Validity of normal model

9 The continuous random variable \(X\) has distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 2 , \\ \frac { 1 } { 8 } x - \frac { 1 } { 4 } & 2 \leqslant x \leqslant 10 , \\ 1 & x > 10 . \end{cases}$$ Find the value of \(k\) for which \(\mathrm { P } ( X \geqslant k ) = 0.6\). The random variable \(Y\) is defined by \(Y = 2 \ln X\). Find the distribution function of \(Y\). Find the probability density function of \(Y\) and sketch its graph.

1 It is given that \(X \sim \mathrm {~N} ( 30,49 ) , Y \sim \mathrm {~N} ( 30,16 )\) and \(Z \sim \mathrm {~N} ( 50,16 )\). On a single diagram, with the horizontal axis going from 0 to 70 , sketch three curves to represent the distributions of \(X , Y\) and \(Z\).

1 Name the distribution and suggest suitable numerical parameters that you could use to model the weights in kilograms of female 18-year-old students.

2. The random variable \(X\) is normally distributed with mean 177.0 and standard deviation 6.4.

1. Find \(\mathrm { P } ( 166 < X < 185 )\). It is suggested that \(X\) might be a suitable random variable to model the height, in cm , of adult males.
2. Give two reasons why this is a sensible suggestion.
3. Explain briefly why mathematical models can help to improve our understanding of real-world problems.

2. The random variable \(X\) is normally distributed with mean 177.0 and standard deviation 6.4.

1. Find \(\mathrm { P } ( 166 < X < 185 )\).
   (4 marks)
   It is suggested that \(X\) might be a suitable random variable to model the height, in cm , of adult males.
2. Give two reasons why this is a sensible suggestion.
   (2 marks)
3. Explain briefly why mathematical models can help to improve our understanding of real-world problems.
   (2 marks)

9 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[max width=\textwidth, alt={}, center]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-08_842_1651_495_207} One of the 150 plants is chosen at random, and its height, \(X \mathrm {~cm}\), is noted.

1. Show that \(\mathrm { P } ( 20 < X < 30 ) = 0.147\), correct to 3 significant figures. Sam suggests that the distribution of \(X\) can be well modelled by the distribution \(\mathrm { N } ( 40,100 )\).
2. 1. Give a brief justification for the use of the normal distribution in this context.
   2. Give a brief justification for the choice of the parameter values 40 and 100 .
3. Use Sam's model to find \(\mathrm { P } ( 20 < X < 30 )\). 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 \(\mathrm { N } \left( m , s ^ { 2 } \right)\) as her model.
4. Use Nina's model to find \(\mathrm { P } ( 20 < X < 30 )\).
5. 1. Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model.
   2. By considering the different ranges of values of \(X\) given in the table, discuss how well the two models fit the original distribution.

4. Automatic coin counting machines sort, count and batch coins. A particular brand of these machines rejects \(2 p\) coins that are less than 6.12 grams or greater than 8.12 grams.

1. The histogram represents the distribution of the weight of UK 2p coins supplied by the Royal Mint. This distribution has mean 7.12 grams and standard deviation 0.357 grams. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Weight of UK two pence coins} \includegraphics[alt={},max width=\textwidth]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-3_602_969_664_589}
   \end{figure} Explain why the weight of 2 p coins can be modelled using a normal distribution.
2. Assume the distribution of the weight of \(2 p\) coins is normally distributed. Calculate the proportion of \(2 p\) coins that are rejected by this brand of coin counting machine.
3. A manager suspects that a large batch of \(2 p\) coins is counterfeit. A random sample of 30 of the suspect coins is selected. Each of the coins in the sample is weighed. The results are shown in the summary statistics table.

   Summary statistics
   Mean	Standard deviation	Minimum	Lower quartile	Median	Upper quartile	Maximum
   6.89	0.296	6.45	6.63	6.88	7.08	7.48

   i) What assumption must be made about the weights of coins in this batch in order to conduct a test of significance on the sample mean? State, with a reason, whether you think this assumption is reasonable.
   ii) Assuming the population standard deviation is 0.357 grams, test at the \(1 \%\) significance level whether the mean weight of the \(2 p\) coins in this batch is less than 7.12 grams.

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 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]

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]