Find p then binomial probability

6 The mass of grapes sold per day by a large shop can be modelled by a normal distribution with mean 28 kg . On \(10 \%\) of days less than 16 kg of grapes are sold.

1. Find the standard deviation of the mass of grapes sold per day.
   The mass of grapes sold on any day is independent of the mass sold on any other day.
2. 12 days are chosen at random. Find the probability that less than 16 kg of grapes are sold on more than 2 of these 12 days.
3. In a random sample of 365 days, on how many days would you expect the mass of grapes sold to be within 1.3 standard deviations of the mean?

7 The times, in minutes, that Karli spends each day on social media are normally distributed with mean 125 and standard deviation 24.

1. 1. On how many days of the year ( 365 days) would you expect Karli to spend more than 142 minutes on social media?
   2. Find the probability that Karli spends more than 142 minutes on social media on fewer than 2 of 10 randomly chosen days.
2. On \(90 \%\) of days, Karli spends more than \(t\) minutes on social media. Find the value of \(t\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\frac { 1 } { 4 } \mu\). It is given that \(\mathrm { P } ( X > 20 ) = 0.04\).

1. Find \(\mu\).
2. Find \(\mathrm { P } ( 10 < X < 20 )\).
3. 250 independent observations of \(X\) are taken. Find the probability that at least 235 of them are less than 20.

7 The time Rafa spends on his homework each day in term-time has a normal distribution with mean 1.9 hours and standard deviation \(\sigma\) hours. On \(80 \%\) of these days he spends more than 1.35 hours on his homework.

1. Find the value of \(\sigma\).
2. Find the probability that, on a randomly chosen day in term-time, Rafa spends less than 2 hours on his homework.
3. A random sample of 200 days in term-time is taken. Use an approximation to find the probability that the number of days on which Rafa spends more than 1.35 hours on his homework is between 163 and 173 inclusive.

6 The lifetimes, in hours, of a particular type of light bulb are normally distributed with mean 2000 hours and standard deviation \(\sigma\) hours. The probability that a randomly chosen light bulb of this type has a lifetime of more than 1800 hours is 0.96 .

1. Find the value of \(\sigma\).
   New technology has resulted in a new type of light bulb. It is found that on average one in five of these new light bulbs has a lifetime of more than 2500 hours.
2. For a random selection of 300 of these new light bulbs, use a suitable approximate distribution to find the probability that fewer than 70 have a lifetime of more than 2500 hours.
3. Justify the use of your approximate distribution in part (ii).

7 The times spent by people visiting a certain dentist are independent and normally distributed with a mean of 8.2 minutes. \(79 \%\) of people who visit this dentist have visits lasting less than 10 minutes.

1. Find the standard deviation of the times spent by people visiting this dentist.
2. Find the probability that the time spent visiting this dentist by a randomly chosen person deviates from the mean by more than 1 minute.
3. Find the probability that, of 6 randomly chosen people, more than 2 have visits lasting longer than 10 minutes.
4. Find the probability that, of 35 randomly chosen people, fewer than 16 have visits lasting less than 8.2 minutes. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

5 The weights of letters posted by a certain business are normally distributed with mean 20 g . It is found that the weights of \(94 \%\) of the letters are within 12 g of the mean.

1. Find the standard deviation of the weights of the letters.
2. Find the probability that a randomly chosen letter weighs more than 13 g .
3. Find the probability that at least 2 of a random sample of 7 letters have weights which are more than 12 g above the mean.

3 Lengths of rolls of parcel tape have a normal distribution with mean 75 m , and 15\% of the rolls have lengths less than 73 m .

1. Find the standard deviation of the lengths. Alison buys 8 rolls of parcel tape.
2. Find the probability that fewer than 3 of these rolls have lengths more than 77 m .

1. The length of pregnancy for a randomly selected pregnant sheep is \(D\) days where

$$D \sim \mathrm {~N} \left( 112.4 , \sigma ^ { 2 } \right)$$ Given that 5\% of pregnant sheep have a length of pregnancy of less than 108 days,

1. find the value of \(\sigma\) Qiang selects 25 pregnant sheep at random from a large flock.
2. Find the probability that more than 3 of these pregnant sheep have a length of pregnancy of less than 108 days. Charlie takes 200 random samples of 25 pregnant sheep.
3. Use a Poisson approximation to estimate the probability that at least 2 of the samples have more than 3 pregnant sheep with a length of pregnancy of less than 108 days.

4. Pieces of ribbon are cut to length \(L \mathrm {~cm}\) where \(L \sim \mathrm {~N} \left( \mu , 0.5 ^ { 2 } \right)\)

1. Given that \(30 \%\) of the pieces of ribbon have length more than 100 cm , find the value of \(\mu\) to the nearest 0.1 cm . John selects 12 pieces of ribbon at random.
2. Find the probability that fewer than 3 of these pieces of ribbon have length more than 100 cm . Aditi selects 400 pieces of ribbon at random.
3. Using a suitable approximation, find the probability that more than 127 of these pieces of ribbon will have length more than 100 cm .

1. A shop sells rods of nominal length 200 cm . The rods are bought from a manufacturer who uses a machine to cut rods of length \(L \mathrm {~cm}\), where \(L \sim \mathrm {~N} \left( \mu , 0.2 ^ { 2 } \right)\)

The value of \(\mu\) is such that there is only a \(5 \%\) chance that a rod, selected at random from those supplied to the shop, will have length less than 200 cm .

1. Find the value of \(\mu\) to one decimal place. A customer buys a random sample of 8 of these rods.
2. Find the probability that at least 3 of these rods will have length less than 200 cm . Another customer buys a random sample of 60 of these rods.
3. Using a suitable approximation, find the probability that more than 5 of these rods will have length less than 200 cm .
   

5. A garden centre sells canes of nominal length 150 cm . The canes are bought from a supplier who uses a machine to cut canes of length \(L\) where \(L \sim \mathrm {~N} \left( \mu , 0.3 ^ { 2 } \right)\).

1. Find the value of \(\mu\), to the nearest 0.1 cm , such that there is only a \(5 \%\) chance that a cane supplied to the garden centre will have length less than 150 cm . A customer buys 10 of these canes from the garden centre.
2. Find the probability that at most 2 of the canes have length less than 150 cm . Another customer buys 500 canes.
3. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm .
   (6)

5. The time taken for a randomly selected person to complete a test is \(M\) minutes, where \(M \sim \mathrm {~N} \left( 14 , \sigma ^ { 2 } \right)\) Given that \(10 \%\) of people take less than 12 minutes to complete the test,

1. find the value of \(\sigma\) Graham selects 15 people at random.
2. Find the probability that fewer than 2 of these people will take less than 12 minutes to complete the test. Jovanna takes a random sample of \(n\) people. Using a normal approximation, the probability that fewer than 9 of these \(n\) people will take less than 12 minutes to complete the test is 0.3085 to 4 decimal places.
3. Find the value of \(n\).

7. The times taken by a large number of people to read a certain book can be modelled by a normal distribution with mean \(5 \cdot 2\) hours. It is found that \(62 \cdot 5 \%\) of the people took more than \(4 \cdot 5\) hours to read the book.

1. Show that the standard deviation of the times is approximately \(2 \cdot 2\) hours.
2. Calculate the percentage of the people who took between 4 and 7 hours to read the book.
3. Calculate the probability that two of the people chosen at random both took less than 5 hours to read the book, stating any assumption that you make.
4. If a number of extra people were taken into account, all of whom took exactly \(5 \cdot 2\) hours to read the book, state with reasons what would happen to (i) the mean, (ii) the variance and explain briefly why the distribution would no longer be normal.

4. The weights of sacks of potatoes are normally distributed. It is known that one in five sacks weigh more than 6 kg and three in five sacks weigh more than 5.5 kg .
i. Find the mean and standard deviation of the weights of potato sacks.
ii. The sacks are put into crates, with twelve sacks going into each crate. What is the probability that a given crate contains two or more sacks that weigh more than 6 kg ? You must explain your reasoning clearly in this question.
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