Expected frequency with unknown parameter

4 A mathematical puzzle is given to a large number of students. The times taken to complete the puzzle are normally distributed with mean 14.6 minutes and standard deviation 5.2 minutes.

1. In a random sample of 250 of the students, how many would you expect to have taken more than 20 minutes to complete the puzzle?
   All the students are given a second puzzle to complete. Their times, in minutes, are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is found that \(20 \%\) of the students have times less than 14.5 minutes and \(67 \%\) of the students have times greater than 18.5 minutes.
2. Find the value of \(\mu\) and the value of \(\sigma\).

4 The heights, in metres, of white pine trees are normally distributed with mean 19.8 and standard deviation 2.4 . In a certain forest there are 450 white pine trees.

1. How many of these trees would you expect to have height less than 18.2 metres?
   The heights, in metres, of red pine trees are normally distributed with mean 23.4 and standard deviation \(\sigma\). It is known that \(26 \%\) of red pine trees have height greater than 25.5 metres.
2. Find the value of \(\sigma\).

5 The heights of school desks have a normal distribution with mean 69 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that 15.5\% of these desks have a height greater than 70 cm .

1. Find the value of \(\sigma\). When Jodu sits at a desk, his knees are at a height of 58 cm above the floor. A desk is comfortable for Jodu if his knees are at least 9 cm below the top of the desk. Jodu's school has 300 desks.
2. Calculate an estimate of the number of these desks that are comfortable for Jodu.

3

1. The volume of soup in Super Soup cartons has a normal distribution with mean \(\mu\) millilitres and standard deviation 9 millilitres. Tests have shown that \(10 \%\) of cartons contain less than 440 millilitres of soup. Find the value of \(\mu\).
2. A food retailer orders 150 Super Soup cartons. Calculate the number of these cartons for which you would expect the volume of soup to be more than 1.8 standard deviations above the mean.

5

1. Give an example of a variable in real life which could be modelled by a normal distribution.
2. The random variable \(X\) is normally distributed with mean \(\mu\) and variance 21.0. Given that \(\mathrm { P } ( X > 10.0 ) = 0.7389\), find the value of \(\mu\).
3. If 300 observations are taken at random from the distribution in part (ii), estimate how many of these would be greater than 22.0.

7

1. A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.
   1. Find on how many days of the year ( 365 days) the daily sales can be expected to exceed 3900 litres. The daily sales at another petrol station are \(X\) litres, where \(X\) is normally distributed with mean \(m\) and standard deviation 560. It is given that \(\mathrm { P } ( X > 8000 ) = 0.122\).
   2. Find the value of \(m\).
   3. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.
2. The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\sigma = \frac { 2 } { 3 } \mu\), find the probability that a random value of \(Y\) is less than \(2 \mu\).

4 The time taken for cucumber seeds to germinate under certain conditions has a normal distribution with mean 125 hours and standard deviation \(\sigma\) hours.

1. It is found that \(13 \%\) of seeds take longer than 136 hours to germinate. Find the value of \(\sigma\).
2. 170 seeds are sown. Find the expected number of seeds which take between 131 and 141 hours to germinate.

4 The heights of students at the Mainland college are normally distributed with mean 148 cm and standard deviation 8 cm .

1. The probability that a Mainland student chosen at random has a height less than \(h \mathrm {~cm}\) is 0.67 . Find the value of \(h\).
   120 Mainland students are chosen at random.
2. Find the number of these students that would be expected to have a height within half a standard deviation of the mean.

6. The waiting time, \(L\) minutes, to see a doctor at a health centre is normally distributed with \(L \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). Given that \(\mathrm { P } ( L < 15 ) = 0.9\) and \(\mathrm { P } ( L < 5 ) = 0.05\)

1. find the value of \(\mu\) and the value of \(\sigma\). There are 23 people waiting to see a doctor at the health centre.
2. Determine the expected number of these people who will have a waiting time of more than 12 minutes.

8. The lifetimes of bulbs used in a lamp are normally distributed. A company \(X\) sells bulbs with a mean lifetime of 850 hours and a standard deviation of 50 hours.

1. Find the probability of a bulb, from company \(X\), having a lifetime of less than 830 hours.
2. In a box of 500 bulbs, from company \(X\), find the expected number having a lifetime of less than 830 hours. A rival company \(Y\) sells bulbs with a mean lifetime of 860 hours and \(20 \%\) of these bulbs have a lifetime of less than 818 hours.
3. Find the standard deviation of the lifetimes of bulbs from company \(Y\). Both companies sell the bulbs for the same price.
4. State which company you would recommend. Give reasons for your answer.

1. The weight, in grams, of beans in a tin is normally distributed with mean \(\mu\) and standard deviation 7.8

Given that \(10 \%\) of tins contain less than 200 g , find

1. the value of \(\mu\)
2. the percentage of tins that contain more than 225 g of beans. The machine settings are adjusted so that the weight, in grams, of beans in a tin is normally distributed with mean 205 and standard deviation \(\sigma\).
3. Given that \(98 \%\) of tins contain between 200 g and 210 g find the value of \(\sigma\).

1. A machine cuts strips of metal to length \(L \mathrm {~cm}\), where \(L\) is normally distributed with standard deviation 0.5 cm .

Strips with length either less than 49 cm or greater than 50.75 cm cannot be used.
Given that 2.5\% of the cut lengths exceed 50.98 cm ,

1. find the probability that a randomly chosen strip of metal can be used. Ten strips of metal are selected at random.
2. Find the probability fewer than 4 of these strips cannot be used. A second machine cuts strips of metal of length \(X \mathrm {~cm}\), where \(X\) is normally distributed with standard deviation 0.6 cm A random sample of 15 strips cut by this second machine was found to have a mean length of 50.4 cm
3. Stating your hypotheses clearly and using a \(1 \%\) level of significance, test whether or not the mean length of all the strips, cut by the second machine, is greater than 50.1 cm

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]