Single sum threshold probability

1 The masses, in grams, of plums of a certain type have the distribution \(\mathrm { N } \left( 40.4,5.2 ^ { 2 } \right)\). The plums are packed in bags, with each bag containing 6 randomly chosen plums. If the total weight of the plums in a bag is less than 220 g the bag is rejected. Find the percentage of bags that are rejected.

3 The mass, in tonnes, of iron ore produced per day at a mine is normally distributed with mean 7.0 and standard deviation 0.46. Find the probability that the total amount of iron ore produced in 10 randomly chosen days is more than 71 tonnes.

1 The masses, in grams, of apples of a certain type are normally distributed with mean 60.4 and standard deviation 8.2. The apples are packed in bags, with each bag containing 8 randomly chosen apples. The bags are checked by Quality Control and any bag containing apples with a total mass of less than 436 g is rejected. Find the proportion of bags that are rejected.

5 The masses, in grams, of large boxes of chocolates and small boxes of chocolates have the distributions \(\mathrm { N } ( 325,6.1 )\) and \(\mathrm { N } ( 167,5.6 )\) respectively.

1. Find the probability that the total mass of 10 randomly chosen large boxes of chocolates is less than 3240 g .
2. Find the probability that the mass of a randomly chosen large box of chocolates is more than twice the mass of a randomly chosen small box of chocolates.

6 The lifetimes, in hours, of Longlive light bulbs and Enerlow light bulbs have the independent distributions \(\mathrm { N } \left( 1020,45 ^ { 2 } \right)\) and \(\mathrm { N } \left( 2800,52 ^ { 2 } \right)\) respectively.

1. Find the probability that the total of the lifetimes of 5 randomly chosen Longlive bulbs is less than 5200 hours.
2. Find the probability that the lifetime of a randomly chosen Enerlow bulb is at least 3 times that of a randomly chosen Longlive bulb.

7 The employees of a certain company have masses which are normally distributed. Female employees have a mean of 66.7 kg and standard deviation 9.3 kg , and male employees have a mean of 78.3 kg and standard deviation 8.5 kg . It may be assumed that all employees' masses are independent. On the ground floor 6 women and 9 men enter the empty staff lift for which it is stated that the maximum load is 1150 kg .

1. Calculate the probability that the maximum load is exceeded. At the first floor all 15 passengers leave and 6 women, 8 men and an unknown employee enter.
2. Assuming that the unknown employee is equally likely to be a woman or a man, calculate the probability that the maximum load is now exceeded.

4 The weights of Avonley Blue cheeses made by a small producer are found to be Normally distributed with mean 10 kg and standard deviation 0.4 kg .

1. Find the probability that a randomly chosen cheese weighs less than 9.5 kg . One particular shop orders four Avonley Blue cheeses each week from the producer. From experience, the shopkeeper knows that the weekly demand from customers for Avonley Blue cheese is Normally distributed with mean 35 kg and standard deviation 3.5 kg . In the interests of food hygiene, no cheese is kept by the shopkeeper from one week to the next.
2. Find the probability that, in a randomly chosen week, demand from customers for Avonley Blue will exceed the supply. Following a campaign to promote Avonley Blue cheese, the shopkeeper finds that the weekly demand for it has increased by \(30 \%\) (i.e. the mean and standard deviation are both increased by \(30 \%\) ). Therefore the shopkeeper increases his weekly order by one cheese.
3. Find the probability that, in a randomly chosen week, demand will now exceed supply.
4. Following complaints, the cheese producer decides to check the mean weight of the Avonley Blue cheeses. For a random sample of 12 cheeses, she finds that the mean weight is 9.73 kg . Assuming that the population standard deviation of the weights is still 0.4 kg , find a \(95 \%\) confidence interval for the true mean weight of the cheeses and comment on the result. Explain what is meant by a 95\% confidence interval. RECOGNISING ACHIEVEMENT

7 At a particular supermarket, the times taken to serve each customer in a queue at a standard checkout may be modelled by a normal distribution with mean 240 seconds and standard deviation 20 seconds. There is a queue of 3 customers at a standard checkout.
Making a reasonable assumption about the times taken to serve these customers,

1. find the probability that the total time taken to serve the 3 customers will be less than 11 minutes.
2. State the assumption you have made in part (a) In the supermarket there is also an express checkout, which is reserved for customers buying 10 or fewer items. The time taken to serve a customer at this express checkout may be modelled by a normal distribution with mean 100 seconds and standard deviation 8 seconds. On a particular day Jiang has 8 items to pay for and has to choose whether to join a queue of 3 customers waiting at a standard checkout or a queue of 7 customers waiting at the express checkout. Using a similar assumption to that made in part (a),
3. find the probability that the total time taken to serve the 3 customers at the standard checkout will exceed the total time taken to serve the 7 customers at the express checkout.

1. A company packs chickpeas into small bags and large bags.

The weight of a small bag of chickpeas is normally distributed with mean 500 g and standard deviation 5 g A random sample of 3 small bags of chickpeas is taken.

1. Find the probability that the total weight of these 3 bags of chickpeas is between 1490 g and 1530 g The weight of a large bag of chickpeas is normally distributed with mean 1020 g and standard deviation 20 g One large bag and one small bag of chickpeas are chosen at random.
2. Calculate the probability that the weight of the large bag of chickpeas is at least 30 g more than twice the weight of the small bag of chickpeas. Show your working clearly.

2 Geoffrey is a university lecturer. He has to prepare five questions for an examination. He knows by experience that it takes about 3 hours to prepare a question, and he models the time (in minutes) taken to prepare one by the Normally distributed random variable \(X\) with mean 180 and standard deviation 12, independently for all questions.

1. One morning, Geoffrey has a gap of 2 hours 50 minutes ( 170 minutes) between other activities. Find the probability that he can prepare a question in this time.
2. One weekend, Geoffrey can devote 14 hours to preparing the complete examination paper. Find the probability that he can prepare all five questions in this time. A colleague, Helen, has to check the questions.
3. She models the time (in minutes) to check a question by the Normally distributed random variable \(Y\) with mean 50 and standard deviation 6, independently for all questions and independently of \(X\). Find the probability that the total time for Geoffrey to prepare a question and Helen to check it exceeds 4 hours.
4. When working under pressure of deadlines, Helen models the time to check a question in a different way. She uses the Normally distributed random variable \(\frac { 1 } { 4 } X\), where \(X\) is as above. Find the length of time, as given by this model, which Helen needs to ensure that, with probability 0.9 , she has time to check a question. Ian, an educational researcher, suggests that a better model for the time taken to prepare a question would be a constant \(k\) representing "thinking time" plus a random variable \(T\) representing the time required to write the question itself, independently for all questions.
5. Taking \(k\) as 45 and \(T\) as Normally distributed with mean 120 and standard deviation 10 (all units are minutes), find the probability according to Ian's model that a question can be prepared in less than 2 hours 30 minutes. Juliet, an administrator, proposes that the examination should be reduced in time and shorter questions should be used.
6. Juliet suggests that Ian's model should be used for the time taken to prepare such shorter questions but with \(k = 30\) and \(T\) replaced by \(\frac { 3 } { 5 } T\). Find the probability as given by this model that a question can be prepared in less than \(1 \frac { 3 } { 4 }\) hours.

5 A particular brand of pasta is sold in bags of two different sizes. The mass of pasta in the large bags is advertised as being 1500 g ; in fact it is Normally distributed with mean 1515 g and standard deviation 4.7 g . The mass of pasta in the small bags is advertised as being 500 g ; in fact it is Normally distributed with mean 508 g and standard deviation 3.3 g .

1. Find the probability that the total mass of pasta in 5 randomly selected small bags is less than 2550 g .
2. Find the probability that the mass of pasta in a randomly selected large bag is greater than three times the mass of pasta in a randomly selected small bag.