Hypothesis Testing (Cambridge International A-Level Mathematics): Exam Questions

A hypothesis test uses a sample of data in an experiment to test a statement made about the value of a population parameter ($p$).

Explain, in the context of hypothesis testing, what is meant by:

(i) ‘sample of data’,

(ii) ‘population parameter’

(iii) ‘null hypothesis’,

(iv) ‘alternative hypothesis’,

(v) ‘a Type I error’,

(vi) ‘a Type II error’.

From previous research, Marta has found that in general there is a 15% chance that any given customer ordering food at her restaurant will choose a salad. She wants to test whether people are more inclined to eat salads when it is sunny out.

(i) Clearly defining the value of the population parameter ($p$), state a suitable null hypothesis that Marta could use for this test.

(ii) State a suitable alternative hypothesis that Marta could use for this test.

(iii) Give an example of a test statistic that Marta could use to carry out this test.

After carrying out the test, Marta had evidence to conclude that people are more likely to eat salads when the sun is out. State whether she accepted or rejected the null hypothesis you have written in part (a)(i).

For the following null and alternative hypotheses, state whether the test is a one-tailed or a two-tailed test and give a suitable example context for each problem.

(i) H _ 0 ∶ p = 0.5 , H _ 1 ∶ p greater than 0.5.

(ii) H _ 0 ∶ p = 1 / 6 , H _ 1 ∶ p != 1 / 6.

(iii) $H_{0} : p = 0.3 , H_{1} : p < 0.3.$

In a quiz, students have to choose the correct answer to each question from three possible options. There is only one correct answer for each question. Ethan got $k$ answers correct, and he claims that he merely guessed the answer to every question but his teacher believes he used some knowledge in the quiz. he uses the null hypothesis H _ 0 ∶ p = 1 third to test her belief at the 10% significance level.

(i) If the teacher wishes to test to see if Ethan was trying to get the answers correct, rather than guessing them at random, write down the alternative hypothesis she should use and explain the conditions under which the null hypothesis would be rejected.

(ii) If the teacher wishes to test to see if Ethan was trying to get the answers incorrect, rather than guessing them at random, write down the alternative hypothesis she should use and explain the conditions under which the null hypothesis would be rejected.

(iii) If the teacher wishes to test to see whether Ethan was not guessing the answers at random, but she is uncertain whether he was using his knowledge to get them right or to get them wrong, write down the alternative hypothesis she should use and explain the conditions under which the null hypothesis would be rejected.

A hypothesis test at the 4% significance level is carried out on a spinner with four sectors using the following hypotheses:

H _ 0 ∶ p = 1 fourth , H _ 1 ∶ p != 1 fourth ,

(i) Describe what the parameter, , could be defined as.

(ii) In the context of this question, explain how the significance level of 4% should be used.

(iii) If the significance level were instead given as 10%, would the probability of incorrectly rejecting the null hypothesis be likely to increase or decrease? Give a reason for your answer.

The spinner is spun 50 times and it is decided to reject the null hypothesis if there are less than 7 or more than 18 successes.

(i) The critical regions for this test are given as $X\leq a$ and $X\geq b$. Write down the values of $a$ and $b$.

(ii) State the set of values for which a Type II error could occur.

Two volunteers at a national park, Owen and Cathy, have begun a campaign to stop people leaving their litter behind after visiting the park. To see whether their campaign has had an effect, Owen conducts a hypothesis test at the 10% significance level, using the following hypotheses:

H _ 0 ∶ p = 0.2 , H _ 1 ∶ p != 0.2

(i) State the percentage of people who left litter behind in the national park before the start of the campaign.

(ii) State whether this is a one-tailed or two-tailed test.

Owen observes a random sample of 100 people at the national park and finds that 14 of them left litter behind. He calculates that if $H_{0}$were true, then the probability of 14 or less people leaving litter would be 0.08044.

With reference to the hypotheses above, state with a reason whether Owen should accept or reject his null hypothesis.

Cathy conducted her own hypothesis test at the 10% significance level, using the same sample data as Owen, but instead she used the following hypotheses:

H _ 0 ∶ p = 0.2 , H _ 1 ∶ p less than 0.2

(i) Explain how Cathy’s hypothesis test is different to Owen’s.

(ii) Using these hypotheses, state whether the sample results given in part (b) should lead Cathy to accept or reject her null hypothesis. Give a reason for your answer.

A drinks manufacturer, BestBubbles, claims that in taste tests more than 50% of people can distinguish between its drinks and those of a rival brand. The company decides to test its claim by having 20 people each taste two drinks and then attempt to determine which was made by BestBubbles and which was made by the rival company. The random variable $X$ represents the number of people who correctly identify the drink that was made by BestBubbles.

(i) State, giving a reason, whether this is a one-tailed or a two-tailed test.

(ii) Write down the null and alternative hypotheses for this test.

Under the null hypothesis, it is given that:

$P(X=13)=0.07393$

$P(X=14)=0.03696$

$P(X>14)=0.02069$

(i) Calculate $P(X\geq14)$ and $P(X\geq13).$

(ii) Given that a 10% level of significance was used, write down the critical value and the critical region for this test.

(iii) State the actual level of significance for this test.

In fact, 15 of the 20 people correctly identify the drink made by BestBubbles.

(i) State whether there is sufficient evidence to reject the null hypothesis at the 10% significance level.

(ii) Write a conclusion for this hypothesis test in the context of the question.

For each of the following statements, write down whether an error has been made, and if so state whether it is a Type I or a Type II error.

(i) $H_{0}$ is true and $H_{0}$ is accepted.

(ii) $H_{0}$ is true and $H_{0}$ is rejected.

(iii) $H_{0}$ is not true and $H_{0}$ is accepted.

(iv) $H_{0}$ is not true and $H_{0}$ is rejected.

Explain why the probability of a Type I error is usually just below the significance level.

Describe how to calculate the probability of a Type II error.

Explain what you understand by a critical region of a test statistic.

Nationally 44% of A Level mathematics students identify as female. The headteacher of a particular school claims that the proportion of A Level mathematics students in the school who identify as female is higher than the national average.

(i) State a suitable null hypothesis to test the headteacher’s claim.

(ii) State a suitable alternative hypothesis to test the headteacher’s claim.

The headteacher takes a random sample of 60 A Level mathematics students and records the number of them who identify as female, $x$. For a test at the 10% significance level the critical region is $X\geq32$.

Given that $x=36$, comment on the headteacher’s claim.

The probability of a chicken laying an egg on any given day is 65%. Two farmers, Amina and Bert, have 30 chickens each. They believe that the probability of their chickens laying an egg on any given day is different to 65%.

(i) State a suitable null hypothesis to test the farmers’ belief.

(ii) State a suitable alternative hypothesis for a two-tailed test.

During a specific day, Amina and Bert each record the number of their 30 chickens that lay an egg. At the 5% significance level the critical regions for this test are $X\leq13$ and $X\geq25.$.

Write down the critical values for the hypothesis test.

(i) Given that for Amina $x=12$, comment on her belief.

(ii) Given that for Bert $x=24$, comment on his belief.

A memory experiment involves having participants read a list of 20 words for two minutes and then recording how many of the words they can recall. Peter, a psychologist, claims that more than 60% of teenagers can recall all the words. Peter takes a random sample of 40 teenagers and records how many of them recall all the words.

(i) State a suitable null hypothesis to test the psychologist’s claim.

(ii) State a suitable alternative hypothesis to test the psychologist’s claim.

Given that the critical value for the test is $x=19$, state the outcome of the test if

(i) 18 out of the 40 teenagers recall all the words

(ii) 19 out of the 40 teenagers recall all the words

(iii) 20 out of the 40 teenagers recall all the words.

A machine produces toys for a company. It was found that 8% of the toys it was producing were faulty. After an engineer works on the machine, she claims that the proportion of faulty toys should now have decreased.

State suitable null and alternative hypotheses to test this claim.

After the engineer is finished, the manager of the company takes a random sample of 100 toys and finds that 2 of them are faulty.

Given that $P(X\leq2)=0.01127$ when $X \text{tilde} B ( 100 , 0.08 )$, determine the outcome of the hypothesis test using a 1% level of significance. Give your conclusion in context.

After it was estimated that only 72% of patients were turning up for their appointments at Pearly Teeth dental surgery, the owner began sending text message reminders to the patients on the day before their appointments. In order to test whether the reminders have increased the proportion of patients turning up to their appointments, the owner decides to conduct a hypothesis test at the 5% level of significance using the next 160 patients scheduled for appointments as a sample.

State suitable null and alternative hypotheses to test this claim.

Given that for this hypothesis test the random variable to be used is $X \text{tilde} B ( 160 , p )$, describe in context what $X$ represents.

Out of the 160 patients used for the sample, 127 turned up for their appointments. Under the assumption that the null hypothesis is true, it is given that $P(X\geq127)=0.02094$.

Determine the outcome of the hypothesis test, giving your conclusion in context.

Chase buys a board game which contains a six-sided dice. He rolls the dice 150 times and obtains the number six on 15 occasions. Chase wishes to test his belief that the dice is not fair.

(i) State a suitable null hypothesis to test Chase’s belief.

(ii) State a suitable alternative hypothesis for a two-tailed test.

Given that $P(X\leq15)=0.01452$ when $X \text{tilde} B ( 150 , \frac{1}{6} )$, test Chase’s belief that the dice is not fair, using a 2% level of significance.

A test of the null hypothesis $H_{0}:p=0.3$ is carried out for the random variable $X \text{tilde} B ( 25 , p )$. The observed value of the test statistic is $x=3$. You are given the following probabilities:

$P(X<3)=0.00896$

$P(X=3)=0.02428$

$P(X\leq3)=0.03324$

Determine the outcome of the test, with reasons, when the alternative hypothesis is:

$H_{1}:p<0.3$ with a 1% level of significance.

$H_{1}:p\neq0.3$ with a 5% level of significance.

A two-tailed test of the null hypothesis $H_{0}:p=0.23$ is carried out for the random variable $X \text{tilde} B ( 60 , p )$.

Write down the alternative hypothesis.

One of the critical regions is $X\geq20$. You are given the following probabilities:

$P(X\leq8)=0.04603$

$P(X\leq9)=0.08932$

$\text{P}(X\leq10)=0.15526$

Given that a 10% level of significance is used, determine the other critical region. Give a reason for your answer by using a relevant probability.

You are also given that $P(X\geq20)=0.04427$.

Find the actual level of significance of this test.

Birds belonging to a certain species have been known to be stealing milk from milk deliveries across Canada. It is known that the foil caps on 3% of the bottles of all milk deliveries are being pecked open by the birds. Residents in a certain urban area believe that more than 3% of their milk bottles are being pecked open by the birds. They decide to test their claim at the 10% significance level by taking a random sample of 100 milk bottles from a delivery and seeing how many have been pecked open by the birds. The residents determine that the null hypothesis will be rejected in favour of the alternative hypothesis if more than 5 of the milk bottles in their sample have been pecked open by the birds.

State suitable null and alternative hypotheses for the residents’ test.

Given that 4 of the bottles are found to have been pecked open by the birds, state which of the two error types, Type I or Type II, could have been made. Justify your answer.

Write down the greatest possible probability of making a Type I error.

Joel is a manager at a swimming pool and claims that less than half of customers wear goggles in the water. Joel forms a sample using the next 100 swimmers and he notes that 42 of them wear goggles.

If $X \text{tilde} B ( 100 , 0.5 )$ then:

$P(X<42)=0.0443$

$P(X\leq42)=0.0666$

$P(X=42)=0.0223$

$P(X\geq42)=0.9557$

$P(X>42)=0.9334$

Stating your hypotheses clearly, test Joel’s claim using a 5% level of significance.

Joel discovers that there was a family of 12 people included in the sample, all of whom wore goggles.

Explain how this information affects the conclusion to the hypothesis test.

At Hilbert’s Hotel three quarters of customers leave feedback upon departure by writing a comment in a book on the reception desk. Karla, the manager, decides to get rid of the feedback book and instead leaves a feedback form in each room. To test whether this new system has made a difference to the proportion of guests who leave feedback, Karla forms a sample using the next 80 room bookings. Once the 80 sets of guests leave Hilbert’s Hotel, Karla counts that 65 feedback forms have been completed.

When $X \text{tilde} B ( 80 , \frac{3}{4} )$the following probabilities are given:

$P(X<65)=0.8792$

$P(X\leq65)=0.9260$

$P(X=65)=0.0468$

$P(X\geq65)=0.1208$

$P(X>65)=0.0740$

Test, using a 10% level of significance, whether there is evidence to suggest that the feedback forms have changed the proportion of guests who leave feedback. State your hypotheses clearly.

Karla repeats the same test, with the same hypotheses, the following week and finds that 53 out of the 80 sets of guests fill in their feedback forms. This leads to the null hypothesis being rejected. Karla claims that this shows that there is evidence that the proportion of guests leaving feedback has decreased.

Explain whether Karla’s claim is valid.

Explain one advantage of using critical regions instead of finding probabilities for a hypothesis test.

A test of the null hypothesis $H_{0}:p=0.2$ against the alternative hypothesis $H_{1}:p<0.2$ is carried out for the random variable $X \sim B \left(n , p\right)$.

The table below shows the probabilities for different values that can take:

Calculate $P(X\leq3)$.

(i) Using a 5% level of significance, find the critical region for the test.

(ii) State the actual level of significance for the test.

A group of high school statistics students are investigating the probability of winning a game called Chi Squares. Their teacher claims that they have more than a 60% chance of winning the game. To test the claim, they play 30 games of Chi Squares and win 80% of them. They perform a hypothesis test using a 5% level of significance. Below are shown the solutions of two students, Gertrude and Nate:

You are given that the students have correctly calculated their probabilities.

Identify and explain the three mistakes made by Gertrude.

Identify and explain the two mistakes made by Nate.

Use the information above to find the correct probability they should have used to test the observed value $x=24$, showing your calculation clearly.

Explain what you understand by the significance level of a hypothesis test.

(i) Give an advantage of using a lower significance level for a hypothesis test.

(ii) Give a disadvantage of using a lower significance level for a hypothesis test.

For each of the following scenarios, explain whether a 1%, 5% or 10% level of significance would be most appropriate.

(i) A shopkeeper takes a sample of 10 cartons of milk to test whether the amount of milk in a carton has decreased.

(ii) A doctor takes a sample of 100 patients to test whether there is an improvement to the recovery rate of an illness when a new drug is used, compared with the current best treatment regime.

(iii) A manager takes a sample of 100 employees to test whether their level of job satisfaction has changed after new working hours have been introduced.

The table below shows the cumulative probabilities for different values that $X \text{tilde} B ( 10 , 0.5 )$ can take:

Kieran collects coins and suspects that one of them is biased. To test his suspicion Kieran flips the coin 10 times and records the number of times, $T$, that it lands on tails.

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

Calculate the probability of making a Type I error.

The coin lands on heads on each of the 10 flips. Kieran claims that the coin is definitely biased.

Comment on the validity of Kieran’s claim.

Describe one adjustment Kieran could make to his test to give a more reliable conclusion.

It is known that historically 40% of all bees in a certain part of the UK belonged to pollinating species. Farmers in the area, however, believe that that percentage has decreased in the past ten years. They design an experiment in which they will safely catch 200 bees in the area, check which species they belong to, and then release them. The farmers carry out a hypothesis test at the 10% significance level.

They calculate that for $X \text{tilde} B ( 200 , 0.4 )$.

$P(X<70)=0.063903\,P(X=70)=0.020495\,P(X=71)=0.025018$

Clearly defining any parameters, state the null and alternative hypotheses for the farmers’ test.

Find the probability of a Type I error. Justify your answer.

It is discovered subsequently that in fact only 30% of bees in the area now belong to pollinating species. Given that for $X \text{tilde} B ( 200 , 0.3 )$.

$P(X\geq70)=0.072786\,P(X=70)=0.018579\,P(X<70)=0.927214$

Find the probability that the farmers’ hypothesis test could have resulted in a Type II error. Justify your answer.

In the context of hypothesis testing, explain the term:

(i) critical region

(ii) critical value

The table below shows the probabilities for different values that $X \text{tilde} B ( 40 , 0.8 )$can take:

A test of the null hypothesis $H_{0}:p=0.8$ against the alternative hypothesis H _ 1 ∶ p greater than 0.8 is carried out for the random variable $X \text{tilde} B ( 40 , p ) .$

Using a 5% level of significance, find the values of $X$ which would lead to the rejection of the null hypothesis.

A second test is carried out with the same null hypothesis against the alternative hypothesis $H_{1} : p \text{not equal} to 0.8.$

Given that $x=38$ is a critical value, find the minimum level of significance for the test.

Meditest is a company manufacturing medical tests which are used to determine whether a patient has a certain illness. Meditest claims that the tests are 95% accurate, however a particular hospital will only purchase the tests if they are more than 95% accurate. Meditest test the accuracy of their product using a sample of 250 patients with the illness and agree on a 1% level of significance. They discover that the tests are accurate for 245 out of the 250 patients.

If $X \text{tilde} B ( 250 , 0.95 )$ then $P(X=245)=0.008515$ and $P(X>245)=0.004571$.

Stating your hypotheses clearly, test whether Meditest’s product is more than 95% accurate using a 1% level of significance.

Meditest notice that they would have had sufficient evidence to reject the null hypothesis using a 5% level of significance. They change the level of significance from 1% to 5% and report to the hospital that their product is more than 95% accurate.

Comment on the validity of Meditest’s report to the hospital.

Frank is the owner of a factory which has recently opened near a school where Hilda is the headteacher. Before the factory opened, the attendance rate at the school was good 90% of the time. Hilda claims that the proportion of days when the attendance rate is good has decreased and she suspects this is due to the fumes from the factory making the children sick. Frank disagrees and claims that the factory has made no difference to the attendance rate. To test their claims a sample of 40 days is taken and on 32 days the attendance rate is good.

If $X \text{tilde} B ( 40 , 0.9 )$ then:

$P(X<32)=0.015495$

$P(X=32)=0.026407$

$P(X>32)=0.958098$

Stating your hypotheses clearly, test Hilda’s claim using a 5% level of significance. Give your answer in context.

Explain whether the outcome of the test supports Hilda’s suspicion.

Stating your hypotheses clearly, test Frank’s claim using a 5% level of significance.

Suggest a reason why Frank might want to use a two-tailed test.

State, with a reason, whether a one-tailed test or a two-tailed test would have been more appropriate for this scenario.

At a certain international school in Thailand 10% of students who drive motorbikes were found to be doing so without proper safety equipment. Mr Roy, the head of school began a ‘protect your head’ campaign. To see if the campaign had worked Mr Roy takes a random sample of 20 students who drive motorbikes and observes them to see if they are wearing proper safety equipment. If all of the students are wearing proper safety equipment Mr Roy will conclude that his campaign has worked.

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

(ii) Unfortunately, one student in Mr Roy’s sample rides to school without proper safety equipment that day. Which of the errors, Type I or Type II is possible? Explain your answer.

Mr Roy enlists the help of some students from the school who have a go at improving the campaign. He then repeats the same hypothesis test a few weeks later using another random sample of 20 students who drive motorbikes.

Given that the proportion of students who drive motorbikes but without proper safety equipment is now 4%, find the probability of a Type II error.

Given that $X \text{tilde} B ( 40 , 0.3 )$ then:

$P(X\leq5)=0.008618$

$P(X=6)=k$

$P(X=7)=0.031522$

When a sample of size 40 is used to test H _ 0 ∶ p = 0.3 against H _ 1 ∶ p less than 0.3, it is known that $x=6$ is the critical value using a 5% level of significance. Use the probabilities above to find upper and lower bounds for the value of $k$.

When a sample of size 40 is used to test H _ 0 ∶ p = 0.3 against H _ 1 ∶ p != 0.3, it is known that $x=6$is one of the two critical values using a 5% level of significance. Use the probabilities above to find an improvement for one of the bounds for the value of $k$.

If $X \sim B \left(n , p\right)$ then $P(X=0)=(1-p)^{n}$ and $P(X=n)=p^{n}$.

A sample of size 30 is used to test the null hypothesis H _ 0 ∶ p = 0.9 against text H end text _ 1 ∶ p greater than 0.9 the alternative hypothesis using a $k$% level of significance.

Given that there is at least one value that leads to the rejection of the null hypothesis, find the range of values for $k$.

A sample of size 100 is used to test the null hypothesis H _ 0 ∶ p = q against the alternative hypothesis H _ 1 ∶ p less than q using a 5% level of significance.

Given that there are no critical values for this test, find the range of values for $q$.

A sample of size $m$ is used to test the null hypothesis H _ 0 ∶ p = 0.2 against the alternative hypothesis H _ 1 ∶ p != 0.2 using a 1% level of significance.

Given that there is exactly one critical region for this test, find the range of values for $m$.