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

\(X\) is a continuous random variable such that \(X \sim N(\mu, \sigma^2)\). On the sketch of this Normal distribution in the Printed Answer Booklet, shade the area bounded by the curve, the \(x\)-axis and the lines \(x = \mu \pm \sigma\). [2]

At the beginning of the academic year, all the pupils in year 12 at a college take part in an assessment. Summary statistics for the marks obtained by the 2021 cohort are given below. \(n = 205 \quad \sum x = 23042 \quad \sum x^2 = 2591716\) Marks may only be whole numbers, but the Head of Mathematics believes that the distribution of marks may be modelled by a Normal distribution.

1. Calculate
   [2]
2. Use your answers to part (a) to write down a possible Normal model for the distribution of marks. [2]

One candidate in the cohort scored less than 105.

1. Determine whether the model found in part (b) is consistent with this information. [3]
2. Use the model to calculate an estimate of the number of candidates who scored 115 marks. [2]

Each day, for many years, the maximum temperature in degrees Celsius at a particular location is recorded. The maximum temperatures for days in October can be modelled by a Normal distribution. The appropriate Normal curve is shown in Fig. 6. \includegraphics{figure_6}

1. 1. Use the model to write down the mean of the maximum temperatures. [1]
   2. Explain why the curve indicates that the standard deviation is approximately 3 degrees Celsius. [1]

Temperatures can be converted from Celsius to Fahrenheit using the formula \(F = 1.8C + 32\), where \(F\) is the temperature in degrees Fahrenheit and \(C\) is the temperature in degrees Celsius.

1. For maximum temperature in October in degrees Fahrenheit, estimate
   [2]

A quality control department checks the lifetimes of batteries produced by a company. The lifetimes, \(x\) minutes, for a random sample of 80 'Superstrength' batteries are shown in the table below.

Lifetime	\(160 \leq x < 165\)	\(165 \leq x < 168\)	\(168 \leq x < 170\)	\(170 \leq x < 172\)	\(172 \leq x < 175\)	\(175 \leq x < 180\)
Frequency	5	14	20	21	16	4

1. Estimate the proportion of these batteries which have a lifetime of at least 174.0 minutes. [2]
2. Use the data in the table to estimate
   [3]

The data in the table on the previous page are represented in the following histogram, Fig 15. \includegraphics{figure_15} A quality control manager models the data by a Normal distribution with the mean and standard deviation you calculated in part (b).

1. Comment briefly on whether the histogram supports this choice of model. [2]
2. 1. Use this model to estimate the probability that a randomly selected battery will have a lifetime of more than 174.0 minutes.
   2. Compare your answer with your answer to part (a). [3]

The company also manufactures 'Ultrapower' batteries, which are stated to have a mean lifetime of 210 minutes.

1. A random sample of 8 Ultrapower batteries is selected. The mean lifetime of these batteries is 207.3 minutes. Carry out a hypothesis test at the 5% level to investigate whether the mean lifetime is as high as stated. You should use the following hypotheses \(\text{H}_0 : \mu = 210\), \(\text{H}_1 : \mu < 210\), where \(\mu\) represents the population mean for Ultrapower batteries. You should assume that the population is Normally distributed with standard deviation 3.4. [5]

Antonio arrives at a train station at a random point in time. The trains to his desired destination are scheduled to depart at 12-minute intervals.

1. Assume that Antonio gets on the next train.
   1. Suggest an appropriate distribution to model his waiting time and give the parameters.
   2. State the mean and the variance of this distribution.
   3. State an assumption you have made in suggesting this distribution. [4]
2. Now assume that the probability that Antonio misses the next available train because he is distracted by his smartphone is \(0 \cdot 12\). If he misses the next available train, he is sure to get on the one after that.
   1. Find the probability that he waits between 9 and 19 minutes.
   2. Given that he waits between 9 and 19 minutes, find the probability that he gets on the first train. [6]

Arwyn collects data about household expenditure on food. He records the weekly expenditure on food for 80 randomly selected households from across Wales.

1. Explain why a normal distribution may be an appropriate model for the weekly expenditure on food for this sample. [1]

Arwyn uses the distribution N(64, 15²) to model expenditure on food.

1. Find the number of households in the sample that this model would predict to have weekly food expenditure in the range
   1. \(60 \leqslant x < 70\),
   2. \(x \geqslant 90\). [4]
2. Use your answers to part (b)
   1. to comment on the suitability of this model,
   2. to explain how Arwyn could improve the model by changing one of its parameters. [2]
3. Arwyn's friend Colleen wishes to use the improved model to predict household expenditure on food in Northern Ireland. Comment on this plan. [1]

A company produces kettlebells whose weights are normally distributed with mean \(16\) kg and standard deviation \(0.08\) kg.

1. Find the probability that the weight of a randomly selected kettlebell is greater than \(16.05\) kg. [2]

The company trials a new production method. It needs to check that the mean is still \(16\) kg. It assumes that the standard deviation is unchanged. The company takes a random sample of 25 kettlebells and it decides to reject the new production method if the sample mean does not round to \(16\) kg to the nearest \(100\) g.

1. Find the probability that the new production method will be rejected if, in fact, the mean is still \(16\) kg. [4]

The company decides instead to use a 5\% significance test. A random sample of 25 kettlebells is selected and the mean is found to be \(16.02\) kg.

1. Carry out the test to determine whether or not the new production method will be rejected. [6]

A farmer uses many identical containers to store four different types of grain: wheat, corn, einkorn and emmer.

1. The mass \(W\), in kg, of wheat stored in each individual container is normally distributed with mean \(\mu\) and standard deviation 0.6. Given that, for containers of wheat, 10\% store less than 19 kg, find the value of \(\mu\). [3]

The mass \(X\), in kg, of corn stored in each individual container is normally distributed with mean 20.1 and standard deviation 1.2.

1. Find the probability that the mean mass of corn in a random sample of 8 containers of corn will be greater than 20 kg. [3]

The mass \(Y\), in kg, of einkorn stored in each individual container is normally distributed with mean 22.2 and standard deviation 1.5. The farmer and his wife need to move two identical wheelbarrows, one of which is loaded with 3 containers of corn, and the other of which is loaded with 3 containers of einkorn. They agree that the farmer's wife will move the heavier wheelbarrow.

1. Calculate the probability that the farmer's wife will move
   1. the einkorn,
   2. the corn. [5]
2. The mass \(E\), in kg, of emmer stored in each individual container is normally distributed with mean 10.5 and standard deviation \(\sigma\). The farmer's son tries to calculate the probability that the mass of corn in a single container will be more than three times the mass of emmer in a single container. He obtains an answer of 0.35208.
   1. Find the value of \(\sigma\) that the farmer's son used.
   2. Explain why the value of \(\sigma\) that he used is unreasonable. [8]

Alun does the crossword in the Daily Bugle every day. The time that he takes to complete the crossword, \(X\) minutes, is modelled by the normal distribution \(\mathrm{N}(32, 4^2)\). You may assume that the times taken to complete the crossword on successive days are independent.

1. 1. Find the upper quartile of \(X\) and explain its meaning in context.
   2. Find the probability that the total time taken by Alun to complete the crosswords on five randomly chosen days is greater than 170 minutes. [7]
2. Belle also does the crossword every day and the time that she takes to complete the crossword, \(Y\) minutes, is modelled by the normal distribution \(\mathrm{N}(18, 2^2)\). Find the probability that, on a randomly chosen day, the time taken by Alun to complete the crossword is more than twice the time taken by Belle to complete the crossword. [6]

The weights of sacks of potatoes are normally distributed. It is known that one in five sacks weigh more than 6kg and three in five sacks weigh more than 5.5kg.

1. Find the mean and standard deviation of the weights of potato sacks. [5]
2. The sacks are put into crates, with twelve sacks going into each crate. What is the probability that a given crate contains two or more sacks that weigh more than 6kg? You must explain your reasoning clearly in this question. [4]

A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes.

1. Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. [1]
2. Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes. The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes. Stating your hypotheses clearly and using a 5% significance level, test whether or not there is evidence to support the patients' complaint. [4]
3. The health centre also claims that the time a dentist spends with a patient during a routine appointment, \(T\) minutes, can be modelled by the normal distribution where \(T \sim N(5, 3.5^2)\) Using this model,
   1. find the probability that a routine appointment with the dentist takes less than 2 minutes [1]
   2. find \(P(T < 2 | T > 0)\) [3]
   3. hence explain why this normal distribution may not be a good model for \(T\). [1]
4. The dentist believes that she cannot complete a routine appointment in less than 2 minutes. She suggests that the health centre should use a refined model only including values of \(T > 2\) Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place. [5]

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]

At the beginning of the academic year, all the pupils in year 12 at a college take part in an assessment. Summary statistics for the marks obtained by the 2021 cohort are given below. \(n = 205\) \(\sum x = 23042\) \(\sum x^2 = 2591716\) Marks may only be whole numbers, but the Head of Mathematics believes that the distribution of marks may be modelled by a Normal distribution.

1. Calculate
   [2]
2. Use your answers to part (a) to write down a possible Normal model for the distribution of marks. [2]

The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram. \includegraphics{figure_2} One of the 150 plants is chosen at random, and its height, \(X\) cm, is noted.

1. Show that P\((20 < X < 30) = 0.147\), correct to 3 significant figures. [2]

Sam suggests that the distribution of \(X\) can be well modelled by the distribution N\((40, 100)\).

1. 1. Give a brief justification for the use of the normal distribution in this context. [1]
   2. Give a brief justification for the choice of the parameter values 40 and 100. [2]
2. Use Sam's model to find P\((20 < X < 30)\). [1]

Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, \(m\) and \(s\), for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution N\((m, s^2)\) as her model.

1. Use Nina's model to find P\((20 < X < 30)\). [4]
2. 1. Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model. [2]
   2. By considering the different ranges of values of \(X\) given in the table, discuss how well the two models fit the original distribution. [2]

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 market gardener records the masses of a random sample of 100 of this year's crop of plums. The table shows his results.

1. Explain why the normal distribution might be a reasonable model for this distribution. [1]

The market gardener models the distribution of masses by \(N(47.5, 10^2)\).

1. Find the number of plums in the sample that this model would predict to have masses in the range:
   1. \(35 \leq m < 45\) [2]
   2. \(m < 25\) [2]
2. Use your answers to parts (b)(i) and (b)(ii) to comment on the suitability of this model. [1]

The market gardener plans to use this model to predict the distribution of the masses of next year's crop of plums.

1. Comment on this plan. [1]

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 firm produces chocolate bars whose weights are normally distributed with mean 120 g and standard deviation 6 g.

1. Bars which weigh more than 114 g are sold at a profit of 15p per bar. The remaining bars are sold at no profit. Show that the expected profit per 100 bars is £12.62. [5]
2. It is subsequently decided that bars which weigh more than \(x\) g should be sold at a profit of 20p per bar. Those which weigh \(x\) g or less are sold to employees at a profit of 3p per bar. The expected profit per 100 bars is £19.17. Find the value of \(x\). [7]

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]

A machine is being used to manufacture ball bearings. The diameters of the ball bearings are normally distributed with mean 8.3 mm and standard deviation 0.20 mm.

1. Find the probability that the diameter of a randomly chosen ball bearing lies between 8.1 mm and 8.5 mm. [5]
2. Following an overhaul of the machine, it is now found that the diameters of 88% of ball bearings are less than 8.5 mm while 10% are less than 8.1 mm. Estimate the new mean and standard deviation of the diameters. [6]

A machine is being used to manufacture ball bearings. The diameters of the ball bearings are normally distributed with mean 8.3 mm and standard deviation 0.20 mm.

1. Find the probability that the diameter of a randomly chosen ball bearing lies between 8.1 mm and 8.5 mm. [5]
2. Following an overhaul of the machine, it is now found that the diameters of 88\% of ball bearings are less than 8.5 mm while 10\% are less than 8.1 mm. Estimate the new mean and standard deviation of the diameters. [6]

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]

The times for a motorist to travel from home to work are normally distributed with a mean of 24 minutes and a standard deviation of 4 minutes. Find the probability that a particular trip from home to work takes

1. more than 27 minutes, [3]
2. between 20 and 25 minutes. [3]

The times for a motorist to travel from home to work are normally distributed with a mean of 24 minutes and a standard deviation of 4 minutes. Find the probability that a particular trip from home to work takes

1. more than 27 minutes, [3]
2. between 20 and 25 minutes. [3]