Poisson & Geometric Hypothesis Testing (Edexcel A Level Further Maths: Further Statistics 1): Exam Questions

Every morning Geethaka repeatedly rolls a fair, six-sided die until he rolls a 3 and then he stops. The random variable $X$ represents the number of times he rolls the die each morning.

Suggest a suitable model for the random variable $X$.

Nira wants to check Geethaka's die to decide whether or not the probability of rolling a 3 with his die is less than $\frac{1}{6}$

Nira rolls the die repeatedly until she rolls a 3.

She obtains $x=16$.

By carrying out a suitable test, determine what Nira's conclusion should be. You should state your hypotheses clearly and use a 5% level of significance.

Telephone calls arrive at a call centre randomly, at an average rate of 1.7 per minute. After the call centre was closed for a week, in a random sample of 10 minutes there were 25 calls to the call centre.

Carry out a suitable test to determine whether or not there is evidence that the rate of calls arriving at the call centre has changed.

Use a 5% level of significance and state your hypotheses clearly.

Only 1.2% of the calls to the call centre last longer than 8 minutes.

One day Tiang has 70 calls.

Find the probability that out of these 70 calls Tiang has more than 2 calls lasting longer than 8 minutes.

The call centre records show that 95% of days have at least one call lasting longer than 30 minutes.

On Wednesday 900 calls arrived at the call centre and none of them lasted longer than 30 minutes.

Use a Poisson approximation to estimate the proportion of calls arriving at the call centre that last longer than 30 minutes.

During the summer, mountain rescue team A receives calls for help randomly with a rate of 0.4 per day.

Find the probability that during the summer, mountain rescue team A receives at least 19 calls for help in 28 randomly selected days.

The leader of mountain rescue team A randomly selects 250 summer days from the last few years. She records the number of calls for help received on each of these days.

Using a Poisson approximation, estimate the probability of the leader finding at least 20 of these days when more than 1 call for help was received by mountain rescue team A.

Mountain rescue team A believes that the number of calls for help per day is lower in the winter than in the summer. The number of calls for help received in 42 randomly selected winter days is 8

Use a suitable test, at the 5% level of significance, to assess whether or not there is evidence that the number of calls for help per day is lower in the winter than in the summer. State your hypotheses clearly.

During the summer, mountain rescue team B receives calls for help randomly with a rate of 0.2 per day, independently of calls to mountain rescue team A.

The random variable $C$ is the total number of calls for help received by mountain rescue teams A and B during a period of $n$ days in the summer.

On a Monday in the summer, mountain rescue teams A and B each receive a call for help.

Given that over the next $n$ days $\text{P}(C=0)<0.001$

calculate the minimum value of $n$

Write down an assumption that needs to be made for the model to be appropriate.

Asha, Davinda and Jerry each have a bag containing a large number of counters, some of which are white and the rest are red.

Each person draws counters from their bag one at a time, notes the colour of the counter and returns it to their bag.

The probability of Asha getting a red counter on any one draw is 0.07

Find the probability that Asha will draw at least 3 white counters before a red counter is drawn.

Find the probability that Asha gets a red counter for the second time on her 9th draw.

The probability of Davinda getting a red counter on any one draw is $p$.

Davinda draws counters until she gets $n$ red counters. The random variable $D$ is the number of counters Davinda draws.

Given that the mean and the standard deviation of $D$ are 4400 and 660 respectively,

find the value of $p$.

Jerry believes that his bag contains a smaller proportion of red counters than Asha’s bag.

To test his belief, Jerry draws counters from his bag until he gets a red counter. Jerry defines the random variable $J$ to be the number of counters drawn up to and including the first red counter.

Stating your hypotheses clearly and using a 10% level of significance, find the critical region for this test.

Jerry gets a red counter for the first time on his 34th draw.

Giving a reason for your answer, state whether or not there is evidence that Jerry’s bag contains a smaller proportion of red counters than Asha’s bag.

Given that the probability of Jerry getting a red counter on any one draw is 0.011

show that the power of the test is 0.702 to 3 significant figures.

On a weekday, a garage receives telephone calls randomly, at a mean rate of 1.25 per 10 minutes.

Show that the probability that on a weekday at least 2 calls are received by the garage in a 30-minute period is 0.888 to 3 decimal places.

Calculate the probability that at least 2 calls are received by the garage in fewer than 4 out of 6 randomly selected, non-overlapping 30-minute periods on a weekday.

The manager of the garage randomly selects 150 non-overlapping 30-minute periods on weekdays.

She records the number of calls received in each of these 30-minute periods.

Using a Poisson approximation show that the probability of the manager finding at least 3 of these 30-minute periods when exactly 8 calls are received by the garage is 0.664 to 3 significant figures.

Explain why the Poisson approximation may be reasonable in this case.

The manager of the garage decides to test whether the number of calls received on a Saturday is different from the number of calls received on a weekday. She selects a Saturday at random and records the number of telephone calls received by the garage in the first 4 hours.

Write down the hypotheses for this test.

The manager found that there had been 40 telephone calls received by the garage in the first 4 hours.

Carry out the test using a 5% level of significance.

The number of customers entering Jeff's supermarket each morning follows a Poisson distribution.

Past information shows that customers enter at an average rate of 2 every 5 minutes.

Using this information,

(i) Find the probability that exactly 26 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning.

(ii) Find the probability that at least 21 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning.

A rival supermarket is opened nearby. Following its opening, the number of customers entering Jeff's supermarket over a randomly selected 40-minute period is found to be 10

Test, at the 5% significance level, whether or not there is evidence of a decrease in the rate of customers entering Jeff's supermarket. State your hypotheses clearly.

A further randomly selected 20-minute period is observed and the hypothesis test is repeated.

Given that the true rate of customers entering Jeff's supermarket is now 1 every 5 minutes,

calculate the probability of a Type II error.

Information was collected about accidents on the Seapron bypass. It was found that the number of accidents per month could be modelled by a Poisson distribution with mean 2.5

Following some work on the bypass, the numbers of accidents during a series of 3-month periods were recorded. The data were used to test whether or not there was a change in the mean number of accidents per month.

Stating your hypotheses clearly and using a 5% level of significance, find the critical region for this test. You should state the probability in each tail.

State $\text{P}\left(\text{Type I error}\right)$ using this test.

Data from the series of 3-month periods are recorded for 2 years.

Find the probability that at least 2 of these 3-month periods give a significant result.

Given that the number of accidents per month on the bypass, after the work is completed, is actually 2.1 per month,

find $\text{P}\left(\text{Type II error}\right)$ for the test in part (a)

Bacteria are randomly distributed in a river at a rate of 5 per litre of water. A new factory opens and a scientist claims it is polluting the river with bacteria. He takes a sample of 0.5 litres of water from the river near the factory and finds that it contains 7 bacteria.

Stating your hypotheses clearly test, at the 5% level of significance, whether there is evidence that the level of pollution has increased.

Sam and Tessa are testing a spinner to see if the probability, $p$, of it landing on red is less than $\frac{1}{5}$

They both use a 10% significance level.

Sam decides to spin the spinner 20 times and record the number of times it lands on red.

Find the critical region for Sam's test.

Write down the size of Sam's test.

Tessa decides to spin the spinner until it lands on red and she records the number of spins.

Find the critical region for Tessa's test.

Find the size of Tessa's test.

(i) Show that the power function for Sam's test is given by

$(1-p{)}^{19}(1+19p)$

(ii) Find the power function for Tessa's test.

With reference to parts (b), (d) and (e), state, giving your reasons, whether you would recommend Sam's test or Tessa's test when $p=0.15$

The number of calls received by a helpline is modelled by a Poisson distribution with a mean of 2 calls per 5-minute period.

After an advertising campaign, the helpline manager wants to know whether the mean number of calls has increased.

The number of calls received in a randomly chosen 20-minute period is recorded.

Stating your hypotheses clearly, and using a 5% level of significance, find the critical region for a suitable test.

At a fairground, each spin of a wheel shows WIN with probability $\frac{1}{5}$, independently of all other spins. A player spins the wheel repeatedly until it shows WIN. The random variable $X$ represents the number of spins needed.

State the distribution of $X$.

A player believes that the probability of the wheel showing WIN is actually less than $\frac{1}{5}$. She spins the wheel repeatedly until it shows WIN, and it takes her 16 spins.

Stating your hypotheses clearly and using a 5% level of significance, carry out a suitable test and state the player's conclusion.