Mixed calculations with boundaries

5 Farmer Jones grows apples. The weights, in grams, of the apples grown this year are normally distributed with mean 170 and standard deviation 25. Apples that weigh between 142 grams and 205 grams are sold to a supermarket.

1. Find the probability that a randomly chosen apple grown by Farmer Jones this year is sold to the supermarket.
   Farmer Jones sells the apples to the supermarket at \(\\) 0.24\( each. He sells apples that weigh more than 205 grams to a local shop at \)\\( 0.30\) each. He does not sell apples that weigh less than 142 grams. The total number of apples grown by Farmer Jones this year is 20000.
2. Calculate an estimate for his total income from this year's apples.
   Farmer Tan also grows apples. The weights, in grams, of the apples grown this year follow the distribution \(\mathrm { N } \left( 182,20 ^ { 2 } \right) .72 \%\) of these apples have a weight more than \(w\) grams.
3. Find the value of \(w\).

3 The weights of oranges can be modelled by a normal distribution with mean 131 grams and standard deviation 54 grams. Oranges are classified as small, medium or large. A large orange weighs at least 184 grams and 20\% of oranges are classified as small.

1. Find the percentage of oranges that are classified as large.
2. Find the greatest possible weight of a small orange.

5 The time in hours that Davin plays on his games machine each day is normally distributed with mean 3.5 and standard deviation 0.9.

1. Find the probability that on a randomly chosen day Davin plays on his games machine for more than 4.2 hours.
2. On 90\% of days Davin plays on his games machine for more than \(t\) hours. Find the value of \(t\).
3. Calculate an estimate for the number of days in a year ( 365 days) on which Davin plays on his games machine for between 2.8 and 4.2 hours.

3 Pia runs 2 km every day and her times in minutes are normally distributed with mean 10.1 and standard deviation 1.3.

1. Find the probability that on a randomly chosen day Pia takes longer than 11.3 minutes to run 2 km .
2. On \(75 \%\) of days, Pia takes longer than \(t\) minutes to run 2 km . Find the value of \(t\).
3. On how many days in a period of 90 days would you expect Pia to take between 8.9 and 11.3 minutes to run 2 km ?

4 Raj wants to improve his fitness, so every day he goes for a run. The times, in minutes, of his runs have a normal distribution with mean 41.2 and standard deviation 3.6.

1. Find the probability that on a randomly chosen day Raj runs for more than 43.2 minutes.
2. Find an estimate for the number of days in a year ( 365 days) on which Raj runs for less than 43.2 minutes.
3. On 95\% of days, Raj runs for more than \(t\) minutes. Find the value of \(t\).

3 A farmer sells eggs. The weights, in grams, of the eggs can be modelled by a normal distribution with mean 80.5 and standard deviation 6.6. Eggs are classified as small, medium or large according to their weight. A small egg weighs less than 76 grams and \(40 \%\) of the eggs are classified as medium.

1. Find the percentage of eggs that are classified as small.
2. Find the least possible weight of an egg classified as large.
   150 of the eggs for sale last week were weighed.
3. Use an approximation to find the probability that more than 68 of these eggs were classified as medium.

4 Melons are sold in three sizes: small, medium and large. The weights follow a normal distribution with mean 450 grams and standard deviation 120 grams. Melons weighing less than 350 grams are classified as small.

1. Find the proportion of melons which are classified as small.
2. The rest of the melons are divided in equal proportions between medium and large. Find the weight above which melons are classified as large.

6 The time in minutes taken by Peter to walk to the shop and buy a newspaper is normally distributed with mean 9.5 and standard deviation 1.3.

1. Find the probability that on a randomly chosen day Peter takes longer than 10.2 minutes.
2. On \(90 \%\) of days he takes longer than \(t\) minutes. Find the value of \(t\).
3. Calculate an estimate of the number of days in a year ( 365 days) on which Peter takes less than 8.8 minutes to walk to the shop and buy a newspaper.

7 The weight of adult female giraffes has a normal distribution with mean 830 kg and standard deviation 120 kg .

1. There are 430 adult female giraffes in a particular game reserve. Find the number of these adult female giraffes which can be expected to weigh less than 700 kg .
2. Given that \(90 \%\) of adult female giraffes weigh between \(( 830 - w ) \mathrm { kg }\) and \(( 830 + w ) \mathrm { kg }\), find the value of \(w\).
   The weight of adult male giraffes has a normal distribution with mean 1190 kg and standard deviation \(\sigma \mathrm { kg }\).
3. Given that \(83.4 \%\) of adult male giraffes weigh more than 950 kg , find the value of \(\sigma\).

1 The time taken, in minutes, by a ferry to cross a lake has a normal distribution with mean 85 and standard deviation 6.8.

1. Find the probability that, on a randomly chosen occasion, the time taken by the ferry to cross the lake is between 79 and 91 minutes.
2. Over a long period it is found that \(96 \%\) of ferry crossings take longer than a certain time \(t\) minutes. Find the value of \(t\).

3 The times taken, in minutes, for trains to travel between Alphaton and Beeton are normally distributed with mean 140 and standard deviation 12.

1. Find the probability that a randomly chosen train will take less than 132 minutes to travel between Alphaton and Beeton.
2. The probability that a randomly chosen train takes more than \(k\) minutes to travel between Alphaton and Beeton is 0.675 . Find the value of \(k\).

3 The distance in metres that a ball can be thrown by pupils at a particular school follows a normal distribution with mean 35.0 m and standard deviation 11.6 m .

1. Find the probability that a randomly chosen pupil can throw a ball between 30 and 40 m .
2. The school gives a certificate to the \(10 \%\) of pupils who throw further than a certain distance. Find the least distance that must be thrown to qualify for a certificate.

3 The times for a certain car journey have a normal distribution with mean 100 minutes and standard deviation 7 minutes. Journey times are classified as follows: \begin{displayquote} 'short' (the shortest \(33 \%\) of times),
'long' (the longest \(33 \%\) of times),
'standard' (the remaining 34\% of times).

1. Find the probability that a randomly chosen car journey takes between 85 and 100 minutes.
2. Find the least and greatest times for 'standard' journeys. \end{displayquote}

5 Gem stones from a certain mine have weights, \(X\) grams, which are normally distributed with mean 1.9 g and standard deviation 0.55 g . These gem stones are sorted into three categories for sale depending on their weights, as follows. Small: under 1.2 g Medium: between 1.2 g and 2.5 g Large: over 2.5 g

1. Find the proportion of gem stones in each of these three categories.
2. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 2.5 ) = 0.8\).

6 The weights of bananas in a fruit shop have a normal distribution with mean 150 grams and standard deviation 50 grams. Three sizes of banana are sold. Small: under 95 grams
Medium: between 95 grams and 205 grams
Large: over 205 grams

1. Find the proportion of bananas that are small.
2. Find the weight exceeded by \(10 \%\) of bananas. The prices of bananas are 10 cents for a small banana, 20 cents for a medium banana and 25 cents for a large banana.
3. (a) Show that the probability that a randomly chosen banana costs 20 cents is 0.7286 .
   (b) Calculate the expected total cost of 100 randomly chosen bananas.

7 The weight, in grams, of pineapples is denoted by the random variable \(X\) which has a normal distribution with mean 500 and standard deviation 91.5. Pineapples weighing over 570 grams are classified as 'large'. Those weighing under 390 grams are classified as 'small' and the rest are classified as 'medium'.

1. Find the proportions of large, small and medium pineapples.
2. Find the weight exceeded by the heaviest \(5 \%\) of pineapples.
3. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 610 ) = 0.3\).

3. The weights of packages that arrive at a factory are normally distributed with a mean of 18 kg and a standard deviation of 5.4 kg

1. Find the probability that a randomly selected package weighs less than 10 kg The heaviest 15\% of packages are moved around the factory by Jemima using a forklift truck.
2. Find the weight, in kg , of the lightest of these packages that Jemima will move. One of the packages not moved by Jemima is selected at random.
3. Find the probability that it weighs more than 18 kg A delivery of 4 packages is made to the factory. The weights of the packages are independent.
4. Find the probability that exactly 2 of them will be moved by Jemima.

1. The heights of females from a country are normally distributed with

• a mean of 166.5 cm
• a standard deviation of 6.1 cm

Given that \(1 \%\) of females from this country are shorter than \(k \mathrm {~cm}\),

1. find the value of \(k\)
2. Find the proportion of females from this country with heights between 150 cm and 175 cm A female, from this country, is chosen at random from those with heights between 150 cm and 175 cm
3. Find the probability that her height is more than 160 cm The heights of females from a different country are normally distributed with a standard deviation of 7.4 cm Mia believes that the mean height of females from this country is less than 166.5 cm
   Mia takes a random sample of 50 females from this country and finds the mean of her sample is 164.6 cm
4. Carry out a suitable test to assess Mia’s belief. You should
   • state your hypotheses clearly
   • use a \(5 \%\) level of significance
   \section*{Question 5 continued.} \section*{Question 5 continued.} \section*{Question 5 continued.}

3. The distance achieved in a long jump competition by students at a school is normally Students who achieve a distance greater than 4.3 metres receive a medal.

1. Find the proportion of students who receive medals. The school wishes to give a certificate of achievement or a medal to the \(80 \%\) of students who achieve a distance of at least \(d\) metres.
2. Find the value of \(d\). Of those who received medals, the \(\frac { 1 } { 3 }\) who jump the furthest will receive gold medals.
3. Find the shortest distance, \(g\) metres, that must be achieved to receive a gold medal. A journalist from the local newspaper interviews a randomly selected group of 3 medal winners.
4. Find the exact probability that there is at least one gold medal winner in the group.
   \includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-08_79_1153_233_251} Students who achieve a distance greater than 4.3 metres receive a medal.
5. Find the proportion of students who receive medals.

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

1. A competition consists of two rounds.

The time, in minutes, taken by adults to complete round one is modelled by a normal distribution with mean 15 minutes and standard deviation 2 minutes.

1. Use standardisation to find the proportion of adults that take less than 18 minutes to complete round one. Only the fastest \(60 \%\) of adults from round one take part in round two.
2. Use standardisation to find the longest time that an adult can take to complete round one if they are to take part in round two. The time, \(T\) minutes, taken by adults to complete round two is modelled by a normal distribution with mean \(\mu\) Given that \(\mathrm { P } ( \mu - 10 < T < \mu + 10 ) = 0.95\)
3. find \(\mathrm { P } ( T > \mu - 5 \mid T > \mu - 10 )\)

1. The weights, \(W\) grams, of kiwi fruit grown on a farm are normally distributed with mean 80 grams and standard deviation 8 grams.

The table shows the classifications of the kiwi fruit by their weight, where \(k\) is a positive constant. One kiwi fruit is selected at random from those grown on the farm.

1. Find the probability that this kiwi fruit is Large. 35\% of the kiwi fruit are Jumbo.
2. Find the value of \(k\) to one decimal place. 75\% of Tiny kiwi fruit weigh more than \(y\) grams.
3. Find the value of \(y\) giving your answer to one decimal place.

4. Past records show that the times, in seconds, taken to run 100 m by children at a school can be modelled by a normal distribution with a mean of 16.12 and a standard deviation of 1.60 A child from the school is selected at random.

1. Find the probability that this child runs 100 m in less than 15 s . On sports day the school awards certificates to the fastest \(30 \%\) of the children in the 100 m race.
2. Estimate, to 2 decimal places, the slowest time taken to run 100 m for which a child will be awarded a certificate.

6. The time, in minutes, taken by men to run a marathon is modelled by a normal distribution with mean 240 minutes and standard deviation 40 minutes.

1. Find the proportion of men that take longer than 300 minutes to run a marathon.
   (3) Nathaniel is preparing to run a marathon. He aims to finish in the first 20\% of male runners.
2. Using the above model estimate the longest time that Nathaniel can take to run the marathon and achieve his aim.
   (3) The time, \(W\) minutes, taken by women to run a marathon is modelled by a normal distribution with mean \(\mu\) minutes. Given that \(\mathrm { P } ( W < \mu + 30 ) = 0.82\)
3. find \(\mathrm { P } ( W < \mu - 30 \mid W < \mu )\)