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1a. [1 mark]

Abhinav carries out a χ2 test at the 1 % significance level to determine whether a person’s gender

impacts their chosen professional field: engineering, medicine or law.

He surveyed 220 people and the results are shown in the table.

State the null hypothesis, H0, for this test.

1b. [2 marks]

Calculate the expected number of male engineers.

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1c. [2 marks]

Find the p-value for this test.

1d. [1 mark]

Abhinav rejects H0.

State a reason why Abhinav is incorrect in doing so.

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2a. [1 mark]

On one day 180 flights arrived at a particular airport. The distance travelled and the arrival status for

each incoming flight was recorded. The flight was then classified as on time, slightly delayed, or heavily

delayed.

The results are shown in the following table.

A χ2 test is carried out at the 10 % significance level to determine whether the arrival status of

incoming flights is independent of the distance travelled.

State the alternative hypothesis.

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2b. [2 marks]

Calculate the expected frequency of flights travelling at most 500 km and arriving slightly delayed.

2c. [1 mark]

Write down the number of degrees of freedom.

2d. [2 marks]

Write down the χ2 statistic.

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2e. [1 mark]

Write down the associated p-value.

2f. [2 marks]

The critical value for this test is 7.779.

State, with a reason, whether you would reject the null hypothesis.

2g. [2 marks]

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A flight is chosen at random from the 180 recorded flights.

Write down the probability that this flight arrived on time.

2h. [2 marks]

Given that this flight was not heavily delayed, find the probability that it travelled between 500 km and

5000 km.

2i. [3 marks]

Two flights are chosen at random from those which were slightly delayed.

Find the probability that each of these flights travelled at least 5000 km.

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3a. [2 marks]

The weight, W, of basketball players in a tournament is found to be normally distributed with a mean of

65 kg and a standard deviation of 5 kg.

Find the probability that a basketball player has a weight that is less than 61 kg.

3b. [2 marks]

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In a training session there are 40 basketball players.

Find the expected number of players with a weight less than 61 kg in this training session.

3c. [2 marks]

The probability that a basketball player has a weight that is within 1.5 standard deviations of the mean

is q.

Sketch a normal curve to represent this probability.

3d. [1 mark]

Find the value of q.

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3e. [2 marks]

Given that P(W > k) = 0.225 , find the value of k.

3f. [1 mark]

A basketball coach observed 60 of her players to determine whether their performance and their

weight were independent of each other. Her observations were recorded as shown in the table.

She decided to conduct a χ 2 test for independence at the 5% significance level.

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For this test state the null hypothesis.

3g. [2 marks]

For this test find the p-value.

3h. [2 marks]

State a conclusion for this test. Justify your answer.

4a. [1 mark]

A group of 800 students answered 40 questions on a category of their choice out of History, Science and

Literature.

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For each student the category and the number of correct answers, , was recorded. The results

obtained are represented in the following table.

State whether is a discrete or a continuous variable.

4b. [1 mark]

Write down, for , the modal class;

4c. [1 mark]

Write down, for , the mid-interval value of the modal class.

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4d. [2 marks]

Use your graphic display calculator to estimate the mean of ;

4e. [1 mark]

Use your graphic display calculator to estimate the standard deviation of .

4f. [2 marks]

A test at the 5% significance level is carried out on the results. The critical value for this test is

12.592.

Find the expected frequency of students choosing the Science category and obtaining 31 to 40 correct

answers.

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4g. [1 mark]

Write down the null hypothesis for this test;

4h. [1 mark]

Write down the number of degrees of freedom.

4i. [1 mark]

Write down the -value for the test;

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4j. [2 marks]

Write down the statistic.

4k. [2 marks]

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

5a. [1 mark]

In a school, students in grades 9 to 12 were asked to select their preferred drink. The choices were

milk, juice and water. The data obtained are organized in the following table.14

A test is carried out at the 5% significance level with hypotheses:

The critical value for this test is 12.6.

Write down the value of .

5b. [1 mark]

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

5c. [2 marks]

Use your graphic display calculator to find the statistic for this test.

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5d. [2 marks]

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

6a. [2 marks]

In a school, all Mathematical Studies SL students were given a test. The test contained four questions,

each one on a different topic from the syllabus. The quality of each response was classified as

satisfactory or not satisfactory. Each student answered only three of the four questions, each on a

separate answer sheet.

The table below shows the number of satisfactory and not satisfactory responses for each question.

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If the teacher chooses a response at random, find the probability that it is a response to the Calculus

question;

6b. [2 marks]

If the teacher chooses a response at random, find the probability that it is a satisfactory response to the

Calculus question;

6c. [2 marks]

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If the teacher chooses a response at random, find the probability that it is a satisfactory response, given

that it is a response to the Calculus question.

6d. [3 marks]

The teacher groups the responses by topic, and chooses two responses to the Logic question. Find the

probability that both are not satisfactory.

6e. [1 mark]

A test is carried out at the 5% significance level for the data in the table.

State the null hypothesis for this test.

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6f. [1 mark]

Show that the expected frequency of satisfactory Calculus responses is 12.

6g. [1 mark]

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

6h. [2 marks]

Use your graphic display calculator to find the statistic for this data.

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6i. [2 marks]

The critical value for this test is 7.815.

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

7a. [2 marks]

A hospital collected data from 1000 patients in four hospital wards to review the quality of its

healthcare. The data, showing the number of patients who became infected during their stay in hospital,

was recorded in the following table.

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A -test was performed at the 5% significance level.

The critical value for this test is 7.815.

The null hypothesis for the test is

: Becoming infected during a stay in the hospital is independent of the ward.

Find the expected frequency of the patients who became infected whilst in Nightingale ward.

7b. [2 marks]

For this test, write down the statistic.

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7c. [2 marks]

State, giving a reason, whether the null hypothesis should be rejected.

8a. [1 mark]

The manager of a travel agency surveyed 1200 travellers. She wanted to find out whether there was a

relationship between a traveller’s age and their preferred destination. The travellers were asked to

complete the following survey.

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A test was carried out, at the significance level, on the data collected.

Write down the null hypothesis.

8b. [2 marks]

Find the number of degrees of freedom.

8c. [1 mark]

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The critical value of this test is .

Use this information to write down the values of the statistic for which the null hypothesis is

rejected.

8d. [2 marks]

From the travellers taking part in the survey, 285 were 61 years or older and 420 preferred Tokyo.

Calculate the expected number of travellers who preferred Tokyo and were 61 years or older.

9a. [2 marks]

A survey was conducted among a random sample of people about their favourite TV show. People were

classified by gender and by TV show preference (Sports, Documentary, News and Reality TV).

The results are shown in the contingency table below.

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Find the expected number of females who prefer documentary shows.

 

9b. [2 marks]

A test at the  significance level is used to determine whether TV show preference

is independent of gender.

Write down the -value for the test.

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9c. [2 marks]

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

10a. [1 mark]

The producer of a TV dancing show asked a group of 150 viewers their age and the type of Latin dance

they preferred. The types of Latin dances in the show were Argentine tango, Samba, Rumba and Cha-

cha-cha. The data obtained were organized in the following table.

A test was carried out, at the 5% significance level.

Write down the null hypothesis for this test.

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10b. [1 mark]

Write down the observed number of viewers who preferred Rumba and were older than 20 years old.

10c. [2 marks]

Use your graphic display calculator to find the -value for this test.

10d. [2 marks]

The producer claims that the type of Latin dance a viewer preferred is independent of their age.

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Decide whether this claim is justified. Give a reason for your decision.

11a. [2 marks]

In a debate on voting, a survey was conducted. The survey asked people’s opinion on whether or not

the minimum voting age should be reduced to 16 years of age. The results are shown as follows.

A test at the 1% significance level was conducted. The critical value of the test is 9.21.

State

(i)     , the null hypothesis for the test;

(ii)     , the alternative hypothesis for the test.

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11b. [1 mark]

Write down the number of degrees of freedom.

11c. [2 marks]

Show that the expected frequency of those between the ages of 26 and 40 who oppose the reduction in

the voting age is 21.5, correct to three significant figures.

11d. [3 marks]

Find

(i)     the statistic;

(ii)     the associated -value for the test.

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11e. [2 marks]

Determine, giving a reason, whether should be accepted.

12a. [1 mark]

Minta surveyed students from her school about their preferred morning snack from a choice of an

apple, a fruit salad or a smoothie.

She surveyed 350 students, of whom 210 are female.

She performed a test at the 5% significance level to determine whether there is a relationship

between the choice of morning snack and gender.

State Minta’s null hypothesis.

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12b. [1 mark]

State the number of degrees of freedom.

12c. [2 marks]

150 students showed a preference for a smoothie.

Calculate the expected number of female students who chose a smoothie.

12d. [2 marks]

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Minta found that the calculated value of the test was 3.576. The critical value at the 5% significance

level is .

State Minta’s conclusion. Give a reason for your answer.

13a. [1 mark]

A study was carried out to determine whether the country chosen by students for their university

studies was influenced by a person’s gender. A random sample was taken. The results are shown in the

following table.

A test was performed at the 1% significance level.

The critical value for this test is 9.210.

State the null hypothesis.

13b. [1 mark]

Write down the number of degrees of freedom.

13c. [2 marks]

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Write down

(i)     the statistic;

(ii)     the associated p-value.

13d. [2 marks]

State, giving a reason, whether the null hypothesis should be accepted.

14a. [2 marks]

A group of 100 students gave the following responses to the question of how they get to school.

A test for independence was conducted at the significance level. The null hypothesis was

defined as

: Method of getting to school is independent of gender.

Find the expected frequency for the females who use public transport to get to school.

14b. [2 marks]

Find the statistic.

14c. [2 marks]

The critical value is at the significance level.

State whether or not the null hypothesis is accepted. Give a reason for your answer.

15a. [1 mark]

spectators at a swimming championship were asked which, of four swimming styles, was the one

they preferred to watch.

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The results of their responses are shown in the table.

A test was conducted at the significance level.

Write down the null hypothesis for this test.

15b. [1 mark]

Write down the number of degrees of freedom.

15c. [2 marks]

Write down the value of .

15d. [2 marks]

The critical value, at the significance level, is .

State, giving a reason, the conclusion to the test.

16a. [1 mark]

A market researcher surveyed men and women about their preferred holiday destination. The holiday

destinations were Antigua, Barbados, Cuba, Guadeloupe and Jamaica. A test for independence was

conducted at the 5 % significance level.

The calculated value was found to be 8.73.

Write down the null hypothesis.

16b. [2 marks]

Find the number of degrees of freedom for this test.

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16c. [1 mark]

Write down the critical value for this test.

16d. [2 marks]

State the conclusion of this test. Give a reason for your decision.

17a. [2 marks]

An agricultural cooperative uses three brands of fertilizer, A, B and C, on 120 different crops. The crop

yields are classified as High, Medium or Low.

The data collected are organized in the table below.

The agricultural cooperative decides to conduct a chi-squared test at the 1 % significance level using

the data.

State the null hypothesis, H0, for the test.

17b. [1 mark]

Write down the number of degrees of freedom.

17c. [1 mark]

Write down the critical value for the test.

17d. [2 marks]

Show that the expected number of Medium Yield crops using Fertilizer C is 17, correct to the nearest

integer.

17e. [3 marks]

Use your graphic display calculator to find for the data

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(i) the calculated value, ;

(ii) the p-value.

17f. [2 marks]

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

18a. [6 marks]

A store recorded their sales of televisions during the 2010 football World Cup. They looked at the

numbers of televisions bought by gender and the size of the television screens.

This information is shown in the table below; S represents the size of the television screen in inches.

The store wants to use this information to predict the probability of selling these sizes of televisions for

the 2014 football World Cup.

Use the table to find the probability that

(i) a television will be bought by a female;

(ii) a television with a screen size of 32 < S ≤ 46 will be bought;

(iii) a television with a screen size of 32 < S ≤ 46 will be bought by a female;

(iv) a television with a screen size greater than 46 inches will be bought, given that it is bought by a

male.

18b. [1 mark]

The manager of the store wants to determine whether the screen size is independent of gender. A Chi-

squared test is performed at the 1 % significance level.

Write down the null hypothesis.

18c. [2 marks]

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The manager of the store wants to determine whether the screen size is independent of gender. A Chi-

squared test is performed at the 1 % significance level.

Show that the expected frequency for females who bought a screen size of 32 < S ≤ 46, is 79, correct to

the nearest integer.

18d. [1 mark]

The manager of the store wants to determine whether the screen size is independent of gender. A Chi-

squared test is performed at the 1 % significance level.

Write down the number of degrees of freedom.

18e. [2 marks]

The manager of the store wants to determine whether the screen size is independent of gender. A Chi-

squared test is performed at the 1 % significance level.

Write down the calculated value.

18f. [1 mark]

The manager of the store wants to determine whether the screen size is independent of gender. A Chi-

squared test is performed at the 1 % significance level.

Write down the critical value for this test.

18g. [2 marks]

The manager of the store wants to determine whether the screen size is independent of gender. A Chi-

squared test is performed at the 1 % significance level.

Determine if the null hypothesis should be accepted. Give a reason for your answer.

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