Hypothesis Testing for Population Parameters (DP IB Applications & Interpretation (AI): HL): Exam Questions

Vasily is a professional chess player. Over many years of competing, the mean number of minutes he has spent per move is 3.71. After beginning to work with a new training partner, Vasily believes that the mean number of minutes he spends per move has decreased. In his next tournament Vasily makes a total of 510 moves in his chess games, and the mean number of minutes per move is found to be 3.62. 

Let the random variable $X$ represent the amount of time Vasily spends per move after beginning to work with his new training partner. It is known from past experience that the standard deviation for the amount of time Vasily spends per chess move is 1.02 minutes.

State the null and alternative hypotheses for a hypothesis test to test Vasily’s belief.

Write down the distribution of the sample mean $\overline{X}$ that may be used to test Vasily’s belief. Justify your answer.

Use the $p$-value to test whether there is sufficient evidence at the 10% significance level to support Vasily’s belief that the mean number of minutes he spends per move has decreased.

Find the critical region for the test.

Gavio is the owner of a company that farms black soldier fly maggots to be made into insect protein powder for use in recipes in trendy restaurants. The weight of the maggots is known to be normally distributed, with a mean of 0.104 grams and a standard deviation of 0.039 grams.

Gavio has begun to use a new type of feed for the maggots on his farm. Because he is curious whether the new feed has resulted in a change in the average weight of his maggots, he selects 100 maggots raised on the new feed. It may be assumed that the use of the new feed has not changed the standard deviation of the maggots’ weights.

Explain why a two-tailed hypothesis test is appropriate in this situation.

The test is to be conducted at the 5% significance level.

(i) Explain what a Type I error is.

(ii) Write down the probability that Gavio’s test will result in a Type I error.

The 100 maggots raised on the new feed have a total weight of 11.158 grams. Conduct a hypothesis test to determine whether there is sufficient evidence at the 5% level to suggest that the mean weight of the maggots has changed.

Suggest one change Gavio could make to his test to reduce the probability of a Type II error without increasing the probability of a Type I error.

The lengths of rock songs follow a normal distribution. Miggy believes that the mean length of a rock song is greater than 253 seconds, the mean length of a metal song. Miggy tests his belief at the 10% significance level, using the null hypothesis ${\mathrm{H}}_0:\mu =253$, where $\mu$ is the mean length, in seconds, of a rock song. He takes a random sample of 9 rock songs and records their lengths, in seconds, in the table below.

(i) State why Miggy should use a $t$-test.

(ii) Write down the alternative hypothesis for Miggy’s test.

Find the $p$-value for the test.

State, with a reason, whether the test supports Miggy’s belief.

Kaarina and Hannu are big fans of Lobfickle brand jelly beans. The weights of individual Lobfickle jelly beans are known to be normally distributed. Hannu insists that the mean weight of a Lobfickle jelly bean is 1.20 grams, claiming that he read that on the internet somewhere. Kaarina suspects that the mean weight is less than that.

To test her suspicion, Kaarina takes a sample of 10 Lobfickle jelly beans and records their weights in grams. The table below shows her results.

Kaarina tests her suspicion at the 5% significance level.

Carry out Kaarina’s test. State your hypotheses and justify your conclusion.

Hannu says that Lobfickle jelly beans are too important to risk making a mistake here. Therefore he claims that a 1% level of significance should have been used for the test instead.

Write down the conclusion of the hypothesis test if Hannu’s proposed level of significance had been used instead.

An alternative wellness and lifestyle coach has developed a new tutoring program which he insists will improve students’ mathematics exam results. Students taking part in the program are encouraged to count the numbers of cute kittens in each of a series of images that are flashed before them while they listen to recordings by the lifestyle coach describing how awesome and clever he is. The coach claims that the practice with counting will improve the students’ mathematics results because mathematics is all about numbers and counting things.

Seven students who participated in the program were tested at the start of the program, and again once the program was completed. Their results in the tests were as follows:

Complete the following table showing the change in the students’ scores from before and after completing the program:

Calculate the $p$-value for a $t$-test at a 10% significance level on the table of differences from part (a), being sure to state your null and alternative hypotheses.

State the conclusion of the test. Justify your answer.

The number of spam emails that Arturo receives per day is modelled by a Poisson distribution with a mean of 25 spam emails per day.

After changing the settings on his spam filter, Arturo decides to test whether the new settings have reduced the number of spam emails he receives. To do this he records the number of spam emails he receives over a period of one week. He decides to use a 5% level of significance for his test.

State the null and alternative hypotheses for the test.

(i) Find the critical value and the critical region for Arturo’s test.

(ii) Hence find the probability that Arturo will make a Type I error in determining the conclusion of his test.

During the 1-week period, Arturo receives 149 spam emails.

State Arturo’s conclusion to his test, being sure to justify your answer.

Based on historical records, the number of shooting stars that may be seen per hour by observers during an annual meteor shower can be modelled by a Poisson distribution with a mean of 60 shooting stars per hour.

Due to recent astronomical events, Zlata believes that this year’s shower will be heavier than normal, with observers thus able to see a greater number of shooting stars per hour. To test her belief, she decides to record the number of shooting stars that she sees over a single 2-hour viewing period. If she observes more than 138 shooting stars during that period she will reject the historical mean.

State the null and alternative hypotheses for the test.

Find the probability that Zlata will make a Type I error in the conclusion of her test.

Zlata’s colleague Lyaksandro believes that this year the actual mean number of shooting stars that an observer may expect to see per hour is 85.

Explain what a Type II error is.

If Lyaksandro is correct, find the probability that Zlata’s test will result in a Type II error.

In order to test the hypotheses ${\mathrm{H}}_0:p=0.65$ and ${\mathrm{H}}_1:p>0.65$, where $p$ is the probability of success for a binomial random variable $X$, 24 observations of $X$ are made.

Assuming the null hypothesis is true, determine the expected number of successes out of 24 observations.

The test is to be conducted at the 10% significance level.

(i) Determine the critical region for the test.

(ii) Write down the probability that the test will result in a Type I error.

Out of the 24 observations, there are 19 successes. 

State the conclusions of the hypothesis test.

Determine what the critical regions for the test would have been had the test instead been conducted at a significance level of

(i) 5%

(ii) 1%.

A national institute of health is attempting to assess the efficacy of a new treatment for a disease.

When given the current best treatment, 87% of patients recover fully from the effects of the disease. The institute wishes to know whether the percentage of patients who fully recover is greater when the new treatment is given instead.

Explain why a significance level of 1% or lower would be appropriate for the institute’s test.

The institute decides to give the new treatment to 1000 patients with the disease and to record the number, $X$, who fully recover. The test is to be conducted with a 1% significance level.

(i) State the null and alternative hypotheses for the test.

(ii) Determine the critical region for the test.

Of the 1000 patients given the new treatment, 903 fully recover.

State the conclusions of the hypothesis test.

Hrothgar is a professional mathematician who believes that there is a strong positive correlation between a person’s happiness and the amount of time that person spends solving complex mathematics problems. To test his theory Hrothgar collects data from 10 people on how much time they spend solving complex mathematics problems, along with each person’s score on a standardised ‘level of happiness’ test. The results are shown in the table below:

Draw a scatter diagram for the data in the table.

Test at the 5% level whether it is reasonable to assume a positive linear correlation between the two variables, being sure to state your null and alternative hypotheses. You may assume that the data are a random sample from a bivariate normal distribution.

State whether or not it would be appropriate to calculate a least squares regression line for the data in the table. Justify your answer.

By interpreting the results of parts (a) and (b) above, remark on what the data set suggests with regard to the validity of Hrothgar’s belief.

A gardener believes that sunflowers growing in full sun grow taller than sunflowers growing in partial sun. To investigate this, the gardener measures the heights, in inches, of sunflowers growing in each condition. The results are shown in the table below.

It is assumed that the heights of sunflowers in both conditions are normally distributed with equal variances.

A $t$-test is carried out at the 5% significance level.

State the null and alternative hypotheses.

Find the $p$-value for this test.

State the conclusion of the test. Give a reason for your answer.

State the conclusion of the test if the significance level had been 1% instead. Give a reason for your answer.

The gardener repeats the investigation. Suggest one way to make the results of the test more valid.

George owns an ice cream store and his previous records suggest that only 15% of customers like his famous chilli chocolate flavour ice cream. To increase the proportion of customers who like the chilli chocolate flavour ice cream, George increases the amount of chilli in the ice cream. George wants to test, using a 5% level of significance, whether the extra chilli has increased the proportion of customers who like his chilli chocolate flavour ice cream.

Write down null and alternative hypotheses for George’s test.

George takes a random sample of 120 customers and gives them a sample of his new chilli chocolate ice cream.

Find the probability that George will make a Type I error in his test conclusion.

George finds that 25 out of 120 customers liked the new chilli chocolate flavoured ice cream.

State George’s conclusion to the test. Justify your answer.

State, with a reason, which error, Type I or Type II, George can be certain that he did not make in his conclusion.

Lucinda is investigating the petrol prices at stations in a town in North Carolina. Lucinda visits 9 random petrol stations and records the price of petrol per gallon, $p$ USD, and the distance between that station and the next closest station, $d$ miles. The results are shown in the table below.

Lucinda believes there is a linear relationship between the two variables. Lucinda uses a hypothesis test with a 5% significance level to test for linear correlation.

(i) State why the hypothesis test should be two-tailed.

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

(iii) Determine whether there is significant evidence of a linear correlation between the price of petrol per gallon and the distance between the station and the next closest station.

Describe one way in which Lucinda could improve the reliability of her test.

Lucinda calculates the equation for the least squares regression line of $p$ on $d$. She uses the equation to predict the price of a gallon of petrol at a station given that the next closest station is 6.5 miles away.

State, with a reason, whether Lucinda’s prediction is valid.

Brock sells bags of sand which he claims weigh on average 10 kg. It is known that the mass, in kg, of a bag of sand can be modelled using a normal distribution with variance 0.215. An inspector takes a sample of 8 bags of sand to test whether the average mass differs significantly from the average mass claimed by Brock. The masses, in kg, of the 8 bags are shown below.

The inspector uses a 5% significance level for his test.

(i) State, with a reason, the name of the test the inspector should use.

(ii) Write down the null and alternative hypotheses for the inspector’s test.

State the conclusion of the inspector’s test. Justify your answer.

The inspector repeats the test a week later using another sample of 8 bags. Brock is considering reducing the significance level to 1%.

Explain how reducing the significance level would affect:

(i) the probability of the inspector making a Type I error in his conclusion.

(ii) the probability of the inspector making a Type II error in his conclusion.

Gina, a member of the US gymnastic committee, is interested in analysing the gymnasts’ results to support future development of the team. In order to analyse the scoring of two gymnastic disciplines, the vault and the horizontal bar, she collated the scores from 10 competitors in both events. The results are shown in the table below.

Gina carries out a hypothesis test, using a 5% significance level, on the correlation coefficient to investigate whether an increase in the vault score is associated with an increase in the horizontal bar score.

(i) Explain why the hypothesis test should be one-tailed.

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

(iii) Calculate the value of $r$, Pearson’s product-moment correlation coefficient, for these scores.

The critical value for this test is 0.5494. Determine whether there is significant evidence of a positive linear correlation between scores in the vault and the horizontal bar. Justify your answer.

(i) Explain why Gina should use a paired $t$-test.

(ii) Perform a paired $t$-test, at the 5% significance level, to test whether competitors score higher, on average, on the horizontal bar than on the vault. State the hypotheses clearly and justify your conclusion.

Chloe, a gym instructor, has two training programs for her clients: The Sweat Inducer and The Calorie Destroyer. At the start of a week, Chloe picks one of the programs and uses that program for all seven sessions that week, however she does not tell her clients which one she picked. It is known that the times taken to complete each program can be modelled using normal distributions with standard deviation 8 minutes. The average time for The Sweat Inducer is 60 minutes and the average time for The Calorie Destroyer is 70 minutes.

Lara, a keen statistician, attends Chloe’s sessions once each day for a week. Lara knows Chloe will use the same program each day and wants to determine which one it is. Lara uses the seven sessions in the week as a random sample to test the null hypothesis ${\mathrm{H}}_0:\mu =60$ against the alternative hypothesis ${\mathrm{H}}_1:\mu =70$. Lara will calculate the mean time for the seven sessions and will reject the null hypothesis if it is greater than 63 minutes.

(i) Explain what a Type I error is in the context of the question.

(ii) Find the probability that Lara makes a Type I error in her test conclusion.

(i) Explain what a Type II error is in the context of the question. 

(ii) Find the probability that Lara makes a Type II error in her test conclusion.

A group of university researchers are concerned that runoff from local factories is affecting the water quality of a river. Water samples were taken next to the factories and from upstream of the factories, and the pH of each sample was recorded. The researchers want to test whether the mean pH is higher next to the factories than upstream, at the 1% significance level. The results are shown in the table below.

The researchers use a two-sample $t$-test, assuming that the two populations have equal variances.

State the distribution that the pH values at each location must be assumed to follow for this test.

State the null and alternative hypotheses.

Find the $p$-value for the test, and state the conclusion of the test. Give a reason for your answer.

Interpret the conclusion of the test for the researchers.

The researchers think that runoff from farms may also be affecting the river. Samples are taken next to a number of farms and compared with the upstream samples, testing whether the mean pH is higher next to the farms. The test is carried out at the same significance level, and the $p$-value is 0.0082.

State the conclusion of this test, and compare it with the result of the first test. Justify your answer.

In order to test the hypotheses ${\mathrm{H}}_0:\mu =330$ and ${\mathrm{H}}_1:\mu >330$, a sample of size 12 is taken from a normally distributed population. The mean of the sample is found to be 332.1, and the standard deviation of the sample, $s_n$, is found to be 2.7.

Using the $p$-value from a $t$-test, test at the 5% significance level whether the sample is from a population with a mean of 330, or from one with a larger mean.

Midge is playing an online game about dragons where the player scores bonus points if a purple dragon appears. One day Midge notices that the game has been updated and she suspects that the purple dragons now appear less frequently. Before the update, it was known that the number of purple dragons appearing in a 10-minute period could be modelled by a Poisson distribution with mean 3.2. To test her suspicion, Midge uses a 10% significance level and plays the game for 60 minutes to see how many purple dragons appear.

(i) Write down null and alternative hypotheses to test Midge’s suspicion. 

(ii) Find the critical value for Midge’s test. Justify your answer.

The game designers did reduce the mean rate at which purple dragons appeared to a mean of 2.5 times in a 10-minute period.

Find the probability that Midge will make a Type II error in her test conclusion.

After Midge plays the game for 60 minutes, she finds that 13 purple dragons appeared.

State Midge’s conclusion to her test. Justify your answer.

Midge repeats this test 100 times in total. Find the expected number of times that Midge makes a Type II error in her test conclusion.

Rachel, Nathan and Hope are three siblings who raise female cows together on a dairy farm. The masses of female cows follow a normal distribution with standard deviation 29.4 kg. It is claimed that the average mass of a female cow is 723 kg. The siblings have 20 cows on their farm.

The three siblings have different beliefs about their cows so they each use a hypothesis test with the same null hypothesis, ${\mathrm{H}}_0:\mu =723$. Rachel believes that their cows weigh less than the average and she tests her belief using a 5% significance level. Nathan believes that their cows weigh more than the average and will reject the null hypothesis if the average mass of their cows is more than 732 kg. Hope believes that the average mass of their cows is different to the claimed average and she tests her belief using a 5% significance level.

(i) Find the critical region for Rachel’s test.

(ii) Find the probability that Nathan’s test will lead to a Type I error.

The three siblings find that the total mass of their cows is 14221 kg.

Show, giving reasons, that only one of the siblings rejects the null hypothesis.

Jars are filled with a large number of jelly bears, the colour of each jelly bear is random. Simon’s favourite colour is red and through extensive research he has found that the probability of picking a red jelly bear from a full jar is 0.13. A new company produces the jelly bears and Simon suspects that the proportion of red jelly bears has decreased. Simon uses a hypothesis test with a 5% significance level to test his suspicion. 

State the null and alternative hypotheses for Simon’s test.

Simon takes a random sample of 20 jelly bears.

Show that this test will never produce a Type I error.

Simon decides his sample was too small, so he repeats his test using a bigger sample with the same significance level. Out of a total of 200 jelly bears, Simon finds that 17 of them are red.

By finding the $p$-value for the test, determine the conclusion of Simon’s test.

A technology company, TechBright, has developed a battery that it claims lasts longer than the battery sold by its main competitor, Elektrik. The battery life, in hours, of 12 batteries from each company was recorded. The results are shown in the table below.

It is assumed that the battery lives for both companies are normally distributed with equal variances.

A $t$-test is carried out to test TechBright's claim.

State the null and alternative hypotheses.

The test is carried out at the 5% significance level.

Find the $p$-value for this test.

State the conclusion of the test. Give a reason for your answer.

State the conclusion of the test if it had been carried out at the 10% significance level instead. Give a reason for your answer.

The test was originally planned at the 5% significance level.

Explain which significance level should be used to decide whether TechBright's claim is supported, and hence determine whether the claim is supported.

Pizza Prince is a fast-food restaurant which is known for its Crown pizza. The weights of Crown pizzas are normally distributed with standard deviation 42 g. It is claimed that the mean weight, $\mu$, is 350 g.

The restaurant receives a complaint from a customer who claims that the average weight of a Crown pizza is less than 350 g. Imogen, an independent inspector, investigates the customer’s claim by visiting the restaurant over a week and randomly sampling five Crown pizzas. Imogen uses the data to test the null hypothesis ${\mathrm{H}}_0:\mu =350$ against the alternative hypothesis ${\mathrm{H}}_1:\mu <350$ using a 5% significance level.

When Imogen writes up her report, she can only find the values for four of the weights. These are shown below.

Imogen remembers that there was insufficient evidence to reject the null hypothesis using a 5% significance level. Calculate the minimum possible value for the missing weight, $w$. Give your answer correct to 1 decimal place.

Imogen also remembers that the result of the hypothesis test would have been different if she had used a 10% significance level. Using this information, find an inequality for $w$. Give the endpoints correct to 1 decimal place.

Gorr and Greer are butchers who buy beef from two suppliers, Beefthor and Zeusbeef, to make 150 gram burgers. Gorr suspects that the mean amount of fat in the beef is different for the two suppliers. Gorr measures the amount of fat, in grams, in a sample of 150 gram burgers made with beef from each supplier. The results are shown in the table below.

Gorr assumes that the population variances are equal, and uses a pooled two-sample $t$-test at the 5% significance level to test his suspicion.

State one further assumption that Gorr has made about the distributions of the amount of fat in burgers made with beef from the two suppliers.

Carry out the test of Gorr's suspicion. State the hypotheses, find the $p$-value, and justify your conclusion.

Greer suspects that beef from Zeusbeef has more fat than beef from Beefthor. Use the data from Gorr's sample to test Greer's suspicion at the 5% significance level. State the hypotheses, find the $p$-value, and justify your conclusion.

Explain why Gorr and Greer’s tests have different conclusions.

Gorr and Greer test their suspicions again, at the same significance level, using a new sample. This time there is sufficient evidence to support Gorr's suspicion. Explain whether Greer's suspicion must also be supported by this sample.

At a funfair, Howard runs a duck game where there is a number of rubber ducks floating in water, each duck has a coloured sticker underneath. Players use a rod to hook one of the toy ducks and check the colour of the sticker. Players only win a prize if the colour of the sticker is purple. The duck is returned to the water after each game. 

Previously it was known that Howard had 80 toy ducks and 28 of them had a purple sticker. Howard increases the number of ducks in the game. Donald, a regular player of the game, wants to test whether the chance of winning a prize has decreased so he conducts a hypothesis test with a 5% significance level. He observes the game being played 150 times and records the number of times a prize is won.

Write down the null and alternative hypotheses for Donald’s test.

Given that Howard added 10 ducks to the game, none of which have a purple sticker, find the probability that Donald will make a Type II error in his test conclusion.

Out of the 150 games that Donald observed, 45 of them resulted in the player winning a prize.

State, with a reason, the conclusion to Donald’s test.

Howard now has 90 ducks, 28 of them with a purple sticker. Donald repeats this test 200 times in total. Find the expected number of times that Donald will make the correct conclusion in his test.

Wei, an economist, is investigating how the value of a vintage car changes over time. The table below shows the age of the car, $t$ years, and its value, $P$ USD.

Wei believes that the relationship between the age and value will mean that there is a positive correlation between the variables $t$ and $\ln P$.

(i) Complete the table below, giving all values correct to 3 decimal places.

(ii) Use a hypothesis test with a 5% significance level to show that there is evidence of a positive linear correlation between the variables $t$ and $\ln P$. State the hypotheses clearly and justify your conclusion.

(iii) Hence state whether an exponential or power model would best represent the value of the car as its age varies.

Find the equation of the least squares regression model for $P$ on $t$ using the model identified in (a) (iii).

Explain one limitation of the model.

Two IB students, Naomi and Harry, are investigating the relationship between performance in Mathematics and Biology exams for students at their school. All students at their school study both Mathematics and Biology. They have recorded the most recent exam percentage scores for a sample of students. The results are shown in the table below.

Harry decides to perform a pooled two-sample $t$-test at the 5% significance level to determine if the mean score is higher in Biology than Mathematics.

(i) State any assumptions that Harry needs to make.

(ii) State the null and alternative hypotheses for Harry’s test.

(iii) State the conclusion of Harry’s test. Justify your answer.

Naomi decides to perform a paired $t$-test at the 5% significance level to determine if a student scores higher in Biology than Mathematics.

(i) State any assumptions that Naomi needs to make.

(ii) State the null and alternative hypotheses for Naomi’s test.

(iii) State the conclusion of Naomi’s test. Justify your answer.

Explain why the two tests seem to suggest different results.

In a fast-food restaurant there is a machine which cooks specialist fries. The number of times the machine breaks down can be modelled by a Poisson distribution with a mean rate of 1.2 times per hour. The manager wants to decrease the rate of breakdowns, so she purchases the newest model of the machine. She conducts a hypothesis test using a specified level of significance by monitoring the new machine for a six-hour period and recording the number of breakdowns. 

(i) Write down the null and alternative hypotheses for the manager’s test. 

(ii) Explain why the probability of a Type I error will be less than the specified level of significance. 

(iii) Given that the probability of the manager making a Type I error is 0.07192 correct to four significant figures, find the critical value for the test.

During the six-hour period, the new machine breaks down 4 times.

State, with a reason, the conclusion to the manager’s test.

Suggest one way in which the manager could improve the reliability of her test.

A company currently gets an average of 25.3 views per minute for their website. Alex, a website developer, has been hired to increase the average number of views per minute for the company. After Alex makes some changes, the director of the company, Reema, wants to test, using a 10% significance level, whether the average number of views per minute has increased. Reema monitors the site for 10 minutes and models the number of views during this time using a Poisson distribution.

(i) State the assumptions that are needed to use a Poisson model.

(ii) Write down null and alternative hypotheses for Reema’s test.

Find the probability that Reema’s test will lead to a Type I error.

Describe one way in which Reema could improve the reliability of her test.

Alex claims the site now gets on average between 26.2 and 31.4 views per minute.

Assuming Alex’s claim is correct, find the maximum probability that Reema’s test will lead to a Type II error.

The amount of sleep, in hours, that Max gets in a night can be modelled using a normal distribution with standard deviation 0.94 hours. Max claims that the mean amount of sleep he gets each night, $\mu$, is 7.24 hours. Max’s three children, Wanda, Pietro, and Lorna, disagree with his claim. They each conduct a hypothesis test, with a 5% level of significance, using the same null hypothesis ${\mathrm{H}}_0:\mu =7.24$.

Wanda uses the alternative hypothesis ${\mathrm{H}}_1:\mu <7.24$.

Pietro uses the alternative hypothesis ${\mathrm{H}}_1:\mu \ne 7.24$.

Lorna uses the alternative hypothesis ${\mathrm{H}}_1:\mu >7.24$.

To perform their tests, the three children take a random sample of 12 nights and calculate the mean amount of sleep, $\overline{x}$ hours, that Max gets per night.

In the case where the test provides sufficient evidence for Pietro to reject the null hypothesis, explain why there is sufficient evidence for exactly one of his sisters to also reject the null hypothesis.

In the case where there is sufficient evidence for only Wanda to reject the null hypothesis, find the range of values for $\overline{x}$. Give the endpoints of the interval to 4 significant figures.

In the case where the actual value of $\mu$ is 8.05, find the probability that Lorna will make a Type II error in her test conclusion.

In the town of Pennyslavania, there is an increase in the number of fake coins being used. It is known that a real coin has a 50% chance of landing on tails when flipped and a fake coin has a 35% chance of landing on tails. Nick, a shopkeeper, suspects one of his coins is fake so he decides to conduct a test using the null hypothesis ${\mathrm{H}}_0:p=0.5$ against the alternative hypothesis ${\mathrm{H}}_1:p=0.35$. Nick will flip the coin 60 times and he will reject the null hypothesis if the coin lands on tails less than 25 times.

(i) Explain what a Type I error is in the context of the question.

(ii) Find the probability that Nick makes a Type I error in his test conclusion.

(i) Explain what a Type II error is in the context of the question.

(ii) Find the probability that Nick makes a Type II error in his test conclusion.

40% of coins in Pennyslavania are fake.

Find the probability that Nick makes an error in his test conclusion.

The mass of an adult pig in England can be modelled using a normal distribution with a mean of 255 kg and a variance of $2000{\text{ kg}}^2$. A pig is labelled as supersized if it weighs more than 350 kg.

Using the model, find the probability that a randomly selected adult pig in England is labelled as supersized.

Ramon, a farmer, raises adult pigs in Scotland. He believes that the probability that one of his pigs is labelled supersized is higher than the probability given by the model for adult pigs in England. To test his belief Ramon randomly selects 12 pigs that he owns and finds that two of them are classed as supersized.

Stating the hypotheses clearly, test Ramon’s belief using a 5% significance level.

The mass of an adult pig in Ramon’s farm can also be modelled by a normal distribution. Ramon believes that the mean mass is higher than 255 kg. Using the 12 pigs in his sample, Ramon calculates that the mean of the sample is 273 kg and the standard deviation of the sample, $s_n$, is 52.1 kg.

(i) State the name of the test that Ramon should use. Justify your answer.

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

(iii) State, with a reason, the conclusion of the test using a 5% significance level.

Ramon claims that the two conclusions contradict each other. Explain why it is possible for both conclusions to be correct.

There is a large cohort of students studying the statistics course at the University of Bernoulli. Students have one of two teachers: Jacob or Daniel. At the end of the term all students sit the same exam and receive a score. The scores of students in Jacob’s class are known to be normally distributed with mean 72 and standard deviation 9. The scores of students in Daniel’s class are normally distributed with mean 51 and standard deviation 5.

After the exams have been marked, they are anonymised so that the student or their teacher is not known. To determine whose class each student is in, Jacob and Daniel devise a hypothesis test. They use the null hypothesis that the student is in Jacob’s class and will reject the null hypothesis if the score of the student is less than $k$.

They want to choose the value of $k$ so that it satisfies the conditions:

• $k$ is an integer,

• the probability of a Type I error is less than 5%,

• the probability of a Type II error is less than 15%.

Show that there is only one value of $k$ that satisfies the three conditions. Find the value of $k$.