2.04f Find normal probabilities: Z transformation

The random variable \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma^2\).

1. If \(2\mu = 3\sigma\), find P\((X < 2\mu)\). [5 marks]
2. If, instead, P\((X < 3\mu) = 0.86\),
   1. find \(\mu\) in terms of \(\sigma\), [4 marks]
   2. calculate P\((X > 0)\). [4 marks]

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 random variable \(X\) has the normal distribution \(N(2, 1.7^2)\).

1. State the standard deviation of \(X\). [1 mark]
2. Find \(P(X < 0)\). [2 marks]
3. Find \(P(0.6 < X < 3.4)\). [4 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 company makes two cars, model \(A\) and model \(B\). The distance that model \(A\) travels on 10 litres of petrol is normally distributed with mean 109 km and variance 72.25 km\(^2\). The distance that model \(B\) travels on 10 litres of petrol is normally distributed with mean 108.5 km and variance 169 km\(^2\). In a trial, one of each model is filled with 10 litres of petrol and sent on a journey of 110 km. Find which model has the greater probability of completing this journey, and state the value of this probability. [8 marks]

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]

The random variable \(X\) is normally distributed with mean 17. The probability that \(X\) is less than 16 is 0.3707.

1. Calculate the standard deviation of \(X\). [4 marks]
2. In 75 independent observations of \(X\), how many would you expect to be greater than 20? [6 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]

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]

A call-centre dealing with complaints collected data on how long customers had to wait before an operator was free to take their call. The lower quartile of the data was 12.7 minutes and the interquartile range was 5.8 minutes.

1. Find the value of the upper quartile of the data. [1 mark]

It is suggested that a normal distribution could be used to model the waiting time.

1. Calculate correct to 3 significant figures the mean and variance of this normal distribution based on the values of the quartiles. [8 marks]

The actual mean and variance of the data were 15.3 minutes and 20.1 minutes\(^2\) respectively.

1. Comment on the suitability of the model. [2 marks]

The continuous random variable \(X\) has the distribution N(\(\mu\), \(\sigma^2\)).

1. Each of the three following sets of probabilities is impossible. Give a reason in each case why the probabilities cannot both be correct. (You should not attempt to find \(\mu\) or \(\sigma\).)
   1. P(\(X > 50\)) = 0.7 and P(\(X < 50\)) = 0.2 [1]
   2. P(\(X > 50\)) = 0.7 and P(\(X > 70\)) = 0.8 [1]
   3. P(\(X > 50\)) = 0.3 and P(\(X < 70\)) = 0.3 [1]
2. Given that P(\(X > 50\)) = 0.7 and P(\(X < 70\)) = 0.7, find the values of \(\mu\) and \(\sigma\). [4]

The random variable \(G\) has a normal distribution. It is known that $$\text{P}(G < 56.2) = \text{P}(G > 63.8) = 0.1.$$ Find P(\(G > 65\)). [6]

The mass, in kilograms, of a packet of flour is a normally distributed random variable with mean 1.03 and variance \(\sigma^2\). Given that 5% of packets have mass less than 1.00 kg, find the percentage of packets with mass greater than 1.05 kg. [6]

It is believed that the number of sets of traffic lights that fail per hour in a particular large city follows a Poisson distribution with a mean of 3. Find the probability that

1. there will be no failures in a one-hour period, [1 mark]
2. there will be more than 4 failures in a 30-minute period. [3 marks]

Using a suitable approximation, find the probability that in a 24-hour period there will be

1. less than 60 failures, [5 marks]
2. exactly 72 failures. [4 marks]

An electronics company purchases two types of resistor from a manufacturer. The resistances of the resistors (in ohms) are known to be Normally distributed. Type A have a mean of 100 ohms and standard deviation of 1.9 ohms. Type B have a mean of 50 ohms and standard deviation of 1.3 ohms.

1. Find the probability that the resistance of a randomly chosen resistor of type A is less than 103 ohms. [3]
2. Three resistors of type A are chosen at random. Find the probability that their total resistance is more than 306 ohms. [3]
3. One resistor of type A and one resistor of type B are chosen at random. Find the probability that their total resistance is more than 147 ohms. [3]
4. Find the probability that the total resistance of two randomly chosen type B resistors is within 3 ohms of one randomly chosen type A resistor. [5]
5. The manufacturer now offers type C resistors which are specified as having a mean resistance of 300 ohms. The resistances of a random sample of 100 resistors from the first batch supplied have sample mean 302.3 ohms and sample standard deviation 3.7 ohms. Find a 95\% confidence interval for the true mean resistance of the resistors in the batch. Hence explain whether the batch appears to be as specified. [4]

1. The manager of a company that employs 250 travelling sales representatives wishes to carry out a detailed analysis of the expenses claimed by the representatives. He has an alphabetical (by surname) list of the representatives. He chooses a sample of representatives by selecting the 10th, 20th, 30th and so on. Name the type of sampling the manager is attempting to use. Describe a weakness in his method of using it, and explain how he might overcome this weakness. [3]

The representatives each use their own cars to drive to meetings with customers. The total distance, in miles, travelled by a representative in a month is Normally distributed with mean 2018 and standard deviation 96.

1. Find the probability that, in a randomly chosen month, a randomly chosen representative travels more than 2100 miles. [3]
2. Find the probability that, in a randomly chosen 3-month period, a randomly chosen representative travels less than 6000 miles. What assumption is needed here? Give a reason why it may not be realistic. [5]
3. Each month every representative submits a claim for travelling expenses plus commission. Travelling expenses are paid at the rate of 45 pence per mile. The commission is 10\% of the value of sales in that month. The value, in £, of the monthly sales has the distribution N(21200, 1100²). Find the probability that a randomly chosen claim lies between £3000 and £3300. [7]

An examiner believes that once she has marked the first 20 papers the time it takes her to mark one paper for a particular exam follows a Normal distribution. Having already marked more than 20 papers for each of the \(P1\), \(M1\) and \(S1\) modules set one summer, the mean and standard deviation, in seconds, of the time it takes her to mark a paper for each module are as shown in the table below.

1. Find the probability that the difference in the time it takes her to mark two randomly chosen \(P1\) papers is less than 5 seconds. [6]
2. Find the probability that it takes her less than 10 hours to mark 45 \(M1\) and 80 \(S1\) papers. [7]

An organic farm produces eggs which it sells through a local shop. The weight of the eggs produced on the farm are normally distributed with a mean of 55 grams and a standard deviation of 3.9 grams.

1. Find the probability that two of the farm's eggs chosen at random differ in weight by more than 4 grams. [5]

The farm sells boxes of six eggs selected at random. The weight of the boxes used are normally distributed with a mean of 28 grams and a standard deviation of 1.2 grams.

1. Find the probability that a randomly chosen box with six eggs in weighs less than 350 grams. [6]

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 school contains 500 students in years 7 to 11 and 250 students in years 12 and 13. A random sample of 20 students is selected to represent the school at a parents' evening. The number of students in the sample who are from years 12 and 13 is denoted by \(X\).

1. State a suitable binomial model for \(X\). [1]

Use your model to answer the following.

1. 1. Write down an expression for \(\text{P}(X = x)\). [1]
   2. State, in set notation, the values of \(x\) for which your expression is valid. [1]
2. Find \(\text{P}(5 \leqslant X \leqslant 9)\). [2]
3. State one disadvantage of using a random sample in this context. [1]

The mass, in kilograms, of a species of fish in the UK has population mean 4.2 and standard deviation 0.25. An environmentalist believes that the fish in a particular river are smaller, on average, than those in other rivers in the UK. A random sample of 100 fish of this species, taken from the river, has sample mean 4.16 kg. Stating a necessary assumption, test at the 5% significance level whether the environmentalist is correct. [8]

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]