CIE A Level Maths: Probability & Statistics 2

Topic Questions

3.3 Hypothesis Testing (Normal Distribution)

1a
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2 marks

A random sample of 25 independent observations of the random variable X space tilde N left parenthesis 50 comma 10 ² right parenthesis is taken. The sample mean, ̅ X is calculated.

Explain why ̅ X space tilde N left parenthesis 50 comma 2 squared right parenthesis.

1b
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4 marks

Find

(i)
P left parenthesis ̅ X less than 45 right parenthesis
(ii)
P left parenthesis ̅ X greater than 54 right parenthesis
1c
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4 marks

Find the value of k such that:

(i)
P left parenthesis ̅ X less than k right parenthesis equals 0.05
(ii)
P left parenthesis ̅ X greater than k right parenthesis equals 0.1

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2a
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1 mark

The random variable S follows a normal distribution with a mean of 40 and a standard deviation of 8.  The mean of 16 independent observations of S is denoted as S with bar on top.

Explain why the standard deviation of S with bar on top is 2.

2b
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4 marks

By standardising and using the table of values for the normal distribution, find:

(i)
straight P left parenthesis S less than 39 right parenthesis

(ii)
P left parenthesis S with bar on top space less than 39 right parenthesis
2c
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3 marks

By standardising and using the table of critical values for the normal distribution, find:

(i)
the value of x such that straight P left parenthesis S greater than x right parenthesis equals 0.1

(ii)

the value of y such that straight P left parenthesis S with bar on top space greater than y right parenthesis equals 0.1.

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3a
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5 marks

The population mean of the random variable X space tilde space N left parenthesis mu comma 6 squared right parenthesis  is being tested using a null hypothesis straight H subscript 0 colon mu equals 15 against the alternative hypothesis straight H subscript 1 colon mu less than 15. A random sample of 10 observations is taken from the population and the sample mean is calculated as x with bar on top.

(i)
Write down the distribution of the sample mean,X with bar on top. 

(ii)
Find the test statistic, z, for ̅ x equals 12 and hence find P left parenthesis ̅ X less than 12 right parenthesis.

(iii)

Find the rejection region for ̅ X, when a 10% significance level is used.

3b
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4 marks

The population mean of the random variable Y tilde N left parenthesis mu comma 10 right parenthesis is being tested using a null hypothesis straight H subscript 0 colon mu equals 0 against an alternative hypothesis.  A random sample of 36 observations is taken from the population and the critical region for the test is Y with bar on top space greater than 0.7778.

(I)
Write down the appropriate alternative hypothesis for the test.

(ii)
Write down the distribution of the sample mean, Y with bar on top. 

(iii)
Find the level of significance that was used in the test.

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4a
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4 marks

A random sample of size 100 is taken from a population given by X tilde N left parenthesis mu comma 9 right parenthesis.

A two-tailed test is used to investigate the null hypothesis H subscript 0 colon mu equals 50 at the 10% level of significance.

(i)
Write down a suitable alternative hypothesis for this test.

(ii)
Find the rejection regions for X with bar on top for this test.
4b
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1 mark

Given that there is insufficient evidence to reject the null hypothesis when x with bar on top equals k comma write down an inequality for the range of values of k.

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5
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5 marks

Write suitable null and alternative hypotheses for each of the following situations.

(i)
A butcher advertises that their burgers weigh 180 g.  A customer believes that the burgers are on average underweight.

(ii)
The CEO of a large multi-academy trust claims that the students in her schools spend on average 150 minutes on homework each night.  A parent wants to test if the claim is true, so he takes a random sample of 20 students and calculates the mean time spent on homework during a specific night.

(iii)
The average weight of an adult male in the UK was known to be 83.6 kg before the country had a lockdown.  After the lockdown ended the standard deviation of weights remained the same.  A fitness instructor is investigating whether adult males in the UK got heavier during the lockdown.

(iv)
The manager of a company buys a hot drink vending machine for his employees.  The machine is supposed to dispense 350 ml of coffee when a customer selects the medium option.  The manager believes that the machine does not dispense enough coffee.  To test this, he takes a sample of 25 medium coffees and calculates the mean as 342 ml.

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6a
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1 mark

The Starlighter is a new brand of flashlight.  It is known that the brightness, B lumens, of the light emitted from a Starlighter follows a normal distribution with a standard deviation of 15 lumens.  Annie, a salesperson, claims that the mean brightness of a Starlighter is greater than 110 lumens.  To test her claim, the null hypothesis straight H subscript 0 ∶ mu equals 110 is used with a 5% level of significance.

Write down a suitable alternative hypothesis to test Annie’s claim.

6b
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3 marks

To test Annie’s claim a random sample of 40 Starlighters is taken and the mean brightness, b with bar on top, is calculated.

(i)
Assuming that the null hypothesis is true write down the distribution of the sample mean, B with bar on top.

(ii)
Find the critical region for the test.
6c
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2 marks

Given that the mean of sample is b with bar on top equals 114.5 spacelumens,

(i)
State whether there is sufficient evidence to reject the null hypothesis at the 5% level of significance

(ii)
Write a conclusion, in context, to the test.

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7a
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2 marks

The wingspan of a small white butterfly, W cm, follows a normal distribution with a standard deviation of 0.8 cm.  A report states that the average wingspan of a small white butterfly is 4.1 cm.  Kenzie, a butterfly enthusiast, wants to conduct a two-tailed hypothesis test, using a 5% level of significance, to investigate the validity of the statement made by the report.

(i)
Write down a suitable null hypothesis for Kenzie’s test.

(ii)
Write down a suitable alternative hypothesis for Kenzie’s test.
7b
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3 marks

Kenzie uses a random sample of 6 small white butterflies and finds that the mean wingspan is 2.65 cm.  Kenzie starts off the hypothesis test as follows:

I f space H subscript 0 i s space t r u e space t h e n space W tilde space N left parenthesis 4.1 comma space 0.8 squared right parenthesis

straight P left parenthesis W less than 2.65 right parenthesis equals P left parenthesis Z space less than negative 1.8125 right parenthesis space space
space space space space space space space space space space space space space space space space space space space space space space equals 1 space minus space 0.9651 space
space space space space space space space space space space space space space space space space space space space space space space equals 0.0349 space less than space 0.05

Identify and explain the two mistakes that Kenzie has made in his hypothesis test.

7c
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4 marks
(i)
Find the test statistic, z, that Kenzie should use.
(ii)
Write down the critical value for the test statistic, z, using the table of critical values.
(iii)
By comparing the test statistic, z, with the critical value, complete the hypothesis test.

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8a
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1 mark

A hypothesis test is used to investigate the population mean of the random variable X tilde N left parenthesis mu comma 10 squared right parenthesis. A random sample of size 16 is used to test the null hypothesis H subscript 0 ∶ mu equals 25.

Write down the probability of a Type I error if a 10% significance level is used.

8b
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3 marks

Find the probability of a Type I error given that the rejection region is

(i)
open parentheses ̅ X less than 21 close parentheses
(ii)
open parentheses ̅ X less than 21 close parentheses space o r space open parentheses ̅ X greater than 29 close parentheses
8c
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2 marks

Given that the rejection region is open parentheses ̅ X greater than 29 close parentheses, find the probability of a Type II error if the true mean is 27.

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1a
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3 marks

A random sample of 25 observations from the random variable X tilde space N left parenthesis mu comma 60 squared right parenthesis is used to test the null hypothesis straight H subscript 0 ∶ mu equals 400 against different alternative hypotheses.

Given that straight H subscript 1 ∶ mu greater than 400, find the rejection region for ̅ X for the test using a 5% level of significance.

1b
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3 marks

Given that H subscript 1 ∶ mu less than 400 and that the rejection region is ̅ X less than 382, find the probability of a Type I error.

1c
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3 marks

Given that H subscript 1 ∶ mu not equal to 400, determine the conclusion to the test using a 5% significance level if the sample mean is ̅ x equals 379.

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2a
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6 marks

The mean time that teenagers in the UK spend on social media is 132 minutes per day and the standard deviation is known to be 24 minutes.  Mr Headnovel, a teacher in the UK, claims that the students at his school spend more time on social media than the country’s average.  He takes a random sample of 15 students and calculates the mean time spent on social media to be 144 minutes.

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

2b
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2 marks

State two assumptions you had to make about the times that teenagers in Mr Headnovel’s school spend on social media?

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3a
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4 marks

Adrenaline is a new rollercoaster at a theme park.  It is known that the time a customer spends in the queue follows a normal distribution with a variance of 52 minutes².  The mean time spent in a queue for other rollercoasters is 41 minutes.  The manager of the theme park wants to use a hypothesis test to investigate whether the mean time in the queue for Adrenaline is different to the mean time for the other rollercoasters.  She takes a sample of 10 customers over a period of several days and records their times spent in the queue for Adrenaline.

Find the critical region for the test at the 10% level of significance. State your hypotheses clearly.

3b
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3 marks

The queuing times for the 10 people in the sample are:

38        49        40        39        49

39        59        32        55        41

State the conclusion of the test in context.

3c
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2 marks

It was discovered that the manager always took her sample during the first opening hour of the day.

Explain the effect this has on the conclusion to the test.

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4a
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4 marks

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

A restaurant inspector believes that the mean weight of the Crown pizza is less than
350 g.  She visits the restaurant over the period of a week, and samples and weighs five randomly selected Crown pizzas.  She uses the data to carry out a hypothesis test at the 5% level of significance.

She tests straight H subscript 0 ∶ mu equals 350  against  straight H subscript 1 ∶ mu less than 350.

When the inspector 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

Given that the result of the hypothesis test is that there is insufficient evidence to reject straight H subscript 0 at the 5% level of significance, calculate the minimum possible value for the missing weight, w. Give your answer correct to 1 decimal place.

4b
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4 marks

The inspector remembers her assistant claiming that if she had used a 10% level of significance then the outcome to the hypothesis test would have been different.

Using this information, write down an inequality for w.

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5a
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1 mark

Given that Z tilde space N left parenthesis 0 comma space 1 squared right parenthesis, find the value of d  such that  straight P left parenthesis Z greater than d right parenthesis equals 0.1, correct to 3 decimal places.

5b
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4 marks

The population mean of the random variable X space tilde N left parenthesis mu comma 5 squared right parenthesis   is being tested using a null hypothesis straight H subscript 0 colon mu equals 20 against the alternative hypothesis  straight H subscript 1 colon mu greater than 20.  A random sample of  observations is taken from the population and the sample mean is calculated as 22.

Using a 10% level of significance, the null hypothesis is rejected. Find the smallest possible value of the sample size n.

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6a
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3 marks

A conker is a seed from a horse chestnut tree.  The masses of conkers are known to be normally distributed with mean mu grams.  Camilla’s teacher claims that the mean mass is 14.5 g but Camilla believes that this is too low.  To test her belief, Camilla takes a random sample of 100 conkers and measures their masses, M.  The results are summarised as follows.

n equals 100 space space space space space space space space space space space sum m equals 1490 space space space space space space space space space space space sum m squared equals 22498

Calculate unbiased estimates for the population mean and variance.

6b
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4 marks

Test at the 5% significance level whether there is evidence to support Camilla’s belief. Clearly state your hypotheses and conclusion.

6c
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1 mark

Would the distribution of the sample mean have been any different had the masses of conkers not followed a normal distribution? Give a reason for your answer.

6d
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2 marks

State, with a reason, which of the errors, Type I or Type II, might have been made in carrying out the test in part (b).

6e
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3 marks

Camilla repeats the test with a different sample of 100 conkers.  The unbiased estimate for the variance is unchanged.

Given that the actual value for the population mean is 15.3 g, find the probability that the test will produce a Type II error.

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7a
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5 marks

Lucy leads a team of question writers. The number of questions which the team write can be modelled as a Poisson distribution with an average rate of 25 questions a day.  Lucy gives her team a one week holiday and when they return, she believes that the rate at which they write questions has changed.  Lucy uses a 5-day period as a sample, during this time the team write 148 questions.

Using an approximating distribution, test at the 5% significance level whether there is evidence to support Lucy’s belief that the writing rate has changed.

7b
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2 marks

Explain which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).

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1a
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2 marks

The mass of a Burmese cat, C, follows a normal distribution with a mean of 4.2 kg and a standard deviation 1.3 kg. Kamala, a cat breeder, claims that Burmese cats weigh more than the average if they live in a household that contains young children. To test her claim, Kamala takes a random sample of 25 cats that live in households containing young children.

The null hypothesis, H subscript 0   colon mu equals 4.2,  is used to test Kamala’s claim.

(i)
Write down the alternative hypothesis to test Kamala’s claim.
(ii)
Write down the distribution of the sample mean̅ C.
1b
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2 marks

Using a 5% significance level, find the rejection region of ̅ C for this test.

1c
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2 marks

Kamala calculates the mean mass of the 25 cats included in her sample to be 4.65 kg.

Determine the outcome of the hypothesis test at the 5% significance level, giving your answer in context.

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2a
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1 mark

The time, X seconds, that it takes Pierre to run a 400 m race can be modelled using X space tilde space N left parenthesis 87 comma 16 right parenthesis. Pierre changes his diet and claims that the time it takes him to run 400 m has decreased.

Write suitable null and alternative hypotheses to test Pierre’s claim.

2b
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4 marks

After changing his diet, Pierre runs 36 separate 400 m races and calculates his mean time on these races to be 86.1 seconds.

Use these 36 races as a sample to test, at the 5% level of significance, whether there is evidence to support Pierre’s claim.

2c
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1 mark

Give a reason to explain why the 36 races might not form a suitable sample for this test.

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3a
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2 marks

The average length, L, of a unicorn’s horn is 91 cm with a variance of 5 cm².  Luna researches unicorns and believes that unicorns that were born beneath a rainbow have longer horns.  To test her belief, Luna takes a random sample of 12 unicorns that were born beneath a rainbow and measures the length of their horns.

(i)
Write suitable null and alternative hypotheses to test Luna’s claim.

(ii)
What assumption do we need to make about the length of unicorn horns so that a normal distribution can be used for the mean of the sample, stack L. with bar on top
3b
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3 marks

Given that the critical value for the hypothesis test is 92.1 cm, calculate the level of significance for the test.

3c
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3 marks

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4a
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1 mark

The IQ of a student at Calculus High can be modelled as a random variable with the distribution straight N left parenthesis 126 comma 50 right parenthesis . The headteacher decides to play classical music during lunchtimes and suspects that this has caused a change in the average IQ of the students.

Write suitable null and alternative hypotheses to test the headteacher’s suspicion.

4b
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5 marks

The headteacher selects 10 students and asks them to complete an IQ test.  Their scores are:

127, 127, 129, 130, 130, 132, 132, 132, 133, 138

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

4c
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1 mark

It was later discovered that the 10 students used in the sample were all in the same advanced classes.

Comment on the validity of the conclusion of the test based on this information.

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5a
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1 mark

Carol is a new employee at a company and wishes to investigate whether there is a difference in pay based on gender, but she does not have access to information for all the employees.  It is known that the average salary of a male employee is £32500, and it can be assumed the salary of a female employee follows a normal distribution with a standard deviation of £6100.  Carol forms a sample using 20 randomly selected female employees.

Write suitable null and alternative hypotheses to test whether the average salary of a female employee is different to the average salary of a male employee.

5b
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3 marks

Using a 5% level of significance, find the critical regions for the mean salary which would lead to the rejection of the null hypothesis.

5c
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2 marks

The total of the salaries of the 20 employees used in the sample is £ 602000.

Use this information to state a conclusion for Carol’s investigation into pay differences based on gender.

5d
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3 marks

Would the outcome of the test have been different if a 10% level of significance had been used?

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6a
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3 marks

The standard normal distribution is denoted by Z space tilde space N left parenthesis 0 comma 1 squared right parenthesis .

(i)
Write down a formula that links the standard normal distribution,Z , to the distribution  X with bar on top tilde space N left parenthesis mu comma sigma squared over n right parenthesis.



(ii)

Find the value of d  such that  straight P left parenthesis Z less than d right parenthesis equals 0.05 , correct to 3 decimal places.

6b
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4 marks

The population mean of the random variable X tilde space N left parenthesis mu comma 10 squared right parenthesis is being tested using a null hypothesis  straight H subscript 0 colon mu equals 30 against the alternative hypothesis  straight H subscript 1 colon mu less than 30.  A random sample of  observations is taken from the population and the sample mean is calculated as 28.

Using a 5% level of significance, there is not enough evidence to reject the null hypothesis.

(i)
Show that square root of n less than 8.2245
(ii)
Hence find the greatest possible value for the sample size, n .

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7a
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2 marks

Carl and Ashleigh are highly competitive siblings.  Ashleigh can complete a crossword in 28 minutes on average.  Carl claims that his mean time for completing a crossword, mu minutes, is different to Ashleigh’s mean time.

Explain why a two-tailed test is needed to test Carl’s claim and write down suitable null and alternative hypotheses.

7b
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3 marks

Carl decides to use the rejection regions ̅ X less than 27 space space and space space ̅ X greater than 29. Carl takes a random sample of 40 crosswords and records the times, X minutes, it takes him to complete each one. The results are summarised as follows.

n equals 40 space space space space space space space space space space space space sum x equals 1103 space space space space space space space space space space sum x squared equals 30870

Calculate unbiased estimates for the mean and variance of X.

7c
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2 marks

Explain how you know that Carl’s test has not resulted in a Type I error.

7d
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3 marks

The true value for mu is subsequently found to be 27.4 minutes.  Find the probability that Carl’s test would have produced a Type II error.

7e
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1 mark

Explain whether the results of the Central Limit theorem were necessary for giving your answer to part (d).

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1a
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2 marks

The population mean of the random variable  X space tilde space N left parenthesis mu comma 5 squared right parenthesis is being tested using a null hypothesis  straight H subscript 0 colon mu equals p against the alternative hypothesis  straight H subscript 1 colon mu not equal to p. A random sample of 16 observations is taken from the population and the sample mean is calculated as x with bar on top equals s. There is insufficient evidence to reject the null hypothesis using a 5% level of significance.

When p equals 30 find the range of values for s.

1b
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3 marks

When space s equals 25 find the range of values for p.

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2a
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6 marks

Margot, a biologist, is researching the lengths of snails that are bred in captivity.  It is known that the standard deviation of the length of a snail in captivity is 7.2 mm.  Margot claims that the mean length of snails is less than 60 mm.  Taking 20 snails as a sample, Margot calculates the sample mean as 56.1 mm.

Stating your hypotheses clearly, test Margot’s claim using a 1% level of significance.

2b
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2 marks

State two assumptions that you made whilst carrying out the test in part (a).

2c
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2 marks

State, with a reason, which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).

2d
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1 mark

Margot repeats the same test but with a different random sample of 20 snails. Write down the probability that the test will produce a Type I error.

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3a
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2 marks

The weight of an adult pig 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 pig is labelled as supersized.

3b
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5 marks

Ramon, a farmer, believes that the probability that his pigs are supersized is higher than the probability given by the model.  To test his belief Ramon randomly selects 12 pigs that he has owned and finds that two of them were classed as supersized.

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

3c
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4 marks

Ramon also claims that the mean weight of the pigs on his farm is higher than the mean weight according to the model.  Using the 12 pigs in his sample, Ramon calculates the sample mean as 273 kg.

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

3d
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2 marks

What do the results from parts (b) and (c) suggest about the variance of the weights of the pigs on Ramon’s farm? Explain your answer.

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4a
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5 marks

Dr Yassin is a newly qualified dentist.  The length of time it takes him to perform a routine tooth extraction is normally distributed with a standard deviation of 41 seconds.  The mean time for a tooth extraction, mu, should be 420 seconds.  His supervisor, Dr Holden, takes a random sample of six patients and records how long it takes Dr Yassin to perform the procedure.  Five of the times are:

 433        381       498       363      419

Dr Holden uses a 5% level of significance to test straight H subscript 0 ∶ mu equals 420  against straight H subscript 1 ∶ mu not equal to 420.

Given that the result of the hypothesis test is that there is insufficient evidence to reject at the 5% level of significance, find an inequality for the length of time, t , for the sixth procedure.

4b
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4 marks

If Dr Holden had instead used the alternative hypothesis straight H subscript 1 ∶ mu less than 420 then the result would have been different.

Using this information, find an improved inequality for t.

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5a
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3 marks

Cyd is a fan of jazz music.  The length of a jazz song, L, follows a normal distribution with a standard deviation of 0.71 minutes. Cyd reads a headline that states that the mean length of a jazz song is 4 minutes, Cyd claims that the mean length of a jazz song is less than 4 minutes.  To test her claim, she takes a random sample of 40 songs and calculates the sample mean.

Stating your hypotheses clearly, find the rejection region for ̅ L for Cyd’s test using a 5% level of significance.

5b
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1 mark

Cyd decides to include more songs in her sample, what effect would this have on the rejection region?

5c
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4 marks

Cyd includes  songs in her sample and calculates the sample mean as 3.95 minutes.

Given that this sample mean is in the rejection region, find the minimum possible value for the sample size n.

5d
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1 mark

Explain whether it was necessary to use the Central Limit theorem for any part of this question.

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6
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8 marks

There is a large cohort of students studying the statistics course at the University of Bernoulli.  The students’ marks on an exam can be modelled by a normal distribution with mean mu and standard deviation 12.  The lecturer, Jacob, is trying to demonstrate the power of hypothesis testing.  He knows the true value of mu to be 59 but he keeps this a secret from the students.

Jacob asks the students to use the mean of a random sample of 25 exam results to test the null hypothesis H subscript 0 ∶ mu equals 55 against the alternate hypothesis H subscript 1 ∶ mu greater than 55.  Jacob calculates the probability of the test producing a Type II error to be 0.325.

(i)
The students calculate the probability that their test will result in a Type I error, find this probability.
(ii)
Write down the actual probability of a Type I error on this test.

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7a
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6 marks

The proportion of dalmatians that have no black spots on their faces is denoted p.  It is claimed that p is 0.2 but Ella suspects that the true value is lower.  Ella takes a random sample of 101 dalmatians and finds that 88 have black spots on their faces.

Using an appropriate approximating distribution, test Ella’s suspicion at the 5% significance level.

7b
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1 mark

Justify your choice of approximating distribution.

7c
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2 marks

State, with a reason, which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).

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