3.2 Hypothesis Testing (Discrete Distribution)

It is known that there is a 0.35% chance of picking up a virus when downloading a file from a particular website.  After a security update, a security analyst believes that the chances of picking up a virus have been reduced.  In a sample of 1542 files downloaded from the website, 1 virus is detected.  The analyst wishes to use a hypothesis test with significance levels  appropriately chosen to allow her to report with as much certainty as possible that her belief is correct.

Test at the 1%, 5% and 10% significance levels and explain which of those significance levels the analyst should use for the test in her report.

At a school it is known that 80% of students obtain 3 or more A* to C grades in their A levels.  The principal claims that another local school has a different proportion of A* to C grades at A level.

Last year the other school had 57 out of 80 students achieve 3 or more A* to C grades at A level.

Using a two-tailed hypothesis test, test the principal’s claim at the 5% significance level stating suitable null and alternative hypotheses.

Suggest a hypothesis test with an alternative significance level that would lead to a different conclusion.

On a particular Pokémon game, it is possible to create offspring through breeding.  Sometimes when two Pokémon are bred together, their offspring possess a hidden ability.

One Pokémon fan website,, claims that the probability of producing offspring with a hidden ability is 0.6.  Another website,, claims the probability is higher.

Maya conducts an experiment and finds that 17 out of 20 of her Pokémon’s offspring possess a hidden ability.

Use Maya’s sample to test the claim of website   against website  at the 5% significance level, stating your null and alternative hypotheses clearly.

Website  claims that the probability is 0.8.

Using a two-tailed hypothesis test, test the claim at the 5% significance level and make a statement about which website is more likely to be correct.

The critical regions of a two-tailed hypothesis test to test claim are . Given that under the null hypothesis , find the probability of the hypothesis test resulting in a Type I error.

A walk-in vaccination centre can model the number of people arriving every 5 minutes by the random variable with distribution .

State two assumptions required for the Poisson model to be valid.

The centre launches a new advertising campaign and conducts a hypothesis test to see if it has increased the number of people arriving for a vaccine. They choose a half hour period at random and find that 7 people arrive for a vaccine during that time.

Clearly stating your hypotheses, test at the 10% significance level whether the number of arrivals has increased.

Find the probability of a Type I error, being sure to justify your answer.

It is known that in the past customers have had a 90% success rate when attempting to order festival tickets from a particular website. The website manager has made some improvements and claims that its customers should now have a greater probability of success. It is decided to conduct a hypothesis test at the 10% significance level to test the manager’s claim, using the next 50 people who attempt to purchase tickets as a sample.

State the null and alternative hypotheses to be used in conducting the test.

Explain, in context, what is meant by

Find the probability of the hypothesis test resulting in a Type I error.

Out of the next 50 people who attempt to purchase tickets, 48 are successful.

Using the hypotheses from part (a), carry out the appropriate hypothesis test to test the manager’s claim at the 10% significance level.

If after the improvements were made to the website the success rate had in fact become 95%, determine the probability that the hypothesis test could have resulted in a Type II error.

Anna likes to make crocheted fish for her friends.  Anna models the number of faults in her crocheted fish as having a Poisson distribution with a mean of 45 faults per square metre of material.  After taking a course she believes that this rate has decreased, and she decides to test her belief at the 10% significance level by counting the number of errors in a randomly chosen crocheted fish.

One of Anna’s crocheted fish uses 1000 cm² of crocheted material.

State what is meant by a Type I error in the context of the question and find the probability that Anna’s test will result in a Type I error.

Given that Anna finds 2 faults in the randomly chosen crocheted fish, carry out the hypothesis test.

Find the probability that a Type II error could have occurred in Anna’s test, if the rate of faults after Anna took her course has actually become 25 faults per square metre.

Harry is using the random variable  to test the hypotheses:

Harry states that the critical regions are  and .

Given that , calculate the probability of incorrectly rejecting the null hypothesis.

State, with a reason, the conclusion of Harry’s test given that the observed value is .

Sally is using the random variable  to test the hypotheses:

Sally observes the value .

Charlie, the owner of a chocolate shop, claims that more than 60% of people can tell the difference between two brands of chocolate.  Charlie takes a random sample of 15 customers and asks them to taste both brands of chocolate.  He records that 12 of them could successfully tell the difference between the two brands of chocolate.

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

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

Nationally it is reported that four out of five people are right-handed.  Edward, an education researcher, takes a random sample of 30 children under the age of 18 years old and records the number of them, , who write with their right hand.

Assuming that the national proportion applies to the sample, write down a suitable distribution for .

Edward believes that the proportion of right-handed children differs from the national proportion for all people. To test his belief, he uses his sample of 30 children.

State suitable null and alternative hypotheses to test Edward’s belief.

Given that , find the critical values for a two-tailed test at the 10% significance level for Edward’s belief.  You should state the probability of rejection for each tail, which should be less than 0.05 for each.

Use your critical values to find the probability of the test resulting in a Type I error.

Out of the 30 children in the sample, Edward recorded that 20 of them write with their right hand.

Comment on Edward’s belief based on this observation.

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. To test her claim she uses the new treatment on a sample of 60 patients with the disease and uses a binomial distribution to model the number of them who are cured.

Explain two assumptions that Dr Sabir has made when using a binomial distribution to model the number of patients cured by the new treatment.

Dr Sabir notes that her treatment was effective for 57 out of the 60 patients used in the sample.

Test, at the 1% level of significance, the validity of Dr Sabir’s claim that her treatment is more effective than the existing one. State your hypotheses clearly.

State the conclusion you would have reached if a 5% level of significance had been used for this test.

The owner of a website claims that his website receives an average of 60 hits per hour. An interested purchaser believes the website receives on average less than 60 hits per hour. The owner chooses a 10 minute period and observes that the website receives 4 hits. Assuming that the number of hits the website receives follows a Poisson Distribution,

A “double yolker” is an egg which contains two yolks. It is known that the probability of a chicken laying a double yolker is 0.1%. A chicken farmer, Paolo, claims that double yolkers are rarer than the stated 0.1%. To test his claim, Paolo records that his chickens lay 1217 eggs in a month and he uses these as his sample. He discovers that none of these eggs are double yolkers.

Test, at the 5% level of significance, whether there is evidence to support Paolo’s claim that double yolkers are rarer than 0.1%. State your hypotheses clearly.

Paolo decides to take a larger sample so extends his test to six months. During this time, a sample of 7300 eggs is formed and two of them are double yolkers.

Use a suitable approximation to show that there is evidence, at the 5% level of significance, to support Paolo’s claim that double yolkers are rarer than 0.1%.

Paolo concludes that the probability of a double yolker is definitely less than 0.1%. Give a reason to explain whether Paolo’s conclusion is justified.

It is known that 61% of male dragons eat more than 20 sheep within a day. Bill, a dragon breeder, suspects that the proportion of female dragons that eat more than 20 sheep within a day is different to males.

State suitable null and alternative hypotheses for a two-tailed test for Bill’s suspicion.

To test his suspicion, Bill observes a random sample of 18 female dragons during a full day and counts how many sheep they eat. He finds that 15 out of the 18 female dragons ate more than 20 sheep within the day.

Test, at the 5% level, whether there is evidence to support Bill’s suspicion.

Determine the outcome of the test, at the 5% level, if Bill had used a one-tailed test to check whether the proportion of females eating more than 20 sheep within a day is greater than the proportion of males.

Zien and her younger sister Ying are playing a game of ‘rock, paper, scissors’. Zien claims that she can predict which out of the three options – rock, paper or scissors – Ying will choose every time they play the game. Ying thinks Zien is just guessing so she decides to conduct a hypothesis test to test Zien’s claim. They play the game 10 times and if Zien is correct in 8 or more games, Ying agrees to accept Zien’s claim.

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

Calculate the probability of a Type I error.

Find the range of values that Ying could have been using for the significance level of her test, and hence write down the most likely significance level she used.

The probability of a wild Asian elephant living past 40 years old is 45%. 

Rosie, a zoologist, obtains data on 25 elephants in captivity and records their ages at death. She suspects that the proportion of Asian elephants that live past 40 years old is smaller for those in captivity than those in the wild.

Using a 1% level of significance, find the critical value of a one-tailed hypothesis test to enable Rosie to test her suspicion. Clearly state your hypotheses.

Rosie finds that only one elephant, out of the 20 in captivity, lived past 40 years old.

Roger’s bunnies are well trained and almost always leave their droppings in a designated area, however, Roger occasionally finds that they have left their droppings in his garden too.

Given that over a six-hour period the probability there are no droppings found in the garden is  correct to 6 significant figures, find the mean number of droppings left per hour in Roger’s garden.

Roger’s bunnies’ new treat craze is popcorn and Roger believes that the bunnies leave less droppings in the garden after he has given them this treat. Roger decides to test his theory by giving his bunnies the popcorn and then seeing how many droppings are left in the garden over a three-hour period.

Given that the probability of a Type I error is 0.0404277 correct to 6 significant figures, find the critical region for Roger’s hypothesis test.

Roger finds that two droppings were left in his garden during the three-hour period. Stating your hypotheses clearly, write down the conclusion of Roger’s hypothesis test.

Historic company data shows that the proportion of customers ordering a large vegetable box from Lakebridge Organics is 0.4.

Last week a random sample of 12 customers’ orders was taken and it was found that 10 of them had ordered a large vegetable box.

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

Historic company data shows that the proportion of customers ordering a medium vegetable box is also 0.4. Lucinda, a delivery driver, suspects that the proportion of current customers ordering medium boxes is different from 0.4. She takes a random sample of 10 customers’ orders.

In actual fact, half of all customers order a medium vegetable box when they shop at Lakebridge Organics. Find the probability of the hypothesis test resulting in a Type II error.

A football team, Dinamo Galacticos, are trying to decide on a colour for their new kit.  They are told by a local sports commentator that the proportion of games won by teams wearing red is 0.6.  The manager takes a random sample of  games where one team wears red and observes how many are won by the team wearing red.  Assuming that the proportion of games won by teams in red really is 0.6, then the probability that all of the games would be won is equal to 0.0279936.

Find the value of .

The manager decides to increase her random sample to include  games. She suspects that the actual proportion of games won by teams in red is lower than 0.6.

The sports commentator later says that he may have gotten mixed up, and that he thinks it is actually teams wearing green that win 60% of their games.

Given that teams wearing green have won 13 games from the last 15 in which they played, test at the 5% significance level whether the commentator’s new claim is justified. You must clearly state your null and alternative hypotheses.

A catering company’s past records show that the proportion of their customers who are vegetarian has previously been 0.2.  The company decides to take a random sample of 30 current customers to test whether that figure 0.2 is still valid for their current customer base.

Let  denote the number of vegetarians in the sample. Given that, if there has been no change to the proportion of customers who are vegetarian, the probability that less than 10 customers are vegetarian is 0.938913 correct to 6 decimal places, find the probability of the test resulting in a Type I error, using a 10% significance level.

The number of vegetarian customers in the sample is 11.  One of the company’s employees used the same sample to suggest that this shows that the proportion of vegetarian customers has not just changed but has in fact increased.

Use this observed value to test, at the 5% level of significance, whether the proportion of vegetarian customers has increased. State your hypotheses clearly.

Given that the proportion of vegetarian customers is in fact 0.25 and that the probability that less than 10 customers are vegetarian is actually 0.803407 correct to 6 decimal places, find the probability that the test in part (a) produces a Type II error.

A random variable has distribution  .

In order to test the hypotheses

a random observation of is taken and found to be 3.

Carry out a hypothesis test at the 2% significance level.

Another random variable has distribution  .

A random observation of is taken and found to be 0. Using the same hypotheses as above, find the maximum value of  for which would not be rejected at the 2% significance level.

Another random observation of  is taken and found to be . For the same hypotheses as above, this provides evidence to reject  at the 5% significance level. 

Using your answer to part (b) as the value for , find the possible values of .