2.04e Normal distribution: as model N(mu, sigma^2)

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]

Some children are asked to mark the centre of a scale 10 cm long. The position they choose is indicated by the variable \(X\), where \(0 \leq X \leq 10\). Initially, \(X\) is modelled as a random variable with a continuous uniform distribution.

1. Find the mean and the standard deviation of \(X\). [3 marks]

It is suggested that a better model would be the distribution with probability density function $$f(x) = cx, \quad 0 \leq x \leq 5, \quad f(x) = c(10-x), \quad 5 < x \leq 10, \quad f(x) = 0 \text{ otherwise}.$$

1. Write down the mean of \(X\). [1 mark]
2. Find \(c\), and hence find the standard deviation of \(X\) in this model. [7 marks]
3. Find P(\(4 < X < 6\)). [3 marks]

It is then proposed that an even better model for \(X\) would be a Normal distribution with the mean and standard deviation found in parts (b) and (c).

1. Use these results to find P(\(4 < X < 6\)) in the third model. [4 marks]
2. Compare your answer with (d). Which model do you think is most appropriate? [1 mark]

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]

A teacher gives each student in his class a list of 30 numbers. All the numbers have been generated at random by a computer from a normal distribution with a fixed mean and variance. The teacher tells the class that the variance of the distribution is 25 and asks each of them to calculate a 95\% confidence interval based on their list of numbers. The sum of the numbers given to one student is 1419.

1. Find the confidence interval that should be obtained by this student. [5]

Assuming that all the students calculate their confidence intervals correctly,

1. state the proportion of the students you would expect to have a confidence interval that includes the true mean of the distribution, [1]
2. explain why the probability of any one student's confidence interval including the true mean is not 0.95 [1]

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]

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]

Elizabeth's Bakery makes brownies. It is known that the mass, \(X\) grams, of a brownie may be modelled by a normal distribution. 10\% of the brownies have a mass less than 30 grams. 80\% of the brownies have a mass greater than 32.5 grams.

1. Find the mean and standard deviation of \(X\). [7 marks]
2. 1. Find P\((X \neq 35)\) [1 mark]
   2. Find P\((X < 35)\) [2 marks]
3. Brownies are baked in batches of 13. Calculate the probability that, in a batch of brownies, no more than 3 brownies are less than 35 grams. You may assume that the masses of brownies are independent of each other. [2 marks]

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]

A factory produces jars of jam and jars of marmalade.

1. The weight, \(X\) grams, of jam in a jar can be modelled as a normal variable with mean 372 and a standard deviation of 3.5
   1. Find the probability that the weight of jam in a jar is equal to 372 grams. [1 mark]
   2. Find the probability that the weight of jam in a jar is greater than 368 grams. [2 marks]
2. The weight, \(Y\) grams, of marmalade in a jar can be modelled as a normal variable with mean \(\mu\) and standard deviation \(\sigma\)
   1. Given that \(P(Y < 346) = 0.975\), show that $$346 - \mu = 1.96\sigma$$ Fully justify your answer. [3 marks]
   2. Given further that $$P(Y < 336) = 0.14$$ find \(\mu\) and \(\sigma\) [4 marks]

\(X \sim \text{N}(14, 0.35)\) Find the standard deviation of \(X\), correct to two decimal places. Circle your answer. [1 mark] 0.12 \quad\quad 0.35 \quad\quad 0.59 \quad\quad 1.78

In a particular year, the height of a male athlete at the Summer Olympics has a mean 1.78 metres and standard deviation 0.23 metres. The heights of 95% of male athletes are between 1.33 metres and 2.22 metres.

1. Comment on whether a normal distribution may be suitable to model the height of a male athlete at the Summer Olympics in this particular year. [3 marks]
2. You may assume that the height of a male athlete at the Summer Olympics may be modelled by a normal distribution with mean 1.78 metres and standard deviation 0.23 metres.
   1. Find the probability that the height of a randomly selected male athlete is 1.82 metres. [1 mark]
   2. Find the probability that the height of a randomly selected male athlete is between 1.70 metres and 1.90 metres. [1 mark]
   3. Two male athletes are chosen at random. Calculate the probability that both of their heights are between 1.70 metres and 1.90 metres. [1 mark]
3. The summarised data for the heights, \(h\) metres, of a random sample of 40 male athletes at the Winter Olympics is given below. $$\sum h = 69.2 \quad\quad \sum (h - \bar{h})^2 = 2.81$$ Use this data to calculate estimates of the mean and standard deviation of the heights of male athletes at the Winter Olympics. [3 marks]
4. Using your answers from part (c), compare the heights of male athletes at the Summer Olympics and male athletes at the Winter Olympics. [2 marks]

The mass of aluminium cans recycled each day in a city may be modelled by a normal distribution with mean 24 500 kg and standard deviation 5 200 kg.

1. State the probability that the mass of aluminium cans recycled on any given day is not equal to 24 500 kg. [1 mark]
2. To reduce costs, the city's council decides to collect aluminium cans for recycling less frequently. Following the decision, it was found that over a 24-day period a total mass of 641 520 kg of aluminium cans was recycled. It can be assumed that the distribution of the mass of aluminium cans recycled is still normal with standard deviation 5 200 kg, and that the 24-day period can be regarded as a random sample. Investigate, at the 5% level of significance, whether the mean daily mass of aluminium cans recycled has changed. [7 marks]
3. A member of the council claims that if a different sample of 24 days had been used the hypothesis test in part (b) would have given the same result. Comment on the validity of this claim. [2 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]

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]

In the South West region of England, 100 households were randomly selected and, for each household, the weekly expenditure, \(£X\), per person on food and drink was recorded. The maximum amount recorded was £40.48 and the minimum amount recorded was £22.00 The results are summarised below, where \(\bar{x}\) denotes the sample mean. $$\sum x = 3046.14 \quad\quad \sum (x - \bar{x})^2 = 1746.29$$

1. 1. Find the mean of \(X\) Find the standard deviation of \(X\) [2 marks]
   2. Using your results from part (a)(i) and other information given, explain why the normal distribution can be used to model \(X\). [2 marks]
   3. Find the probability that a household in the South West spends less than £25.00 on food and drink per person per week. [1 mark]
2. For households in the North West of England, the weekly expenditure, \(£Y\), per person on food and drink can be modelled by a normal distribution with mean £29.55 It is known that \(P(Y < 30) = 0.55\) Find the standard deviation of \(Y\), giving your answer to one decimal place. [3 marks]