Approximating one distribution by another

A company sells a variety of bars of chocolate including Pearchoc bars. In an attempt to increase the sales of Pearchoc bars, the company runs a promotion where any Pearchoc wrapper that has the word WINNER printed on the inside wins a prize. The company prints the word WINNER on 5% of Pearchoc bar wrappers. Pearchoc bars are sold in packets of 12

Find the probability that in a randomly selected packet of Pearchoc bars, the number of wrappers with the word WINNER printed on them will be

exactly one,

more than two.

A customer bought 4 packets of Pearchoc bars.

Find the probability that only 1 of the packets contained exactly 1 wrapper with the word WINNER printed on it.

Before the promotion, 4% of all bars of chocolate sold in a supermarket were Pearchoc bars. After the promotion, the company selects a random sample of 150 sales of bars of chocolate from the supermarket database and finds that 10 of these sales were Pearchoc bars.

Using a suitable approximation and stating your hypotheses clearly, test whether or not the company’s promotion has been successful. Use a 5% level of significance.

State the condition under which the normal distribution may be used as an approximation to the Poisson distribution.

Explain why a continuity correction must be incorporated when using the normal distribution as an approximation to the Poisson distribution.

Each weekend during the ski season, requests to hire pairs of skis from Skisea occur randomly. The mean number of requests to hire pairs of skis for a weekend is 205. During the ski season Skisea only has 220 pairs of skis that can be hired for a weekend.

Using a suitable approximation, find the probability that the requests to hire pairs of skis cannot all be met on a particular weekend during the ski season.

During the ski season there are 20 weekends on which a pair of skis can be hired.

Estimate the number of weekends that the company will not be able to meet all the requests to hire pairs of skis.

In a call centre, the number of telephone calls, X, received during any 10-minute period follows a Poisson distribution with mean 9

Find

P(X 5)

P(4 X 10) The length of a working day is 7 hours.

Using a suitable approximation, find the probability that there are fewer than 370 telephone calls in a randomly selected working day. A week, consisting of 5 working days, is selected at random.

Find the probability that in this week at least 4 working days have fewer than 370 telephone calls.

Past records show that the proportion of customers buying organic vegetables from Tesson supermarket is 0.35 During a particular day, a random sample of 40 customers from Tesson supermarket was taken and 18 of them bought organic vegetables.

Test, at the 5% level of significance, whether or not this provides evidence that the proportion of customers who bought organic vegetables has increased. State your hypotheses clearly. The manager of Tesson supermarket claims that the proportion of customers buying organic eggs is different from the proportion of those buying organic vegetables. To test this claim the manager decides to take a random sample of 50 customers.

Using a 5% level of significance, find the critical region to enable the Tesson supermarket manager to test her claim. The probability for each tail of the region should be as close as possible to 2.5% During a particular day, a random sample of 50 customers from Tesson supermarket is taken and 8 of them bought organic eggs.

Using your answer to part (b), state whether or not this sample supports the manager’s claim. Use a 5% level of significance.

State the actual significance level of this test. The proportion of customers who buy organic fruit from Tesson supermarket is 0.2 During a particular day, a random sample of 200 customers from Tesson supermarket is taken. Using a suitable approximation, the probability that fewer than n of these customers bought organic fruit is 0.0465 correct to 4 decimal places.

Find the value of n.

A company receives telephone calls at random at a mean rate of 2.5 per hour.

Find the probability that the company receives

at least 4 telephone calls in the next hour,

exactly 3 telephone calls in the next 15 minutes.

Find, to the nearest minute, the maximum length of time the telephone can be left unattended so that the probability of missing a telephone call is less than 0.2

The company puts an advert in the local newspaper. The number of telephone calls received in a randomly selected 2 hour period after the paper is published is 10

Test at the 5% level of significance whether or not the mean rate of telephone calls has increased. State your hypotheses clearly.

The time taken for a randomly selected person to complete a test is M minutes, where M ~ N (14, ²) Given that 10% of people take less than 12 minutes to complete the test,

find the value of

Graham selects 15 people at random.

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.

Find the value of n.

In a large school, 20% of students own a touch screen laptop. A random sample of n students is chosen from the school. Using a normal approximation, the probability that more than 55 of these n students own a touch screen laptop is 0.0401 correct to 3 significant figures. Find the value of n.

In a survey it is found that barn owls occur randomly at a rate of 9 per 1000 km².

Find the probability that in a randomly selected area of 1000 km² there are at least 10 barn owls.

Find the probability that in a randomly selected area of 200 km² there are exactly 2 barn owls.

Using a suitable approximation, find the probability that in a randomly selected area of 50000 km² there are at least 470 barn owls.

The continuous random variable L represents the error, in metres, made when a machine cuts poles to a target length. The distribution of L is a continuous uniform distribution over the interval [0, 0.5]

Find P(L < 0.4).

Write down E(L).

Calculate Var(L).

A random sample of 30 poles cut by this machine is taken.

Find the probability that fewer than 4 poles have an error of more than 0.4 metres from the target length.

When a new machine cuts poles to a target length, the error, X metres, is modelled by the cumulative distribution function F(x) where

Using this model, find P(X 0.4)

A random sample of 100 poles cut by this new machine is taken.

Using a suitable approximation, find the probability that at least 8 of these poles have an error of more than 0.4 metres.

Sammy manufactures wallpaper. She knows that defects occur randomly in the manufacturing process at a rate of 1 every 8 metres. Once a week the machinery is cleaned and reset. Sammy then takes a random sample of 40 metres of wallpaper from the next batch produced to test if there has been any change in the rate of defects.

Stating your hypotheses clearly and using a 10% level of significance, find the critical region for this test. You should choose your critical region so that the probability of rejection is less than 0.05 in each tail.

State the actual significance level of this test.

Thomas claims that his new machine would reduce the rate of defects and invites Sammy to test it. Sammy takes a random sample of 200 metres of wallpaper produced on Thomas’ machine and finds 19 defects.

Using a suitable approximation, test Thomas’ claim. You should use a 5% level of significance and state your hypotheses clearly.

The number of defects per metre in a roll of cloth has a Poisson distribution with mean 0.25 Find the probability that

a randomly chosen metre of cloth has 1 defect,

the total number of defects in a randomly chosen 6 metre length of cloth is more than 2

A tailor buys 300 metres of cloth.

Using a suitable approximation find the probability that the tailor’s cloth will contain less than 90 defects.

An online shop sells a computer game at an average rate of 1 per day.

Find the probability that the shop sells more than 10 games in a 7 day period.

Once every 7 days the shop has games delivered before it opens.

Find the least number of games the shop should have in stock immediately after a delivery so that the probability of running out of the game before the next delivery is less than 0.05

In an attempt to increase sales of the computer game, the price is reduced for six months. A random sample of 28 days is taken from these six months. In the sample of 28 days, 36 computer games are sold.

Using a suitable approximation and a 5% level of significance, test whether or not the average rate of sales per day has increased during these six months. State your hypotheses clearly.

Frugal bakery claims that their packs of 10 muffins contain on average 80 raisins per pack. A Poisson distribution is used to describe the number of raisins per muffin. A muffin is selected at random to test whether or not the mean number of raisins per muffin has changed.

Find the critical region for a two-tailed test using a 10% level of significance. The probability of rejection in each tail should be less than 0.05

Find the actual significance level of this test.

The bakery has a special promotion claiming that their muffins now contain even more raisins. A random sample of 10 muffins is selected and is found to contain a total of 95 raisins.

Use a suitable approximation to test the bakery’s claim. You should state your hypotheses clearly and use a 5% level of significance.

As part of a selection procedure for a company, applicants have to answer all 20 questions of a multiple choice test. If an applicant chooses answers at random the probability of choosing a correct answer is 0.2 and the number of correct answers is represented by the random variable X.

Suggest a suitable distribution for X.

Each applicant gains 4 points for each correct answer but loses 1 point for each incorrect answer. The random variable S represents the final score, in points, for an applicant who chooses answers to this test at random.

Show that S = 5 X – 20

Find E(S) and Var(S).

An applicant who achieves a score of at least 20 points is invited to take part in the final stage of the selection process.

Find P(S 20)

Cameron is taking the final stage of the selection process which is a multiple choice test consisting of 100 questions. He has been preparing for this test and believes that his chance of answering each question correctly is 0.4

Using a suitable approximation, estimate the probability that Cameron answers more than half of the questions correctly.

A telesales operator is selling a magazine. Each day he chooses a number of people to telephone. The probability that each person he telephones buys the magazine is 0.1

Suggest a suitable distribution to model the number of people who buy the magazine from the telesales operator each day.

On Monday, the telesales operator telephones 10 people. Find the probability that he sells at least 4 magazines.

Calculate the least number of people he needs to telephone on Tuesday, so that the probability of selling at least 1 magazine, on that day, is greater than 0.95

A call centre also sells the magazine. The probability that a telephone call made by the call centre sells a magazine is 0.05 The call centre telephones 100 people every hour.

Using a suitable approximation, find the probability that more than 10 people telephoned by the call centre buy a magazine in a randomly chosen hour.

Write down the conditions under which the Poisson distribution can be used as an approximation to the binomial distribution.

The probability of any one letter being delivered to the wrong house is 0.01 On a randomly selected day Peter delivers 1000 letters.

Using a Poisson approximation, find the probability that Peter delivers at least 4 letters to the wrong house. Give your answer to 4 decimal places.

In a village, power cuts occur randomly at a rate of 3 per year.

Find the probability that in any given year there will be

exactly 7 power cuts,

at least 4 power cuts.

Use a suitable approximation to find the probability that in the next 10 years the number of power cuts will be less than 20

Defects occur at random in planks of wood with a constant rate of 0.5 per 10 cm length. Jim buys a plank of length 100 cm.

Find the probability that Jim’s plank contains at most 3 defects.

Shivani buys 6 planks each of length 100 cm.

Find the probability that fewer than 2 of Shivani’s planks contain at most 3 defects.

Using a suitable approximation, estimate the probability that the total number of defects on Shivani’s 6 planks is less than 18.

A shopkeeper knows, from past records, that 15% of customers buy an item from the display next to the till. After a refurbishment of the shop, he takes a random sample of 30 customers and finds that only 1 customer has bought an item from the display next to the till.

Stating your hypotheses clearly, and using a 5% level of significance, test whether or not there has been a change in the proportion of customers buying an item from the display next to the till.

During the refurbishment a new sandwich display was installed. Before the refurbishment 20% of customers bought sandwiches. The shopkeeper claims that the proportion of customers buying sandwiches has now increased. He selects a random sample of 120 customers and finds that 31 of them have bought sandwiches.

Using a suitable approximation and stating your hypotheses clearly, test the shopkeeper’s claim. Use a 10% level of significance.

A disease occurs in 3% of a population.

State any assumptions that are required to model the number of people with the disease in a random sample of size n as a binomial distribution.

Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people.

Find the mean and variance of the number of people with the disease in a random sample of 100 people.

A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.

Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination.

The continuous random variable X is uniformly distributed over the interval [–1,3]. Find

E(X)

Var(X)

P(X 1.4)

A total of 40 observations of X are made.

Find the probability that at least 10 of these observations are negative.

Cars arrive at a motorway toll booth at an average rate of 150 per hour.

Suggest a suitable distribution to model the number of cars arriving at the toll booth, X, per minute.

State clearly any assumptions you have made by suggesting this model.

Using your model,

find the probability that in any given minute

no cars arrive,

more than 3 cars arrive.

In any given 4 minute period, find m such that P(X m) = 0.0487

Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period.

Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 Find the probability that, in 9 games, Bhim loses

exactly 3 of the games,

fewer than half of the games.

Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games.

Calculate the mean and variance for the number of these 60 games that Bhim loses.

Using a suitable approximation calculate the probability that Bhim loses more than 4 games.

A company has a large number of regular users logging onto its website. On average 4 users every hour fail to connect to the company’s website at their first attempt.

Explain why the Poisson distribution may be a suitable model in this case.

Find the probability that, in a randomly chosen 2 hour period,

all users connect at their first attempt,

at least 4 users fail to connect at their first attempt.

The company suffered from a virus infecting its computer system. During this infection it was found that the number of users failing to connect at their first attempt, over a 12 hour period, was 60.

Using a suitable approximation, test whether or not the mean number of users per hour who failed to connect at their first attempt had increased. Use a 5% level of significance and state your hypotheses clearly.

A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes. Find the probability that

fewer than 9 customers arrive for breakfast on a Monday morning between 10 am and 11 am.

The café serves breakfast every day between 8 am and 12 noon.

Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.

A bag contains a large number of counters of which 15% are coloured red. A random sample of 30 counters is selected and the number of red counters is recorded.

Find the probability of no more than 6 red counters in this sample.

A second random sample of 30 counters is selected and the number of red counters is recorded.

Using a Poisson approximation, estimate the probability that the total number of red counters in the combined sample of size 60 is less than 13.

An administrator makes errors in her typing randomly at a rate of 3 errors every 1000 words.

In a document of 2000 words find the probability that the administrator makes 4 or more errors.

The administrator is given an 8000 word report to type and she is told that the report will only be accepted if there are 20 or fewer errors.

Use a suitable approximation to calculate the probability that the report is accepted.

A factory produces components of which 1% are defective. The components are packed in boxes of 10. A box is selected at random.

Find the probability that the box contains exactly one defective component.

Find the probability that there are at least 2 defective components in the box.

Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components.

A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.

Test, at the 10% level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.

State the minimum number of visits required to obtain a significant result.

State an assumption that has been made about the visits to the server.

In a random two minute period on a Saturday the web server is visited 20 times.

Using a suitable approximation, test at the 10% level of significance, whether or not the rate of visits is greater on a Saturday.