Hypothesis test for a Poisson mean

Customers arrive at a bank at a rate of 10 customers per 5 minutes. John suggests that a Poisson distribution would be a suitable model to describe the number of customers arriving at the bank in a 1-minute interval.

Write down two assumptions that John has made.

Given that the Poisson distribution is a suitable model,

find the probability that in any given 1-minute interval

exactly 2 customers arrive,

more than 4 customers arrive.

When there is sufficient evidence that the rate of customers arriving at the bank is higher than 10 customers per 5 minutes, the manager must open another service till to avoid long queues. In a 4-minute interval, the manager observes 9 customers arriving.

Test, at the 5% significance level, whether or not the manager has sufficient evidence to open another service till. State your hypotheses clearly.

Using a 1% significance level, determine the smallest number of customers the manager must observe in a 1-minute interval in order to have sufficient evidence to open another service till.

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.

Liftsforall claims that the lift they maintain in a block of flats breaks down at random at a mean rate of 4 times per month. To test this, the number of times the lift breaks down in a month is recorded.

Using a 5% level of significance, find the critical region for a two-tailed test of the null hypothesis that ‘the mean rate at which the lift breaks down is 4 times per month’. The probability of rejection in each of the tails should be as close to 2.5% as possible.

Over a randomly selected 1 month period the lift broke down 3 times.

Test, at the 5% level of significance, whether Liftsforall ’s claim is correct. State your hypotheses clearly.

State the actual significance level of this test.

The residents in the block of flats have a maintenance contract with Liftsforall. The residents pay Liftsforall £500 for every quarter (3 months) in which there are at most 3 breakdowns. If there are 4 or more breakdowns in a quarter then the residents do not pay for that quarter. Liftsforall installs a new lift in the block of flats. Given that the new lift breaks down at a mean rate of 2 times per month,

find the probability that the residents do not pay more than £500 to Liftsforall in the next year.

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.

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.

A traffic officer monitors the rate at which vehicles pass a fixed point on a motorway. When the rate exceeds 36 vehicles per minute he must switch on some speed restrictions to improve traffic flow.

Suggest a suitable model to describe the number of vehicles passing the fixed point in a 15 s interval.

The traffic officer records 12 vehicles passing the fixed point in a 15 s interval.

Stating your hypotheses clearly, and using a 5% level of significance, test whether or not the traffic officer has sufficient evidence to switch on the speed restrictions.

Using a 5% level of significance, determine the smallest number of vehicles the traffic officer must observe in a 10 s interval in order to have sufficient evidence to switch on the speed restrictions.

Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard’s claim at the 5% level of significance. State your hypotheses clearly.

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.

An effect of a certain disease is that a small number of the red blood cells are deformed. Emily has this disease and the deformed blood cells occur randomly at a rate of 2.5 per ml of her blood. Following a course of treatment, a random sample of 2 ml of Emily’s blood is found to contain only 1 deformed red blood cell. Stating your hypotheses clearly and using a 5% level of significance, test whether or not there has been a decrease in the number of deformed red blood cells in Emily’s blood.

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.