Further Hypothesis Testing (DP IB Maths: AI HL)

In order to test the hypotheses  , 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 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.

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.

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.

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,  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.

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

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

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. 

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

Lucinda calculates the equation for the least squares regression line of  on . 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. 

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:

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.

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.

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  against the alternative hypothesis  . Lara will calculate the mean time for the seven sessions and will reject the null hypothesis if it is greater than 63 minutes.

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.

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, μ, 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   against the alternative hypothesis 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. 

325.2            356.1            319.7            300.5 

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, write down an inequality for w. Give the endpoints correct to 1 decimal place.

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 hour for the company. After Alex make some changes, the director of the company, Reema, wants to test, using a 10% significance level, whether the average number of views per hour has increased. Reema monitors the site for 10 minutes and models the number of visits during this time using a Poisson distribution. 

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.

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.

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 .

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

Find the equation of the least squares regression model for  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.

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

Explain why the two tests seem to suggest different results.

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, , 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   . 

Wanda uses the alternative hypothesis 

Pietro uses the alternative hypothesis

Lorna uses the alternative hypothesis

To perform their tests, the three children take a random sample of 12 nights and calculate the mean amount of sleep,  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 . Give the endpoints of the interval to 4 significant figures.

In the case where the actual value of  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 against the alternative hypothesis . Nick will flip the coin 60 times and he will reject the null hypothesis if the coin lands on tails less than 25 times.

40% of coins in Pennyslavania are fake.

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

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. 

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.

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 kg². 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 is 52.1 kg. 

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 . 

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

• 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  that satisfies the three conditions. Find the value of .

There are 35 students in Jacob’s class and 45 students in Daniel’s class.

Estimate the number of students whose teacher will be correctly identified using Jacob and Daniel’s test.