DP IB Maths: AI SL

Topic Questions

4.7 Hypothesis Testing

1a
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2 marks

A new technology company, TechBright, has developed a battery that they claim has a longer lifespan than the product sold by its main competitor, Elektrik. A survey has been completed recording the battery life, in hours, of 12 batteries from each company. The results are shown in the table below.

TechBright 10.2 13.8 12.6 13.5 11.8 15.3 12.9 13.2 13.1 12.1 12.3 13.2
Elektrik 11.1 12.4 12.2 13.6 13.5 9.5 12.6 13.0 11.8 12.2 12.4 12.1

State the null and alternative hypotheses.

1b
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2 marks

Perform a t-test at the 5% significance level.

1c
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2 marks

State the conclusion of this test, giving a reason for your answer.

1d
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2 marks

If the test had instead been conducted at a significance level of 10%, state the conclusion for the t-test, giving a reason for your answer.

1e
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2 marks

Determine if TechBright is correct in claiming that their batteries have a longer lifespan than those created by Elektrik. Justify your answer.

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2a
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2 marks

A football coach wants to know if the number of hours spent training by a team in the week before a match affects the outcome of the game.

Data is collected for 100 matches, with the number of hours spent training during the preceding week the match, h, recorded alongside the result of win, lose or draw.  Some of this data is included in the following table:

  Win Lose Draw Total
bold 0 bold less or equal than bold italic h bold less than bold 8 4   11 31
 bold 8 bold less or equal than bold italic h bold less than bold 16   19 12  
bold 16 bold less or equal than bold italic h bold less than bold 24 18 4 1 23
Total 37 39   100

Complete the table of observed data.

2b
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1 mark

State the null hypothesis.

2c
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2 marks

Calculate the expected number of losses for a team that trains between 16 and 24 hours in the week before a match if training time and match results are independent of each other.

2d
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2 marks

Find the number of degrees of freedom.

2e
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2 marks

Calculate the chi squared spacetest statistic for this data, testing at the 10% significance level.

2f
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2 marks

The critical value is 7.779. State the conclusions obtained and justify your answer.

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3a
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1 mark

A spinner used on a game show is divided into 6 equal parts coloured red, yellow, green, blue, purple and orange.  There are reports that the spinner is not fair so an experiment is conducted to determine if this is the case.  The spinner is spun 720 times.

Calculate the number of times you would expect the spinner to land on red.

3b
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1 mark

The results of the experiment are shown in the table below:

  Red Yellow Green Blue Purple Orange
Number of spins 117 143 122 98 115 125

Write down the statistical test that can be performed to determine if the spinner is fair by comparing the observed results with the expected results.

3c
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2 marks

Write down the null and alternative hypotheses.

3d
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2 marks

Perform the test, using technology, and find the  p-value.

3e
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2 marks

If the test is performed at the 1% significance level, state the conclusion of the test. Give a reason for your answer.

3f
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2 marks

Comment on the suitability of the significance level. Justify your answer.

3g
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2 marks

A second spinner undergoes the same experiment, which results in a p-value of 0.082.

Explain how the difference in the p-values could be used to comment upon the fairness of the two spinners.

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4a
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2 marks

150 people are surveyed as part of an investigation into the relationship between gender and sport.  The researcher asks people to choose their preferred sport for viewing from a choice of football, basketball, tennis, gymnastics and swimming.

The results of the survey are shown in the table below

  Football Basketball Tennis Gymnastics Swimming
Female 23 10 15 18 20
Male 30 14 12 1 7

A chi-squared test is performed on the data at a significance level of 5%

State the null and alternative hypotheses.

4b
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1 mark

Using the table of critical values for a significance level of 5%, given below, write down the appropriate critical value.

Degrees of Freedom Critical Value
1 3.841
2 5.991
3 7.815
4 9.488
5 11.070

4c
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2 marks

Calculate the chi squared statistic.

4d
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2 marks

State the conclusion to the test, giving a reason for your answer.

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5a
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2 marks

A gardener believes that sunflowers situated in areas of full sun will grow taller than sunflowers planted in partial sun.  To investigate this, the gardener measures the heights, in inches, of sunflowers growing under both conditions.  The results are shown in the table below.

Full sun Partial sun
72.4 85.4
86.0 71.5
91.1 86.3
83.2 73.4
112.8 74.1
106.8 70.8
87.9 89.4
  93.1
  73.3
  76.4

A t-test is performed at a significance level of 5%.

State the null and alternative hypotheses.

5b
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2 marks

Calculate the p-value for the data.

5c
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2 marks

Comment on the results of the test.

5d
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2 marks

Comment on the result of the test if the significance level had instead been 1%.

5e
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1 mark

If the experiment were to be repeated, suggest one step that could be taken to increase the validity of the results.

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6a
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2 marks

A student is investigating the relationship between a country’s GDP and its literacy rate for his IA.  He has classified 140 countries into low, medium and high GDP and their literacy rates into low, medium and high.  His results are displayed in the table below.

  Low literacy Medium literacy High literacy
Low GDP 26 15 3
Medium GDP 13 21 14
High GDP 1 11 36

Perform a chi-squared test on the data at a 5% significance level to test the hypothesis that the literacy rate of a country is dependent on its GDP.  The critical value is 9.488.

State the null and alternative hypotheses.

6b
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1 mark

Write down the number of degrees of freedom.

6c
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5 marks

Calculate the p-value and the chi squared statistic. You should justify any conclusions found.

6d
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2 marks

Describe one possible issue with the way the data is presented, that might make it difficult to interpret the validity or precise implications of the test’s conclusions.

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7a
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3 marks

A computer game has 5 levels.  At the end of each level a magic star may appear, doubling your score from that level. The game is played through all levels 200 times by a focus group.  The number of times in total that the magic star appears for each game of 5 levels is recorded, and the results for all 200 games are summarised in the table below.

Number of magic stars 0 1 2 3 4 5
Frequency 7 24 70 59 38 2

The game developer wants to investigate whether the results can be modelled using the binomial distribution B invisible function application left parenthesis 5 comma 0.5 right parenthesis .

State the null and alternative hypotheses.

7b
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3 marks

Draw a table of expected frequencies.

7c
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2 marks

A chi-squared goodness of fit test is performed at the 10% significance level. State the chi squared statistic.

7d
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2 marks

The critical value for the test is 9.236. Comment on the results of the test, justifying your answer.

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8a
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3 marks

In healthy adults, systolic blood pressure is normally distributed with a mean of 112 mmHg and standard deviation of 10 mmHg.  A group of 200 patients take part in a clinical trial and their systolic blood pressure, p, is measured and recorded below.

Systolic blood pressure (mmHg) Frequency
p less or equal than 95 6
95 less than p less or equal than 105 59
105 less than p less or equal than 115 65
115 less than p less or equal than 125 63
p greater than 125 7

A chi-squared goodness of fit test at a significance level of 5% is used to determine if the sample of patients is representative of the general population in terms of their blood pressure. 

State the null and alternative hypotheses.

8b
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3 marks

Complete the table of expected frequencies below, giving the frequencies to 4 decimal places.

Systolic blood pressure (mmHg) Expected frequency
p less or equal than 95  
95 less than p less or equal than 105 39.4796
 105 less than p less or equal than 115 75.1896
115 less than p less or equal than 125  
p greater than 125  

8c
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1 mark

State the number of degrees of freedom.

8d
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3 marks

Calculate the p-value and comment on the results of the test.

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1a
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2 marks

At a school in Copenhagen, it is believed that favourite music genre is related to gender.  400 students were asked to indicate their favourite music genre from a selection and the results are indicated in the table below.

  Pop Rock Classical Rap
Female 58 63 17 44
Male 23 96 12 87

It is decided to test this hypothesis by using a chi squared test at the 5% significance level.  

The critical value is 7.815.

State the null and alternative hypotheses for this test.

1b
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1 mark

Write down the number of degrees of freedom for this table.

1c
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2 marks

Calculate the chi squared test statistic for this data.

1d
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2 marks

What conclusion can be drawn from this test? Give a reason for your answer.

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2a
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2 marks

An environmental organisation is trying to establish if altitude affects the growth of pine needles. A number of needles have been taken from trees at both high and low altitudes and their lengths, in inches, recorded. The results are shown in the table below.

Low altitude 6.1 8.2 7.7 8.0 11.9 6.9 7.5 7.1 8.1
High altitude 7.4 7.9 8.3 6.6 9.5 7.9 8.2 8.1 8.5

Perform a t-test to compare the mean lengths of the pine needles.

Write down the null and alternative hypotheses.

2b
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1 mark

State whether this is a one-tailed test or a two-tailed test.

2c
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2 marks

Perform a t-test at the 10% significance level. Write down the p-value.

2d
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2 marks

Write down the conclusion of the test. Give a reason for your answer.

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3a
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1 mark

A carpet salesman in interested how his sales are distributed and records his sales results over a period of six months. The data is shown in the table.

Month January February March April May June
Number of sales 16 12 14 20 15 19

A chi-squared goodness of fit test is to be performed on the data at the 5% significance level to find out whether the data fits a uniform distribution.

Find an estimate of how many carpets the salesman expects to sell each month.

3b
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2 marks

Write down the null and alternative hypotheses.

3c
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1 mark

Write down the number of degrees of freedom for this test.

3d
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2 marks

Calculate the p-value.

3e
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2 marks

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

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4a
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1 mark

A supermarket is interested in how the applications for its loyalty scheme are distributed throughout the working week. It is expected that the distribution of the data will be uniform. Over the course of one week the number of applications has been collected and recorded in the table.

Month Monday Tuesday Wednesday Thursday Friday
Number of sales 473 405 512 467 503

A goodness of fit test at the 10% significance level is to be performed.

The critical value is 7.779.

Calculate the expected value of the number of applications on any given workday.

4b
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2 marks

State the null and alternative hypotheses.

4c
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3 marks

Write down the

(i)

chi squared statistic

(ii)
p-value.
4d
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2 marks

State whether the data fits a uniform distribution, giving a reason for your answer.

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5a
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2 marks

A chi-squared test is performed to see if there is any dependence between eye colour (blue, green or brown) and hair colour (blond, red, brown or black). The test is completed at a significance level of 10%.

State the null and alternative hypotheses.

5b
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2 marks

Find the number of degrees of freedom for this test.

5c
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2 marks

The p-value for this test is 0.0726. State the conclusion that can be drawn and justify your answer.

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6a
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2 marks

It is claimed that women from Japan are taller on average than women from India. The heights, in cm, of 11 women from each country have been collated in the table below.

Japan India
173.0 155.2
158.2 157.8
148.5 156.0
150.6 142.7
168.7 149.6
149.8 150.1
158.8 152.6
155.3 148.2
159.2 151.3
158.9 147.6
166.0 168.0

Write down the type of test that can be used to compare the means of two sets of data.

6b
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2 marks

State the null and alternative hypotheses.

6c
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2 marks

Perform the appropriate test at the 5% significance level.

6d
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2 marks

State whether or not the initial claim is justified. Give a reason for your answer.

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7a
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2 marks

The average weight of a newborn baby born at 38 weeks is expected to be less than the average weight of a newborn born at full term (40 weeks). The weights of several babies, in kg, born at 38 weeks and 40 weeks in one hospital are recorded.

38 weeks 3.12 2.87 3.53 3.08 2.86 3.15 3.03 2.99    
40 weeks 3.08 3.59 3.49 3.61 2.99 3.58 3.42 3.55 3.66 3.58

A t-test is to be performed at a significance level of 10%.

State the null and alternative hypotheses.

7b
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1 mark

State whether a one-tailed or a two-tailed test should be used.

7c
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2 marks

Calculate the p-value statistic.

7d
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2 marks

State whether the initial expectation is confirmed by the test result. Justify your answer.

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8a
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3 marks

A survey was conducted to establish whether particular colours were favoured by people in different age groups. A group of 280 children, teenagers and adults were asked to pick their favourite colour from a choice of red, yellow, blue, green and pink.

The observed values are recorded in the table below.

  Red Yellow Blue Green Pink Total
Children 20 11 18 8 15 72
Teenagers 22 14 23 20 6 85
Adults 26 21 30 26 20 123
Total 68 46 71 54 41 280

A chi-squared test is to be performed at the 5% significance level. The critical value for the test is 15.507.

Complete the contingency table for the expected values below.

  Red Yellow Blue Green Pink Total
Children            
Teenagers            
Adults            
Total            
8b
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2 marks

State the null and alternative hypotheses.

8c
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1 mark

Write down the number of degrees of freedom.

8d
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2 marks

Write down the chi-squared test statistic for this data.

8e
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2 marks

Comment on your results within the context of the question.

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9a
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3 marks

The heights of giraffes are normally distributed with a mean of 3.8 m and a standard deviation of 0.7 m. As part of a conservation project in Kenya, the heights of 350 giraffes are measured and the results of the survey are seen in the table below. A chi-squared test at a significance level of 10% is to be performed to determine if the surveyed giraffes fit the normal distribution stated.

Height (cm) Frequency
h less than 3 50
3 less or equal than h less than 4 160
4 less or equal than h less than 5 119
5 less or equal than h less than 6 21

Complete the following table of expected heights.

Height (cm) Probability Expected frequency
h less than 3    
3 less or equal than h less than 4    
4 less or equal than h less than 5    
5 less or equal than h less than 6    

9b
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2 marks

Write down the null and alternative hypotheses.

9c
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1 mark

State the number of degrees of freedom.

9d
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2 marks

Calculate the p-value.

9e
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2 marks

State whether the results of the chi-squared test support the null hypothesis. Justify your answer.

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1a
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1 mark

An IB student is investigating the study habits of her peers.   She decides to conduct a study to find out if the time that a student prefers to study is affected by their gender.  She selects 20 boys and 20 girls at random from her year group and asks them to pick their preferred time of study from the options of morning, afternoon or evening.

State the type of sampling used in her investigation.

1b
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2 marks

State the null and alternative hypotheses.

1c
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4 marks

The results of the survey are listed below.

Female students Male students
morning, evening, evening, afternoon, morning, morning, afternoon, evening, afternoon, morning, morning, evening, afternoon, morning, evening, evening, morning, afternoon, morning, morning morning, afternoon, morning, evening, evening, evening, afternoon, morning, evening, evening, afternoon, evening, evening, afternoon, evening, morning, afternoon, evening, afternoon, evening

Create a contingency table of observed values.

1d
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2 marks

Show that if the preferred study time and gender are independent then the expected number of  female students that prefer to study in the morning is 6.5.

1e
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4 marks

The critical value for a chi-squared test performed at a significance level of 5% is 5.991. Using technology, calculate the chi squared space statistic and comment on the conclusions of the test in the context of the question, giving a reason for your answer.

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2a
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1 mark

A group of university researchers are concerned that pollution from industrial activity is having an effect on the local river system.  In order to investigate whether runoff from local factories is affecting the water quality, samples were taken from next to the factories as well as from upstream.  The data from both locations is compared to test whether the average pH is higher next to the factories than it is upstream. The test is conducted at a significance level of 1%.  The pH of the samples is recorded in the table below.

Factory
Location (pH)
7.8 8.1 8.0 7.5 7.9 8.1 7.8 7.8 7.8 7.7
Upstream location (pH) 7.2 7.7 7.6 7.8 7.6 7.5 7.9 7.8 7.3 7.6

State the type of distribution of the data.

2b
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2 marks

State the null and alternative hypotheses.

2c
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5 marks

Calculate the p-value and state any conclusions of the test, giving a reason for your answer.

2d
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1 mark

Interpret the implications of the conclusion to the test for the researchers.

2e
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2 marks

It is thought that agricultural runoff will also negatively affect the water quality.  Samples have been taken near a number of farms and compared to the upstream samples at the same significance level.  The p-value for this test is 0.0082.

Comment on the result of this test and compare it to the initial test. Justify your answer.

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3a
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2 marks

A cat shelter looks after 328 cats that have been abandoned or neglected.  As part of the initial health check of a new cat, its weight, W, in pounds is measured and recorded.  The table below collects together the data for the cat weights.

Weight Frequency
8 less or equal than W less than 9 17
9 less or equal than W less than 10 91
10 less or equal than W less than 11 141
11 less or equal than W less than 12 72
12 less or equal than W less than 13 7

The weight distribution of the general cat population can be described by

W~N(10.4, 0.92).  The shelter wants to see if the weight distribution of the cats in their care fits that of the general cat population.

A chi-squared goodness of fit test is to be performed at the 10% significance level.

Write down the null and alternative hypotheses.

3b
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4 marks

Draw a new table showing the expected frequencies if the shelter cat population were to match the general cat population weight distribution.

3c
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2 marks

Calculate the  p-value.

3d
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2 marks

Comment on the results of the test and the conclusions that can be drawn, giving a reason for your answer.

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4a
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1 mark

A lottery machine contains a set of coloured balls: three red, five yellow, two green and two blue.  When a button is pressed a ball is selected at random.  After noting its colour, the same ball is returned to the machine.

Calculate the probability of the machine selecting a yellow ball at random.

4b
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2 marks

The machine is to be tested for bias and so the procedure of selecting a ball at random is repeated 180 times and the results are recorded in the table below.

Colour Red Yellow Blue Green
Frequency 41 72 29 38

Draw a table showing the expected frequencies for the test.

4c
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4 marks

Perform a goodness of fit test at a significance level of 5% and comment on your results, giving a reason for your answer.

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5a
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3 marks

Approximately 1.6 million pairs of twins are born each year world-wide

Calculate the probability that both  babies in any random  set of twins are male .State any assumptions you have made.

5b
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2 marks

Complete the probability table for the number of male babies born in a given set of twins.

Number of male babies 0 1 2
Probability      
5c
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2 marks

In one particular hospital, there are 236 sets of twins born in one year.

Complete the expected frequency table.

Number of male babies 0 1 2
Expected frequency      
5d
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4 marks

The actual number of male babies born as part of a set of twins in this hospital during the year is shown in the table below.

Number of male babies 0 1 2
Expected frequency 46 149 41

Perform a chi-squared goodness of fit test, given that the critical value for the test is is 5.991. Comment on the results of the test, giving a reason for your answer.

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6a
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3 marks

A group of 14 students are studying reaction times in their science lesson.  They use a computer program to measure the time taken, in milliseconds, for a button to be pressed after a green light is shown.  One of the students has compared the group’s results to those of the expected normal distribution of reaction times and completed a chi-squared goodness of fit test at a significance level of 10%.  The calculated p-value is 0.130.

Comment on her results, justifying your answer.

6b
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3 marks

The student noted that in the first test, the dominant hand was used to press the button. It was decided to repeat the experiment using the non-dominant hand.  The calculated p-value for this test, which was also conducted with a significance level of 10%, is 0.0854.

Compare the results of both tests. Justify your answer.

6c
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3 marks

The data from both tests are then compared using a two-tailed t-test at a significance level of 10%.  The p-value for this test is 0.0667.

Explain the result of this test.

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7a
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3 marks

A study has noted that the top 20% of phone users spend more than 4.5 hours per day on their phones.  An independent health charity wishes to investigate whether heavy phone usage is linked to the age of the user.  A survey of 85 people is commissioned where the age group (child, teenager or adult) and the average number of hours, h, that are spent on the phone each day are noted.

The results are displayed in the table below.

  Low usage bold left parenthesis bold 0 bold less or equal than bold italic h bold less than bold 2 bold right parenthesis Medium usage bold left parenthesis bold 2 bold less or equal than bold italic h bold less than bold 4 bold. bold 5 bold right parenthesis  High usagebold left parenthesis bold 4 bold. bold 5 bold less or equal than bold italic h bold right parenthesis Total
Child 15 8 3 26
Teenager 6 18 8 32
Adult 9 12 6 27
Total 30 38 17 85

Draw a contingency table of expected values.

7b
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7 marks

Perform a chi-squared test and comment on the results. You should state the null and alternative hypotheses and comment on any conclusions found.

Select the appropriate critical value from the table below.

Degrees of freedom

Critical value (5%)
1 3.841
2 5.991
3 7.815
4 9.488
5 11.070
6 12.592
7c
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2 marks

State two steps that could be taken to increase the validity of the conclusions drawn from the test.

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8a
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2 marks

Gorr and Greer are butchers and purchase beef from two different suppliers: Beefthor and Zeusbeef. Gorr and Greer use the beef to make 150 gram burgers. Gorr suspects that the mean amount of fat in the beef is different between the two suppliers. Gorr measures the amount of fat, in grams, in a sample of 150 gram burgers using beef from the two suppliers. The data is shown in the table below.

Beefthor 28.1 29.3 27.2 27.5 30.1 27.0 28.1 27.2
Zeusbeef 29.5 28.9 30.3 28.3 28.8 27.9 29.3  

Gorr uses a pooled two-sample t-test to test his suspicion using a 5% significance level.

State two assumptions that Gorr has made about the distribution of the amount of fat in the burgers using beef from each of the two suppliers.

8b
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4 marks

Perform the hypothesis test to test Gorr’s suspicion. State the hypotheses clearly and justify your conclusion.

8c
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2 marks

Greer suspects that the beef from Zeusbeef has more fat than the beef from Beefthor. Use the data from Gorr’s sample to test Greer’s suspicion using a 5% significance level. State the hypotheses clearly and justify your conclusion.

8d
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2 marks

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

8e
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2 marks

Gorr and Greer use another sample to test their different suspicions using the same significance level. There is sufficient evidence from this sample to support Gorr’s suspicion. Explain whether Greer’s suspicion will also be supported using this sample.

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