Percentages or proportions given

1 The volume of milk in millilitres in cartons is normally distributed with mean \(\mu\) and standard deviation 8. Measurements were taken of the volume in 900 of these cartons and it was found that 225 of them contained more than 1002 millilitres.

1. Calculate the value of \(\mu\).
2. Three of these 900 cartons are chosen at random. Calculate the probability that exactly 2 of them contain more than 1002 millilitres.

7 The heights that children of a particular age can jump have a normal distribution. On average, 8 children out of 10 can jump a height of more than 127 cm , and 1 child out of 3 can jump a height of more than 135 cm .

1. Find the mean and standard deviation of the heights the children can jump.
2. Find the probability that a randomly chosen child will not be able to jump a height of 145 cm .
3. Find the probability that, of 8 randomly chosen children, at least 2 will be able to jump a height of more than 135 cm .

6 The lengths, in cm, of trout in a fish farm are normally distributed. 96\% of the lengths are less than 34.1 cm and 70\% of the lengths are more than 26.7 cm .

1. Find the mean and the standard deviation of the lengths of the trout. In another fish farm, the lengths of salmon, \(X \mathrm {~cm}\), are normally distributed with mean 32.9 cm and standard deviation 2.4 cm .
2. Find the probability that a randomly chosen salmon is 34 cm long, correct to the nearest centimetre.
3. Find the value of \(t\) such that \(\mathrm { P } ( 31.8 < X < t ) = 0.5\).

4 It is known that 20\% of male giant pandas in a certain area weigh more than 121 kg and \(71.9 \%\) weigh more than 102 kg . Weights of male giant pandas in this area have a normal distribution. Find the mean and standard deviation of the weights of male giant pandas in this area.

7. One event at Pentor sports day is throwing a tennis ball. The distance a child throws a tennis ball is modelled by a normal distribution with mean 32 m and standard deviation 12 m . Any child who throws the tennis ball more than 50 m is awarded a gold certificate.

1. Show that, to 3 significant figures, 6.68\% of children are awarded a gold certificate. A silver certificate is awarded to any child who throws the tennis ball more than \(d\) metres but less than 50 m . Given that 19.1\% of the children are awarded a silver certificate,
2. find the value of \(d\). Three children are selected at random from those who take part in the throwing a tennis ball event.
3. Find the probability that 1 is awarded a gold certificate and 2 are awarded silver certificates. Give your answer to 2 significant figures.

4. The heights of the students at a university are assumed to follow a normal distribution. \(1 \%\) of the students are over 200 cm tall and 76\% are between 165 cm and 200 cm tall. Find

1. the mean and the variance of the distribution,
2. the percentage of the students who are under 158 cm tall.
3. Comment briefly on the suitability of a normal distribution to model such a population. \section*{STATISTICS 1 (A) TEST PAPER 3 Page 2}

5. The ages of the residents of a retirement community are assumed to be normally distributed. \(15 \%\) of the residents are under 60 years old and \(5 \%\) are over 90 years old.

1. Using this information, find the mean and the standard deviation of the ages.
2. If there are 200 residents, find how many are over 80 years old. \section*{STATISTICS 1 (A) TEST PAPER 5 Page 2}

6. 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. The geologist also estimated that only \(2 \%\) of the longest diameters were smaller than 1 mm .
2. Calculate the corresponding percentage that would be predicted by a normal distribution with the parameters you calculated in part (a).
3. Hence, comment on the suitability of the normal distribution as a model in this situation.
   (2 marks)

3. Hei and Tang are designing some pieces of art. They collected a large number of sticks. The random variable \(L\) represents the length of a stick in centimetres and has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). They sorted the sticks into lengths and painted them.
They found that \(60 \%\) of the sticks were longer than 45 cm and these were painted red, whilst \(15 \%\) of the sticks were shorter than 35 cm and these were painted blue. The remaining sticks were painted yellow.

1. Show that \(\mu\) and \(\sigma\) satisfy $$45 + 0.2533 \sigma = \mu$$
2. Find a second equation in \(\mu\) and \(\sigma\).
3. Hence find the value of \(\mu\) and the value of \(\sigma\).
4. Find
   1. \(\mathrm { P } ( L > 35 \mid L < 45 )\)
   2. \(\mathrm { P } ( L < 45 \mid L > 35 )\) Hei created her piece of art using a random selection of blue and yellow sticks.
      Tang created his piece of art using a random selection of red and yellow sticks.
      Hei and Tang each used the same number of sticks to create their piece of art.
      George is viewing Hei's and Tang's pieces of art. He finds a yellow stick on the floor that has fallen from one of these pieces.
5. With reference to your answers to part (d), state, giving a reason, whether the stick is more likely to have fallen from Hei's or Tang's piece of art.

17 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.
17

1. Find the mean and standard deviation of \(X\).
2. 1. Find \(\mathrm { P } ( X \neq 35 )\)
      17
3. (ii) Find \(\mathrm { P } ( X < 35 )\)
   17
4. 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.
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