2.2 Hypothesis Testing (Discrete Distributions)

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 its significance level  appropriately chosen to allow her to report with as much certainty as possible that her belief is correct.

Consider tests 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 45 out of 50 students achieve 3 or more A* to C grades at A level.

Test the principal’s claim at the 5% significance level stating suitable null and alternative hypotheses.

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

In 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 0.4.  Another website, , claims the probability is higher.

Maya conducts an experiment and finds that 28 out of 50 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 0.45.

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 ’s claim are  and .  Find the actual level of significance.

A walk-in vaccination centre can model the number of people arriving every 5 minutes by the random variable  with distribution Po(0.6).

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 this time.

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

Find the probability of incorrectly rejecting the null hypothesis.

A single observation is taken from a discrete random variable  to test the hypotheses

where  is a positive integer to be found.

Given that at the 10% significance level one of the critical regions is , find the value of .

Find the other critical region for the test at the 10% significance level.

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.

Find the probability that Anna might incorrectly conclude that her rate of faults has decreased.

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

It is subsequently discovered that following her course Anna’s fault rate has actually become 25 faults per square metre. Find the probability that the hypothesis test outlined above would have resulted in Anna reaching an incorrect conclusion.

Chloe’s sister likes to play a game in which she hides a coin in one of her hands and asks Chloe to guess which hand it is in.  Chloe claims that she can predict which hand the coin is hidden in more often than not.  Her sister thinks Chloe is just guessing so she decides to conduct a hypothesis test to test Chloe’s claim.

State suitable null and alternative hypotheses for Chloe’s sister’s test.

They play the game 100 times, and Chloe guesses correctly 60 times.

Using a suitable approximation, test at the 5% significance level whether or not Chloe’s claim is justified.

Justify the use of your approximation in part (b).

A college rugby team usually experiences injuries at a rate of 4 per week.  During a 12 week period they are forced to play at a different training ground whilst their regular ground is undergoing maintenance.  During this time the team experiences 65 injuries.

Using a suitable approximation, test at the 5% significance level whether or not there is evidence that the average rate of injuries per week is different at the temporary training ground.  Clearly state your sampling distribution and any assumptions you have made.

Harry is using the random variable to test the hypotheses:

Harry states that the critical regions are  and 

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

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 and , 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.

Find the actual level of significance for a test based on your critical values.

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 6 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 45% 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 15 female dragons during a full day and counts how many sheep they eat.  He finds that 11 out of the 15 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.

A high school principal claims that lost phones are handed in to the school office at an average rate of 1.3 phones per day.  The school secretary believes that the rate is higher than this, so she conducts a hypothesis test at the 10% significance level over a 5-day period and finds that 10 phones are handed in to the school office during that time.

State suitable null and alternative hypotheses for the secretary’s test.

Test, at the 10% level of significance, whether there is evidence to support the secretary’s belief.

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

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 25 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 that Roger incorrectly rejects the null hypothesis 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 40 customers’ orders.

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 0.5.

The manager takes a random sample of 20 games where one team wears red and finds that  of them are won by the team in red.  Assuming that the proportion of games won by teams in red really is 0.5, then the probability that  or fewer games out of 20 would be won is equal to 0.1316 correct to 4 decimal places.

Find the value of .

The manager decides to increase her random sample to include 50 games.  She suspects that the actual proportion of games won by teams in red is lower than 0.5.

The sports commentator later says that he may have gotten mixed up, and that he thinks it is actually teams wearing green that win 50% of their games.

Given that teams wearing green have won 14 games from the last 40 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. Find the critical region for a two-tailed test using a 10% significance level, stating the actual probability in each tail.

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

A random variable has distribution .

In order to test the hypotheses

a random observation of  is taken and found to be 5.

Carry out a hypothesis test at the 10% significance level.

Another random variable has distribution .

A random observation of  is taken and found to be 1. Using the same hypotheses as above, find the maximum value of  for which  would not be rejected at the 10% 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 .

A newspaper article claims that 60% of animal species have become extinct since 1970.  Iggy, a taxonomic biologist, has been studying animal extinction statistics and believes that the proportion is different to the newspaper’s claim.

Iggy collects a random sample of 500 species that were recorded as being in existence in 1970 and records the number that are now extinct.  He uses multiple sources to verify the species’ taxonomic status and finds that the number of extinct species could be as low as 319 or as high as 329.

Iggy wants to use his findings to test his belief, but he also wants to make sure that the test will support his belief for all possible values of the number of extinct species. Using a suitable approximation, find the smallest possible integer percentage value he could use for a significance level to make sure that the test supports his belief

Iggy repeats this process with a different random sample of 1000 species for which the taxonomic status is more certain, and in this second sample he finds the total number of extinct species to be 650.

Show that the evidence from this second sample offers better support for Iggy’s hypothesis than the evidence from his first sample.

It is thought that 2% of all giant huntsman spiders have a leg span of over 30 cm when fully grown.  Residents in a village in Laos believe that in their village the proportion of these spiders with a leg span of over 30 cm is different to 2%.  To test their theory, they gather a sample of 500 fully grown giant huntsman spiders, measure their leg spans and then release them back into the wild.

Using an approximating distribution, find the critical region for the residents’ hypothesis test using a 10% significance level. Clearly state your sampling distribution and hypotheses

Using the approximating distribution, estimate the probability that the residents will incorrectly conclude that the proportion of spiders in their village, with a leg span of over 30 cm, is not 2%.