Hypothesis Testing (Binomial & Poisson Distributions) (Cambridge (CIE) A Level Maths: Probability & Statistics 2): Exam Questions

Alina is standing for election as student council president at her school. She claims that 60% of the students in the school support her. A rival candidate, John, suspects that Alina is overestimating her support.

John chooses a random sample of 100 students and records the number, $X$, who say that they support Alina. He carries out a hypothesis test at the 10% significance level.

(i) State, with a reason, whether John should use a one-tailed or a two-tailed test.

(ii) State suitable null and alternative hypotheses for John's test.

In John's sample, 55 students say that they support Alina.

(i) Assuming that Alina's claim is true, a student correctly calculates that $\text{P}(X=55)=0.0478$, correct to 3 significant figures. He says that, because this value is less than 0.1, the null hypothesis should be rejected.

Explain why this statement is incorrect.

(ii) Use a suitable approximating distribution to carry out John's test.

Previously, 75% of the teachers at a school met the deadline for writing reports. The headteacher introduces a new brand of coffee in the staffroom and believes that the proportion of teachers who meet the deadline has increased.

She chooses a random sample of 15 teachers and notes the number who meet the deadline for the next set of reports. She carries out a test at the 5% significance level.

State suitable null and alternative hypotheses and find the critical region for the test.

A survey found that 65% of the ladybirds in a certain region are of the seven-spot species. Alex believes that more than 65% of the ladybirds in his garden are of the seven-spot species.

He collects a random sample of 25 ladybirds from his garden and records the number, $X$, that are seven-spot ladybirds.

State suitable null and alternative hypotheses for a test of Alex's belief.

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

Use a binomial distribution to carry out the test at the 5% significance level.

In the past, 40% of the visitors to a museum bought a guidebook. After the guidebook is redesigned, the manager suspects that the proportion of visitors who buy one has decreased.

She chooses a random sample of 15 visitors and records the number, $X$, who buy a guidebook. She carries out a test at the 5% significance level, using the following hypotheses, where $p$ is the proportion of visitors who buy a guidebook.

${\text{H}}_0:p=0.4$

${\text{H}}_1:p<0.4$

Find the critical region for the test.

In the sample, 4 visitors buy a guidebook.

State the conclusion of the test, giving a reason for your answer.

In the past, 15% of the items bought from an online clothes shop were returned. Harry, the manager, introduces a new sizing guide and suspects that the proportion of items returned has decreased.

Harry chooses a random sample of 40 items bought after the change and records the number, $X$, that are returned. He tests the hypotheses ${\text{H}}_0:p=0.15$ and ${\text{H}}_1:p<0.15$ at the 5% significance level, where $p$ is the proportion of items returned. The critical region for the test is $X\le 2$.

(i) Find the probability of a Type I error.

(ii) In Harry's sample, 4 items are returned. State the conclusion of the test, giving a reason for your answer.

A factory finds that 3% of the light bulbs it makes are faulty. Sally suspects that a new machine has increased the proportion of faulty bulbs.

She tests a random sample of 35 bulbs made by the new machine and finds that 3 of them are faulty. She tests the hypotheses ${\text{H}}_0:p=0.03$ and ${\text{H}}_1:p>0.03$, where $p$ is the proportion of faulty bulbs.

Carry out the test at the 5% significance level.

Charlie, the owner of a chocolate shop, claims that more than 60% of people can tell the difference between two brands of chocolate.

Charlie chooses a random sample of 15 customers and asks each of them to taste both brands. He records that 12 of them can tell the difference between the two brands.

State suitable null and alternative hypotheses to test Charlie's claim.

Test, at the 10% significance level, whether Charlie's claim is justified.

It is reported that four out of every five people are right-handed. Edward, an education researcher, chooses a random sample of 30 children under the age of 18 and records the number, $X$, who write with their right hand.

Assuming that the reported proportion applies to children, state the distribution of $X$, giving the values of any parameters.

Edward believes that the proportion of children who are right-handed is higher than the reported proportion for all people. He plans to test his belief at the 5% significance level.

State suitable null and alternative hypotheses for Edward's test.

Find the critical region for Edward's test.

State the probability of a Type I error.

In Edward's sample, 29 of the 30 children write with their right hand.

State the conclusion of the test, giving a reason for your answer.

The existing treatment for a disease is known to be effective in 85% of cases. Dr Sabir develops a new treatment which she claims is more effective than the existing one.

She uses the new treatment on a random sample of 60 patients with the disease, and it is effective for 57 of them.

Test, at the 1% significance level, whether there is evidence to support Dr Sabir's claim. State your hypotheses clearly.

State the conclusion that Dr Sabir would have reached if she had carried out the test at the 5% significance level.

The owner of a website claims that the website receives an average of 60 hits per hour. A possible buyer of the website believes that the average number of hits is less than 60 per hour.

The number of hits during a randomly chosen 10-minute period is found to be 4. You should assume that a Poisson model is appropriate.

(i) State suitable null and alternative hypotheses for a test of the buyer's belief.

(ii) Carry out the test at the 1% significance level.

A "double yolker" is an egg which contains two yolks. It is known that the probability that an egg is a double yolker is 0.001. A chicken farmer, Paolo, claims that double yolkers are rarer than this among the eggs his chickens lay.

Paolo takes a random sample of 1217 eggs laid by his chickens and finds that none of them are double yolkers.

Test, at the 5% significance level, whether there is evidence to support Paolo's claim. State your hypotheses clearly.

Paolo then takes a larger random sample of 4800 eggs and finds that 1 of them is a double yolker.

(i) Justify the use of a Poisson approximation for the number of double yolkers in this sample.

(ii) Use a Poisson approximation to show that there is now evidence, at the 5% significance level, to support Paolo's claim.

Paolo concludes that the probability that an egg laid by his chickens is a double yolker is definitely less than 0.001.

State, with a reason, whether Paolo's conclusion is justified.

It is known that 61% of male dragons eat more than 20 sheep in a day. Bill, a dragon breeder, suspects that the proportion of female dragons that eat more than 20 sheep in a day is greater than this.

State suitable null and alternative hypotheses for a test of Bill's suspicion.

Bill observes a random sample of 18 female dragons for a full day and finds that 15 of them eat more than 20 sheep.

Carry out the test at the 5% significance level.

Over many years, Harbour Town Cricket Club has won 45% of the series it has played. Rory believes that the proportion of series the club wins has decreased recently. To test this, he looks at the results of the club's last 17 series.

State suitable null and alternative hypotheses, and find the critical region for a test at the 5% significance level.

The club won 5 of its last 17 series.

State, with a reason, whether this supports Rory's belief.

It is known that there is a 0.35% chance of a computer picking up a virus when a file is downloaded from a particular website. After a security update, an analyst believes that this chance has been reduced.

In a random sample of 1542 files downloaded from the website after the update, 1 virus is detected.

(i) State suitable null and alternative hypotheses for the analyst's test.

(ii) Show that the result is significant at the 5% level but not at the 1% level.

(iii) State the conclusion of the test at the 5% significance level, in context.

At a particular school, it is known that 80% of students achieve grade C or better in their final examinations. The principal claims that, at another local school, the proportion of students who achieve grade C or better is different.

Last year, a random sample of 80 students at the other school was taken, and 57 of them achieved grade C or better.

Use a suitable approximating distribution to test the principal's claim at the 5% significance level, stating your hypotheses clearly.

State, with a reason, whether the conclusion would be different if the test were carried out at the 10% significance level.

The probability that a wild Asian elephant lives past 40 years old is 0.45.

Rosie, a zoologist, suspects that the proportion of Asian elephants that live past 40 years old is smaller for those in captivity than for those in the wild. She obtains the ages at death of a random sample of 20 elephants that lived in captivity.

State suitable null and alternative hypotheses and find the critical region for a test at the 1% significance level.

Rosie finds that only 1 of the 20 elephants lived past 40 years old.

(i) State the conclusion of the test, giving a reason for your answer.

(ii) Find the probability of a Type I error.

In a particular video game, players can breed two creatures to produce an offspring. Sometimes the offspring has a hidden ability.

One fan website, $A$, claims that the probability that an offspring has a hidden ability is 0.6. Another website, $B$, claims that the probability is higher than 0.6.

Maya breeds a random sample of 20 offspring and finds that 17 of them have a hidden ability.

Use Maya's sample to test the claim of website $A$ against the claim of website $B$ at the 5% significance level. State your hypotheses clearly.

Website $B$ states that the probability is 0.8.

Use Maya's sample to test this claim, at the 5% significance level, against the alternative that the probability is less than 0.8. Hence state which website's claim is better supported by Maya's sample.

The number of people arriving at a walk-in vaccination centre in a 5-minute period is modelled by the distribution $\text{Po}(0.6)$.

State two assumptions that are needed for the Poisson model to be valid in this context.

The centre runs an advertising campaign and wishes to test whether the mean number of people arriving has increased. During a randomly chosen 30-minute period, 7 people arrive.

Test, at the 10% significance level, whether the mean number of arrivals has increased. State your hypotheses clearly.

Find the probability of a Type I error for this test. You must justify your answer.

In the past, 90% of attempts to buy festival tickets from a particular website were successful. The website manager makes some improvements and claims that the probability of success is now higher.

To test the manager's claim at the 10% significance level, a random sample of 50 attempts to buy tickets is taken, and the number, $X$, that are successful is recorded.

State suitable null and alternative hypotheses for the test.

Explain what is meant, in this context, by

(i) a Type I error,

(ii) a Type II error.

Find the critical region for the test and the probability of a Type I error.

In the sample, 48 of the 50 attempts are successful.

Carry out the test.

After the improvements, the probability that an attempt is successful is in fact 0.95.

Find the probability of a Type II error.

Anna crochets toy fish. She models the number of faults in her crocheted material as having a Poisson distribution with mean 45 faults per square metre.

After taking a course, Anna believes that the mean number of faults has decreased. She tests her belief at the 10% significance level by counting the number of faults in one randomly chosen fish. Each fish uses 1000 ${\text{cm}}^2$ of material.

State what is meant by a Type I error in this context, and find the probability that Anna's test results in a Type I error.

Anna finds 2 faults in her randomly chosen fish.

Carry out the test, stating your hypotheses clearly.

After Anna's course, the mean number of faults is in fact 25 per square metre.

Find the probability that a Type II error is made in Anna's test.

Roger's rabbits are well trained and almost always leave their droppings in a designated area. Occasionally, however, Roger finds droppings in his garden. The number of droppings left in the garden is modelled by a Poisson distribution.

The probability that no droppings are found in the garden over a 6-hour period is $4.53999\times {10}^{-5}$, correct to 6 significant figures.

Find the mean number of droppings left in the garden per hour.

Roger starts giving his rabbits popcorn as a treat and believes that this reduces the mean number of droppings left in the garden. To test his belief, he will count the droppings left in the garden during a randomly chosen 3-hour period after giving them the treat. He carries out the test at the 5% significance level.

State suitable null and alternative hypotheses, find the critical region for the test and state the probability of a Type I error.

During the chosen 3-hour period, Roger finds 2 droppings in the garden.

State the conclusion of the test, giving a reason for your answer.

Past data show that the proportion of customers of Lakebridge Organics who order a large vegetable box is 0.4.

Last week a random sample of 12 customers' orders was taken, and 10 of them were for a large vegetable box.

Test, at the 5% significance level, whether this suggests that the proportion of customers ordering a large vegetable box has increased. State your hypotheses clearly.

Past data also show that the proportion of customers who order a medium vegetable box is 0.4. Lucinda, a delivery driver, suspects that the proportion of current customers who order a medium box is higher than 0.4. She takes a random sample of 10 customers' orders.

(i) State suitable null and alternative hypotheses, and find the critical region for a test at the 5% significance level.

(ii) State the probability of a Type I error.

In fact, half of all customers order a medium vegetable box.

Find the probability that Lucinda's test results in a Type II error.

Zien and her younger sister Ying play the game "rock, paper, scissors", in which each player chooses one of three options. Zien claims that she can predict which option Ying will choose. Ying thinks that Zien is just guessing.

To test Zien's claim, they play the game 10 times and Zien makes a prediction each time. Ying decides that if Zien's prediction is correct in 8 or more of the 10 games, she will conclude that Zien is not just guessing.

State suitable null and alternative hypotheses for Ying's test.

Find the probability of a Type I error.

Ying's rule is the same as a test at the $\alpha$% significance level with critical region $X\ge 8$.

Find the set of possible values of $\alpha$.

A football team, Dinamo Galacticos, is choosing a colour for its new kit. A local sports commentator says that the proportion of games won by teams wearing red is 0.6.

The manager takes a random sample of $n$ games in which one team wears red, and observes how many are won by the team in red. If the commentator is correct, the probability that all $n$ games are won by the team in red is 0.0279936.

Find the value of $n$.

The manager increases her random sample to $n+4$ games. She suspects that the proportion of games won by teams in red is lower than 0.6.

(i) State suitable null and alternative hypotheses, and find the critical region for a test at the 5% significance level.

(ii) Explain, in context, what a Type I error would be, and find the probability of a Type I error.

(iii) Instead, the manager carries out the test at the 10% significance level. Find the largest number of games won by teams in red for which a Type II error could not be made.

The commentator later says that he may have been mistaken, and that it is teams wearing green that win 60% of their games. A fan believes that teams wearing green win more than 60% of their games.

In a random sample of 15 games played by teams wearing green, 13 were won.

Test the fan's belief at the 5% significance level. State your hypotheses clearly.

A seed company claims that 40% of its wildflower seeds germinate. A gardener suspects that the proportion of these seeds that germinate is less than 40%.

She plants a random sample of 20 of the seeds and finds that 3 of them germinate.

Carry out the test at the 1% significance level.

Later, a second gardener plants a random sample of $m$ of the company's wildflower seeds and carries out a similar test at the 1% significance level. None of the $m$ seeds germinate, and the company's claim is not rejected.

Find the largest possible value of $m$.

The second gardener's sample size is equal to the value of $m$ found in part (b).

Find the critical region for a test of the same hypotheses at the 10% significance level.