cm, in January of that year is denoted by \(R\). Assuming that \(R\) can be modelled by a normal distribution with standard deviation \(12.6\), and given that P\((R > 100) = 0.0764\),

1. find the mean of \(R\), [4 marks]
2. calculate P\((75 < R < 80)\). [5 marks]

An athlete believes that her times for running 200 metres in races are normally distributed with a mean of 22.8 seconds.

1. Given that her time is over 23.3 seconds in 20\% of her races, calculate the variance of her times. [5]
2. The record over this distance for women at her club is 21.82 seconds. According to her model, what is the chance that she will beat this record in her next race? [3]

The lifetime of Zaple smartphone batteries, \(X\) hours, is normally distributed with mean 8 hours and standard deviation 1.5 hours.

1. 1. Find P(\(X \neq 8\)) [1 mark]
   2. Find P(\(6 < X < 10\)) [1 mark]
2. Determine the lifetime exceeded by 90\% of Zaple smartphone batteries. [2 marks]
3. A different smartphone, Kaphone, has its battery's lifetime, \(Y\) hours, modelled by a normal distribution with mean 7 hours and standard deviation \(\sigma\). 25\% of randomly selected Kaphone batteries last less than 5 hours. Find the value of \(\sigma\), correct to three significant figures. [4 marks]

In 2019, the lengths of new-born babies at a clinic can be modelled by a normal distribution with mean 50 cm and standard deviation 4 cm.

1. This normal distribution is represented in the diagram below. Label the values 50 and 54 on the horizontal axis. [2 marks] \includegraphics{figure_17a}
2. State the probability that the length of a new-born baby is less than 50 cm. [1 mark]
3. Find the probability that the length of a new-born baby is more than 56 cm. [1 mark]
4. Find the probability that the length of a new-born baby is more than 40 cm but less than 60 cm. [1 mark]
5. Determine the length exceeded by 95% of all new-born babies at the clinic. [2 marks]
6. In 2020, the lengths of 40 new-born babies at the clinic were selected at random. The total length of the 40 new-born babies was 2060 cm. Carry out a hypothesis test at the 10% significance level to investigate whether the mean length of a new-born baby at the clinic in 2020 has **increased** compared to 2019. You may assume that the length of a new-born baby is still normally distributed with standard deviation 4 cm. [7 marks]

The weights of pineapples on sale at a wholesaler are normally distributed with mean \(1.349\) kg and standard deviation \(0.236\) kg. Before going on sale the pineapples are classified as 'Small', 'Medium', 'Large' and 'Extra Large'.

1. A pineapple is classified as 'Small' if it weighs less than \(1.100\) kg. Find the probability that a randomly chosen pineapple will be classified as 'Small'. [5]
2. \(10\%\) of pineapples are classified as 'Extra Large'. Find the minimum weight required for a pineapple to be classified as 'Extra Large'. [3]