2.04f Find normal probabilities: Z transformation

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]

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]

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]

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]

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]

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]

The rules for the weight of a cricket ball state: ``The ball, when new, shall weigh not less than 155.9 g, nor more than 163 g.'' A company produces cricket balls whose weights are normally distributed. It wants 99\% of the balls it produces to be an acceptable weight.

1. What is the largest acceptable standard deviation? [3]

The weights of the cricket balls are in fact normally distributed with mean 159.5 grams and standard deviation 1.2 grams. The company also produces tennis balls. The weights of the tennis balls are normally distributed with mean 58.5 grams and standard deviation 1.3 grams.

1. Find the probability that the weight of a randomly chosen cricket ball is more than three times the weight of a randomly chosen tennis ball. [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]

In a town, 54% of the residents are female and 46% are male. A random sample of 200 residents is chosen from the town. Using a suitable approximation, find the probability that more than half the sample are female. [4 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 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]

The mass \(J\) kg of a bag of randomly chosen Jersey potatoes is a normally distributed random variable with mean 1.00 and standard deviation 0.06. The mass \(K\) kg of a bag of randomly chosen King Edward potatoes is an independent normally distributed random variable with mean 0.80 and standard deviation 0.04.

1. Find the probability that the total mass of 6 bags of Jersey potatoes and 8 bags of King Edward potatoes is greater than 12.70 kg. [3]
2. Find the probability that the mass of one bag of King Edward potatoes is more than 75\% of the mass of one bag of Jersey potatoes. [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 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]