Direct comparison of probabilities

2 In a certain country, the heights of the adult population are normally distributed with mean 1.64 m and standard deviation 0.25 m .

1. Find the probability that an adult chosen at random from this country will have height greater than 1.93 m .
   \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-04_2716_35_143_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-05_2724_35_136_20} In another country, the heights of the adult population are also normally distributed. \(33 \%\) of the adult population have height less than \(1.56 \mathrm {~m} .25 \%\) of the adult population have height greater than 1.86 m .
2. Find the mean and the standard deviation of this distribution.

5 The distance the Zotoc car can travel on 20 litres of fuel is normally distributed with mean 320 km and standard deviation 21.6 km . The distance the Ganmor car can travel on 20 litres of fuel is normally distributed with mean 350 km and standard deviation 7.5 km . Both cars are filled with 20 litres of fuel and are driven towards a place 367 km away.

1. For each car, find the probability that it runs out of fuel before it has travelled 367 km .
2. The probability that a Zotoc car can travel at least \(( 320 + d ) \mathrm { km }\) on 20 litres of fuel is 0.409 . Find the value of \(d\).

7. The weights, \(G\), of a particular breed of gorilla are normally distributed with mean 180 kg and standard deviation 15 kg .

1. Find the proportion of these gorillas whose weights exceed 174 kg .
2. Find, to 1 decimal place, the value of \(k\) such that \(\mathrm { P } ( k < G < 174 ) = 0.3196\) The weights, \(B\), of a particular breed of buffalo are normally distributed with mean 216 kg and standard deviation 30 kg . Given that \(\mathrm { P } ( G > w ) = \mathrm { P } ( B < w ) = p\)
3. 1. find the value of \(w\)
   2. find the value of \(p\) and standard deviation 15 kg .
4. Find the proportion of these gorillas whose weights exceed 174 kg .
5. Find, to 1 decimal place, the value of \(k\) such that \(\mathrm { P } ( k < G < 174 ) = 0.3196\)

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   Q7

   
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7

1. Electra is employed by E \& G Ltd to install electricity meters in new houses on an estate. Her time, \(X\) minutes, to install a meter may be assumed to be normally distributed with a mean of 48 and a standard deviation of 20 . Determine:
   1. \(\mathrm { P } ( X < 60 )\);
   2. \(\mathrm { P } ( 30 < X < 60 )\);
   3. the time, \(k\) minutes, such that \(\mathrm { P } ( X < k ) = 0.9\).
2. Gazali is employed by E \& G Ltd to install gas meters in the same new houses. His time, \(Y\) minutes, to install a meter has a mean of 37 and a standard deviation of 25 .
   1. Explain why \(Y\) is unlikely to be normally distributed.
   2. State why \(\bar { Y }\), the mean of a random sample of 35 gas meter installations, is likely to be approximately normally distributed.
   3. Determine \(\mathrm { P } ( \bar { Y } > 40 )\).

2

1. Tim rings the church bell in his village every Sunday morning. The time that he spends ringing the bell may be modelled by a normal distribution with mean 7.5 minutes and standard deviation 1.6 minutes. Determine the probability that, on a particular Sunday morning, the time that Tim spends ringing the bell is:
   1. at most 10 minutes;
   2. more than 6 minutes;
   3. between 5 minutes and 10 minutes.
2. June rings the same church bell for weekday weddings. The time that she spends, in minutes, ringing the bell may be modelled by the distribution \(\mathrm { N } \left( \mu , 2.4 ^ { 2 } \right)\). Given that 80 per cent of the times that she spends ringing the bell are less than 15 minutes, find the value of \(\mu\).
   [0pt] [4 marks]
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1. (a) Briefly explain what is meant by a sample space.
   (b) State two properties which a function \(f ( x )\) must have to be a probability function.
2. A company makes two cars, model \(A\) and model \(B\). The distance that model \(A\) travels on 10 litres of petrol is normally distributed with mean 109 km and variance \(72.25 \mathrm {~km} ^ { 2 }\). The distance that model \(B\) travels on 10 litres of petrol is normally distributed with mean 108.5 km and variance \(169 \mathrm {~km} ^ { 2 }\).
   In a trial, one of each model is filled with 10 litres of petrol and sent on a journey of 110 km . Find which model has the greater probability of completing this journey, and state the value of this probability.
3. \(A , B\) and \(C\) are three events such that \(\mathrm { P } ( A ) = x , \mathrm { P } ( B ) = y\) and \(\mathrm { P } ( C ) = x + y\).

It is known that \(\mathrm { P } ( A \cup B ) = 0.6\) and \(\mathrm { P } ( B \mid A ) = 0.2\).
(a) Show that \(4 x + 5 y = 3\). It is also known that \(B\) and \(C\) are mutually exclusive and that \(\mathrm { P } ( B \cup C ) = 0.9\)
(b) Obtain another equation in \(x\) and \(y\) and hence find the values of \(x\) and \(y\).
(c) Deduce whether or not \(A\) and \(B\) are independent events.

2 The length of aluminium baking foil on a roll may be modelled by a normal distribution with mean 91 metres and standard deviation 0.8 metres.

1. Determine the probability that the length of foil on a particular roll is:
   1. less than 90 metres;
   2. not exactly 90 metres;
   3. between 91 metres and 92.5 metres.
2. The length of cling film on a roll may also be modelled by a normal distribution but with mean 153 metres and standard deviation \(\sigma\) metres. It is required that \(1 \%\) of rolls of cling film should have a length less than 150 metres.
   Find the value of \(\sigma\) that is needed to satisfy this requirement.
   [0pt] [4 marks]
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7

1. A greengrocer displays apples in trays. Each customer selects the apples he or she wishes to buy and puts them into a bag. Records show that the weight of such bags of apples may be modelled by a normal distribution with mean 1.16 kg and standard deviation 0.43 kg . Determine the probability that the mean weight of a random sample of 10 such bags of apples exceeds 1.25 kg .
2. The greengrocer also displays pears in trays. Each customer selects the pears he or she wishes to buy and puts them into a bag. A random sample of 40 such bags of pears had a mean weight of 0.86 kg and a standard deviation of 0.65 kg .
   1. Construct a \(\mathbf { 9 6 \% }\) confidence interval for the mean weight of a bag of pears.
   2. Hence comment on a claim that customers wish to buy, on average, a greater weight of apples than of pears.
      [0pt] [2 marks]

7

1. The weight of a sack of mixed dog biscuits can be modelled by a normal distribution with a mean of 10.15 kg and a standard deviation of 0.3 kg . A pet shop purchases 12 such sacks that can be considered to be a random sample.
   Calculate the probability that the mean weight of the 12 sacks is less than 10 kg .
2. The weight of dry cat food in a pouch can also be modelled by a normal distribution. The contents, \(x\) grams, of each of a random sample of 40 pouches were weighed. Subsequent analysis of these weights gave $$\bar { x } = 304.6 \quad \text { and } \quad s = 5.37$$
   1. Construct a \(99 \%\) confidence interval for the mean weight of dry cat food in a pouch. Give the limits to one decimal place.
   2. Comment, with justification, on each of the following two claims. Claim 1: The mean weight of dry cat food in a pouch is more than 300 grams.
      Claim 2: All pouches contain more than 300 grams of dry cat food.
      [0pt] [4 marks]
      \includegraphics[max width=\textwidth, alt={}]{6fbb8891-e6de-42fe-a195-ea643552fdcf-24_2288_1705_221_155}