Edexcel International A Level Maths: Statistics 2

Topic Questions

2.1 Sampling Distributions & Hypothesis Testing

1
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5 marks

A hypothesis test uses a sample of data in an experiment to test a statement made about the value of a population parameter open parentheses p close parentheses.

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)
‘test statistic’.

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2a
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2 marks

Write down one advantage and one disadvantage of taking a census rather than a sample.

2b
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2 marks

A candidate for a college student council president wants to collect data on whether students at their college are happy with the current facilities. The candidate decides to carry out a sample survey to get the opinion of students at the college.

Describe the population and identify the sampling units.

2c
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1 mark

Suggest a suitable sampling frame.

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3a
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4 marks

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 open parentheses p close parentheses, 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.
3b
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1 mark

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).

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4
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6 marks

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 subscript 0 ∶ p equals 0.5 comma space space H subscript 1 ∶ p greater than 0.5.
(ii)
H subscript 0 ∶ p equals begin inline style 1 over 6 end style comma space space H subscript 1 ∶ p not equal to begin inline style 1 over 6 end style.
(iii)
H subscript 0 ∶ p equals 0.3 comma space space H subscript 1 ∶ p less than 0.3.

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5
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6 marks

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.  She uses the null hypothesis H subscript 0 ∶ p equals 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.

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6a
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4 marks

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

H subscript 0 ∶ p equals 1 fourth comma space space space space space H subscript 1 ∶ p not equal to 1 fourth comma

(i)
Describe what the parameter, p, 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.
6b
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2 marks

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 less or equal than a and  X greater or equal than b.  Write down the values of a space and space b.
(ii)
State the set of values for which the null hypothesis would be accepted.

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7a
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2 marks

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 subscript 0 ∶ p equals 0.2 comma space space space space space space space space H subscript 1 ∶ p not equal to 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.
7b
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2 marks

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 straight H subscript 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.

7c
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2 marks

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 subscript 0 ∶ p equals 0.2 comma space space space space space space space space space H subscript 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.

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8a
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2 marks

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.
8b
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5 marks

Under the null hypothesis, it is given that:

P left parenthesis X equals 13 right parenthesis equals 0.07393

P left parenthesis X equals 14 right parenthesis equals 0.03696

P left parenthesis X greater than 14 right parenthesis equals 0.02069

(i)
Calculate P left parenthesis X greater or equal than 14 right parenthesis and P left parenthesis X greater or equal than 13 right parenthesis.
(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.
8c
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2 marks

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.

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1a
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4 marks

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 space X tilde B left parenthesis 100 comma 0.5 right parenthesis then:

P left parenthesis X less than 42 right parenthesis equals 0.0443
P left parenthesis X less or equal than 42 right parenthesis equals 0.0666 space
P left parenthesis X equals 42 right parenthesis equals 0.0223 space
P left parenthesis X greater or equal than 42 right parenthesis equals 0.9557 space
P left parenthesis X greater than 42 right parenthesis equals 0.9334

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

1b
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2 marks

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.

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2a
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4 marks

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 space X tilde B open parentheses 80 comma 3 over 4 close parentheses the following probabilities are given:

P left parenthesis X less than 65 right parenthesis equals 0.8792 space
P left parenthesis X less or equal than 65 right parenthesis equals 0.9260 space
P left parenthesis X equals 65 right parenthesis equals 0.0468 space
P left parenthesis X greater or equal than 65 right parenthesis equals 0.1208 space
P left parenthesis X greater than 65 right parenthesis equals 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.

2b
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2 marks

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.

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3a
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2 marks

A spinner with seven equal sections labelled 0, 1, 1, 3, 3, 3 and 3 is being used in a game show where contestants spin the spinner twice and the two individual values it lands on, S subscript 1 and S subscript 2, are recorded.

q3a-2-1-ial-s2-hard-statistics-2

List all the possible samples that could result when a contestant spins the spinner twice.

3b
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4 marks

Assuming that the spinner is fair, find the sampling distribution for the sum of S subscript 1 space and space S subscript 2 open parentheses sum S close parentheses.

3c
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2 marks

If the sum of the two spins is 0, the contestant wins a grand prize.  After a day of playing the game, the grand prize has been won six times and the game show host thinks that someone may have tampered with the spinner so as to make a grand prize win more likely.  A hypothesis test is carried out at the 5% level of significance to test if the spinner is fair.  

Write suitable null and alternative hypotheses for the test.

3d
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3 marks

Given that the game was played 100 times on the day in question,

(i)
suggest a suitable approximating distribution that could be used to carry out the hypothesis test
(ii)
describe the circumstances under which the null hypothesis would be rejected.

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4a
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3 marks

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:

Gertrude’s solution

H subscript 0 ∶ p equals 0.6
H subscript 1 ∶ p greater or equal than 0.6

Nate’s solution

H subscript 0 ∶ p equals 0.6
H subscript 1 ∶ p greater than 24 over 30 equals 0.8

Let  be the number of games won,

X tilde B left parenthesis 30 comma 0.6 right parenthesis

P left parenthesis X equals 24 right parenthesis equals 0.0115

0.0115 less than 0.05so do not reject H subscript 0

Let  be the number of games won,

X tilde B left parenthesis 30 comma 0.6 right parenthesis

P left parenthesis X greater than 24 right parenthesis equals 0.0057

0.0057 less than 0.05 so reject H subscript 0


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

Identify and explain the three mistakes made by Gertrude.

4b
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2 marks

Identify and explain the two mistakes made by Nate.

4c
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1 mark

Use the information above to find the p-value for the test statistic x equals 24, showing your calculation clearly.

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5a
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1 mark

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

5b
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2 marks
(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.
5c
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6 marks

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.

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6a
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1 mark

A chocolatier produces his trademark homemade chocolate truffle, the Deliciously Decadent, in batches of 50.  He always tastes two randomly-selected chocolates from each batch to check the quality of the batch.

Explain why the chocolatier takes a sample rather than a census to check for quality.

6b
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2 marks

Suggest a suitable sampling frame and identify the sampling units.

6c
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2 marks

If a chocolate is up to standard, the chocolatier assigns it the number 1.  If not then the chocolate is assigned the number 0.

Using this numbering system, list all of the possible samples that could result when the chocolatier quality tests a batch of chocolates.

6d
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3 marks

After conducting many quality tests the chocolatier finds that 5% of chocolates are not up to standard, regardless of which batch they have come from.

Using your answer from part (c), and clearly defining any random variables you use, find the sampling distribution of the mean value of the sample when quality testing a single batch of chocolates.

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7a
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1 mark

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 tilde B left parenthesis 200 comma 0.4 right parenthesis

P left parenthesis X less than 70 right parenthesis equals 0.063903 space
P left parenthesis X equals 70 right parenthesis equals 0.020495 space
P left parenthesis X equals 71 right parenthesis equals 0.025018

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

7b
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2 marks

Find the probability of incorrectly rejecting the null hypothesis.

7c
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2 marks

It is discovered subsequently that in fact only 30% of bees in the area now belong to pollinating species. Given that for X tilde B left parenthesis 200 comma 0.3 right parenthesis

P left parenthesis X greater or equal than 70 right parenthesis equals 0.072786 space
P left parenthesis X equals 70 right parenthesis equals 0.018579 space
P left parenthesis X less than 70 right parenthesis equals 0.927214

find the probability that the farmers’ hypothesis test results in incorrectly accepting the null hypothesis.

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1a
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3 marks

Scientists at an animal health organisation are looking to collect data on a particular non-contagious disease in sheep throughout New Zealand.  The chance of any individual sheep in the country having the disease is believed to be constant and independent of location.

(i)
Give two reasons why the scientists would not choose to use a census for their data collection.
(ii)
Suggest a suitable sampling frame for the population.
1b
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2 marks

For purposes of the scientists’ study, every sheep in the total population may be classified simply as either ‘has the disease’ or ‘does not have the disease’.  A random sample of  sheep is taken from the population and each sheep is tested to see whether it has the disease.

For a given random sample of size n, write down an expression for the total number of possible samples that could occur as a result of the testing.

1c
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2 marks

The random variable X is defined as

X space equals space open curly brackets table row cell 1 comma end cell cell space if space the space sheep space has space the space disease comma end cell row cell 0 comma end cell cell if space the space sheep space does space not space have space the space disease. end cell end table close

Write down the sampling distribution of the statistic

Y equals sum from 1 to n of X

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2a
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2 marks

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

2b
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2 marks

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.
2c
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2 marks

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 space X greater or equal than 32.

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

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3a
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2 marks

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.
3b
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1 mark

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 less or equal than 13 and space X greater or equal than 25.

Write down the critical values for the hypothesis test.

3c
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4 marks
(i)
Given that for Amina space x equals 12, comment on her belief.
(ii)
Given that for Bert space x equals 24, comment on his belief.

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4a
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2 marks

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.
4b
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3 marks

Given that the critical value for the test is space x equals 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.

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5a
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2 marks

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.

5b
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2 marks

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 space P left parenthesis X less or equal than 2 right parenthesis equals 0.01127 whenspace X tilde B left parenthesis 100 comma 0.08 right parenthesis, determine the outcome of the hypothesis test using a 1% level of significance.  Give your conclusion in context.

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6a
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2 marks

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.

6b
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1 mark

Given that for this hypothesis test X tilde B left parenthesis 160 comma p right parenthesis describe, in context, the random variable X.

6c
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2 marks

Out of the 160 patients used for the sample, 127 turned up for their appointments.
It is given that, assuming the null hypothesis is true, P left parenthesis X greater or equal than space 127 right parenthesis space equals space 0.02094.

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

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7a
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2 marks

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.
7b
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3 marks

Given that P left parenthesis X less or equal than 15 right parenthesis equals 0.01452 when  X tilde B open parentheses 150 comma begin inline style 1 over 6 end style close parentheses, test Chase’s belief that the dice is not fair, using a 2% level of significance.

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8
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5 marks

Flight delays at a certain small airport are found to happen randomly and independently at an average rate of 32 delays per 7-day week. The airport manager puts a new scheme into place to reduce delays. The next day there are only 2 delayed flights.  A hypothesis test is carried out at the 5% significance level.

(i)
Suggest a suitable test statistic.
(ii)
Write down a suitable sampling distribution to use along with your test statistic from part (i).
(iii)
Write down suitable null and alternative hypotheses the airport manager could use to see if the average rate of flight delays per week has been reduced.
(iv)
Explain the condition under which the null hypothesis would be rejected.

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9a
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1 mark

A two-tailed test of the null hypothesis H subscript 0 colon p equals 0.23 is carried out for the random variable space X tilde B left parenthesis 60 comma p right parenthesis

Write down the alternative hypothesis.

9b
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2 marks

One of the critical regions is space X greater or equal than 20.  You are given the following probabilities:

P left parenthesis X less or equal than 8 right parenthesis equals 0.04603

P left parenthesis X less or equal than 9 right parenthesis equals 0.08932

P left parenthesis X less or equal than 10 right parenthesis equals 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.

9c
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2 marks

You are also given that space P left parenthesis X greater or equal than 20 right parenthesis equals 0.04427.

Find the actual level of significance of this test.

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1a
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3 marks

In the context of hypothesis testing, explain the term:

(i)
critical region
(ii)
critical value.
1b
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3 marks

The table below shows the probabilities for different values that space X tilde B left parenthesis 40 comma 0.8 right parenthesis can take:

x P left parenthesis X equals x right parenthesis
40 0.000133
39 0.001329
38 0.006480
37 0.020520
36 0.047452

A test of the null hypothesis H subscript 0 colon p equals 0.8 against the alternative hypothesis H subscript 1 ∶ p greater than 0.8 is carried out for the random variable space X tilde B left parenthesis 40 comma p right parenthesis.

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

1c
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3 marks

A second test is carried out with the same null hypothesis against the alternative hypothesis H subscript 1 ∶ p not equal to 0.8.

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

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2a
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5 marks

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 space X tilde B left parenthesis 250 comma 0.95 right parenthesis then P left parenthesis X equals 245 right parenthesis equals 0.008515 and P left parenthesis X greater than 245 right parenthesis equals 0.004571.

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

2b
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1 mark

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.

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3a
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4 marks

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 space X tilde B left parenthesis 40 comma 0.9 right parenthesis then:

P left parenthesis X less than 32 right parenthesis equals 0.015495 space
P left parenthesis X equals 32 right parenthesis equals 0.026407 space
P left parenthesis X greater than 32 right parenthesis equals 0.958098

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

3b
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1 mark

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

3c
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3 marks

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

3d
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1 mark

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

3e
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2 marks

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

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4a
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4 marks

A teacher keeps bronze, silver and gold star stickers as prizes for his students.  The teacher takes a sample of five stickers at random from a full pack.

(i)
Identify the population and the sampling units.
(ii)
Write down the number of possible samples and find how many distinct samples the teacher could take.
4b
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6 marks

The school office contains a large number of the bronze, silver and gold star stickers in the ratio 5:3:2. At the end of each term the students are awarded 50 points for a gold star, 20 points for a silver star and 10 points for a bronze star. Their final score is the product of the points their stars are worth. The teacher takes a sample of 3 stickers at random to show his students how to find the product of the points their stickers are worth.

Given that the teacher chose his sample only from bronze and silver stars, write down the sampling distribution of the product of the values the stickers are worth. Clearly define your random variable and statistic.

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5a
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4 marks

Given that space X tilde B left parenthesis 40 comma 0.3 right parenthesis then:

P left parenthesis X less or equal than 5 right parenthesis equals 0.008618 space
P left parenthesis X equals 6 right parenthesis equals k space
P left parenthesis X equals 7 right parenthesis equals 0.031522

When a sample of size 40 is used to test H subscript 0 ∶ p equals 0.3 against H subscript 1 ∶ p less than 0.3, it is known that space x equals 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.

5b
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2 marks

When a sample of size 40 is used to test  H subscript 0 ∶ p equals 0.3 against H subscript 1 ∶ p not equal to 0.3, it is known that x equals 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.

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6a
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2 marks

If space X tilde B left parenthesis n comma p right parenthesis then P left parenthesis X equals 0 right parenthesis equals left parenthesis 1 minus p right parenthesis to the power of n and P left parenthesis X equals n right parenthesis equals p to the power of n.

A sample of size 30 is used to test the null hypothesis H subscript 0 ∶ p equals 0.9 against the alternative hypothesis H subscript 1 ∶ p greater than 0.9 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.

6b
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3 marks

A sample of size 100 is used to test the null hypothesis H subscript 0 ∶ p equals q against the alternative hypothesis H subscript 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.

6c
Sme Calculator
4 marks

A sample of size m is used to test the null hypothesis H subscript 0 ∶ p equals 0.2 against the alternative hypothesis H subscript 1 ∶ p not equal to 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.

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