 # Exam-Style Questions on Significance

## Problems on Significance adapted from questions set in previous Mathematics exams.

### 1.

IB Studies

A group of students were asked how they get to school each day. The results are shown in the following table

 Walk Bicycle Car Bus Total Male 22 36 35 56 149 Female 32 46 45 50 173 Total 54 82 80 106 322

A $$\chi^2$$ test was preformed to see if there was a connection between gender and the method of travelling to school.

(a) Write down what the null hypothesis, Ho, might have been for this test.

(b) Find the expected frequency for the females who cycle to school.

(c) Find the $$\chi^2$$ statistic.

(d) State whether or not the null hypothesis is accepted at a 5% significance level. Give a reason for your answer.

### 2.

IB Studies

One hundred and sixty students are asked how many portions of fruit or vegetables on average they have per day. The data collected is organized in the following table.

 2 or fewer 3 or 4 More than 4 Total Male 25 35 25 85 Female 22 30 23 75 Total 47 65 48 160

A $$\chi ^2$$ test is carried out at the 5% level of significance.

(a) Write down the null hypothesis, Ho, for this test.

(b) Write down the number of degrees of freedom for this test.

(c) Find the $$\chi ^2_{calc}$$ value for this data.

(d) Find the expected number of females that have more than 4 portions per day (correct to the nearest integer).

(e) Decide whether Ho must be accepted.

### 3.

IB Studies

The older students from Glee High School are required to follow a two year IB Mathematics course. Data were gathered from a sample of 242 students regarding their choice of course. The following data were recorded.

 Gender Studies Standard Higher Total Male 35 15 21 71 Female 60 30 81 171 Total 95 45 102 242

A $$\chi ^2$$ test was carried out at the 5% significance level to analyse the relationship between gender and choice of mathematics course.

(a) Write down the null hypothesis, Ho, for this test.

(b) Find the expected value of female students on the Studies course.

(c) Write down the number of degrees of freedom.

(d) Use your graphic display calculator to determine the $$\chi ^2$$calc value.

One student is chosen at random from the 242 students.

(f) Find the probability that this student is male.

(g) Find the probability that the student chosen at random is on the Standard course.

Two students are chosen at random from the 242 students.

(h) Find the probability that both are on the Studies course.

(i) Find the probability that neither are on the Higher course.

### 4.

IB Studies

The staff of a shop kept records of who bought smart phones during the month of February one year. They looked at the numbers of phones bought by gender and the size of the screens. This information is shown in the table below; S represents the size of the screen in centimetres.

S ≤ 12 12 < S ≤ 1616 < S ≤ 20S > 20Total
Female781135328272
Male337815368332
Total11119120696604

The shop manager wants to use this information to predict the probability of selling these sizes of phone screens for the following month.

(a) Use the table to find the probability that a phone will be bought by a female.

(b) Find the probability that a phone with a screen size of 12cm < S ≤ 16cm will be bought.

(c) Find the probability that a phone with a screen size of 12cm < S ≤ 16cm will be bought by a female.

(d) Find the probability that a phone with a screen size greater than 20cm will be bought given that it is bought by a male.

The manager wants to determine whether the screen size is independent of gender so a chi-squared test is performed at the 1% significance level.

(e) Write down the null hypothesis.

(f) Find the expected frequency for females who bought a screen size of 12cm < S ≤ 16cm to the nearest integer.

(g) Write down the number of degrees of freedom.

(h) Write down the $$\chi ^2$$ calculated value.

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

### 5.

IB Studies

A survey was conducted asking people's opinions on whether mouthwash should be used before or after brushing teeth. The results are shown as follows.

 Age below 18 Age 18 to 30 Age over 30 Total Before 28 29 33 90 After 24 21 15 60 Total 52 50 48 150

A $$\chi ^2$$ test at the 1% significance level was conducted.

(a) State $$H_0$$, the null hypothesis for the test;

(b) State $$H_1$$, the alternative hypothesis for the test.

(c) Write down the number of degrees of freedom.

(d) Find the expected frequency of people between the age 18 to 30 who believe that mouthwash should be used before brushing.

(e) Find the $$\chi ^2$$ statistic;

(f) Find the associated p-value for the test.

(g) Determine, giving a reason, whether $$H_0$$ should be accepted.

The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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