Hypothesis Testing (Discrete Distributions) (Edexcel International A Level Maths: Statistics 2)

For the random variable  calculate

A hypothesis test is to be carried out using an observation of the random variable  to test the hypotheses:

Before carrying out the test, a significance level of 5% is chosen. The critical region is defined as 

Explain why the critical region is defined as

A mathematics teacher makes silly mistakes at the average rate of 20 per day and she teaches 5 lessons, each of equal length, per day.

Alina is running for student council president at her school.  She claims she has the support of 60% of students in the school.  A rival candidate, John, wants to test at the 10% level of significance whether Alina is overestimating her support.

In mathematical terms, the experiment John conducts uses an observation of the random variable  to test the hypotheses:

State how many people are in John’s sample and explain what the parameter  means in the context of this question. Explain why

Assuming Alina’s claim is true, John calculates the following cumulative probabilities:

In John’s survey, 55 people say that they will support Alina.

Calculate .

State whether John should compare the value of or  with his significance level of 10%.

Write down the critical value and the critical region for John’s test.

A hypothesis test at the 6% significance level is carried out on a coin using the following hypotheses:

Give an example of what the parameter, , could represent.

In the context of your answer from part (i), explain what is meant by .

A single observation  is to be taken from a binomial distribution  to test the hypotheses for the coin. One tail of the critical region is found to be .

Using your knowledge of the binomial distribution , write down the other tail of the critical region.

Write down the set of values for which would lead to the acceptance of the null hypothesis.

In a school where all the teachers drink coffee in the staffroom, a headteacher wants to see if using a new brand of coffee in the staffroom improves teachers’ report writing punctuality.  Previously, 75% of teachers in her school would meet the deadline for writing reports.  After the new brand of coffee is introduced the headteacher takes a random sample of 15 teachers, and conducts an experiment to test whether the proportion of teachers meeting the deadline on the next set of reports has improved.  The test is conducted at the 5% significance level, using the following hypotheses:

Write down a suitable distribution for the random variable , the number of teachers in the headteacher’s sample who meet the deadline.

A study of ladybirds in the UK found that 65% of all ladybirds are found to be of the seven-spot species.  Alex believes that more than 65% of the ladybirds in his garden are of the seven-spot species.  He conducts a hypothesis test at the 5% significance level by collecting a sample of 25 ladybirds from his garden and counting the number of them that are seven-spot ladybirds.

Alex uses the random variable to represent the number of seven-spot ladybirds in his sample.  Explain what  and  represent in the context of Alex’ experiment.

Alex finds that 21 out of the 25 ladybirds in his sample are seven-spot ladybirds.

The probability of Alex’s observed value or that of a more extreme value is 0.03205. Write this in the form .

The probability of a student in a primary school library returning his or her books on time had been found to be 0.35. Joanna, the school librarian, has started a new incentive scheme and believes that more students are now returning their books on time because of it.  She conducts a hypothesis test using the null hypothesis to test her belief.

Joanna takes a random sample of 30 students who have checked out books and finds that under the new incentive scheme 15 of them return their books on time.  She calculates the following probabilities for the random variable :

Write down the values of  and 

Write a conclusion for the hypothesis test, in context, if Joanna had chosen a significance level of:

The number of leaves on the grass in Danny’s garden on October 1^st every year has a Poisson distribution with mean 280. 

Given that Danny’s garden is 56 square metres,

Danny’s neighbour has cut down a tree and Danny believes this will decrease the number of leaves in his garden this year.  He is devising a hypothesis test to test his theory at the 10% significance level.  On October 1^st he will choose a square metre of his garden at random and count the number of leaves.

Write down suitable null and alternative hypotheses Danny can use for his test.

Use your answer to part (a)(ii) to determine if finding 2 leaves will lead Danny to accept or reject his null hypothesis. Justify your answer.

Determine the critical region for this hypothesis test and hence, write down the maximum number of leaves that will lead Danny to reject his null hypothesis.