Hypothesis Testing using the Chi-squared Distribution (DP IB Applications & Interpretation (AI): HL): Exam Questions

A school in Copenhagen wants to find out whether students' favourite music genre is independent of whether they play a musical instrument. 400 students were asked whether they play a musical instrument and to choose their favourite genre from four options. The results are shown in the table below.

A ${\chi }^2$ test for independence is carried out at the 5% significance level. The critical value for this test is 7.815.

State the null and alternative hypotheses for this test.

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

Find the ${\chi }^2$ statistic for this data.

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

A carpet salesman is interested in how his sales are distributed over the months of the year. He records the number of carpets he sells in each of six months. The data are shown in the table.

A ${\chi }^2$ goodness of fit test is carried out at the 5% significance level to test whether the data fit a uniform distribution.

Write down the expected number of carpets sold in each month if the data fit a uniform distribution.

State the null and alternative hypotheses.

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

Find the $p$-value for this test.

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

A supermarket is interested in how applications for its loyalty scheme are spread over the working week. The numbers of applications received on each day of one week are recorded in the table.

The supermarket expects the applications to follow a uniform distribution. A ${\chi }^2$ goodness of fit test is carried out at the 10% significance level. The critical value for this test is 7.779.

Write down the expected number of applications on each day.

State the null and alternative hypotheses.

Write down

(i) the ${\chi }^2$ statistic

(ii) the $p$-value.

State whether the data fit a uniform distribution. Give a reason for your answer.

A survey was carried out to find out whether favourite colour is independent of age group. A group of 239 children, teenagers and adults were each asked to choose their favourite colour from red, yellow, blue and green.

The observed frequencies are recorded in the table below.

A ${\chi }^2$ test for independence is carried out at the 5% significance level. The critical value for this test is 12.592.

Complete the table of expected frequencies below. Give each value correct to one decimal place.

State the null and alternative hypotheses.

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

Find the ${\chi }^2$ statistic for this data.

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

Juan is investigating reaction times for his Mathematics IA. He wants to find out whether a student's reaction time is independent of whether the student plays video games regularly. He selects 60 students at random from his year group, asks each student whether they play video games regularly, and records the mean time, $R$ milliseconds, that the student takes to complete a reaction test.

Juan decides to carry out a ${\chi }^2$ test for independence on the data. The results are shown in the table below.

State the null and alternative hypotheses.

Calculate the expected frequency for the number of students who do not play video games regularly and have a "Quick" reaction time.

Explain why a ${\chi }^2$ test carried out on the data as it is currently organised would not be valid.

Suggest how Juan could reorganise the data so that the ${\chi }^2$ test is valid.

An international school of 294 students in Ho Chi Minh City holds a mock election alongside the UK general election. The age, $a$ years, of each voter is recorded on their ballot paper, along with their vote for the Conservative, Labour or Liberal party. The results are shown in the table below.

A ${\chi }^2$ test for independence is to be carried out on the data at the 10% significance level.

State the null and alternative hypotheses.

The table below shows the expected frequencies for the data.

(i) Explain why the table of observed frequencies should be redrawn before the test is carried out.

(ii) Redraw the table of observed frequencies by combining the results for $a\le 12$ with those for $12<a\le 14$.

Use the redrawn table from part (b) to find the $p$-value for the test, and state the conclusion of the test. Give a reason for your answer.

126 people are surveyed as part of an investigation into whether a person's preferred sport to watch is independent of their age group. Each person is asked to choose their preferred sport to watch from football, tennis, gymnastics and swimming.

The results of the survey are shown in the table below.

A ${\chi }^2$ test for independence is carried out on the data at the 5% significance level.

State the null and alternative hypotheses.

Use the table of critical values for a significance level of 5%, given below, to write down the critical value for this test.

Find the ${\chi }^2$ statistic.

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

A student is investigating the relationship between a country's GDP and its literacy rate for his IA. He has classified 140 countries as having low, medium or high GDP, and low, medium or high literacy rates. His results are shown in the table below.

He carries out a ${\chi }^2$ test for independence at the 5% significance level. The critical value for this test is 9.488.

State the null and alternative hypotheses.

Write down the number of degrees of freedom.

Find

(i) the ${\chi }^2$ statistic

(ii) the $p$-value.

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

Describe one issue with the way the data are presented that could make the conclusion of the test difficult to interpret.

A lottery machine contains 12 coloured balls: three red, five yellow, two blue and two green. When a button is pressed, a ball is selected at random. After its colour is noted, the ball is returned to the machine.

Write down the probability that the machine selects a yellow ball.

The machine is tested for bias. A ball is selected 180 times, and the results are recorded in the table below.

Complete the table of expected frequencies below, if the machine is not biased.

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

Find the $p$-value for the test, and state whether the machine is biased. Give a reason for your answer.

A computer game has 5 levels. At the end of each level a magic star may appear, doubling the score for that level. A focus group plays through all 5 levels 200 times, and the number of magic stars that appear in each game is recorded. The results are shown in the table below.

The game developer wants to investigate whether the number of magic stars in a game can be modelled by the binomial distribution $\mathrm{B}(5,0.5)$. A ${\chi }^2$ goodness of fit test is carried out at the 10% significance level.

State the null and alternative hypotheses.

Complete the table of expected frequencies below.

Write down the ${\chi }^2$ statistic.

The critical value for this test is 9.236.

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

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

A ${\chi }^2$ goodness of fit test is carried out at the 5% significance level to determine whether the blood pressures of the patients fit the distribution for healthy adults.

State the null and alternative hypotheses.

Complete the table of expected frequencies below. Give each frequency correct to four decimal places.

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

Find the $p$-value for this test, and state the conclusion of the test. Give a reason for your answer.

The heights of giraffes are normally distributed with mean 3.8 m and standard deviation 0.7 m. As part of a conservation project in Kenya, the heights of 350 giraffes are measured. The results are shown in the table below.

A ${\chi }^2$ goodness of fit test is carried out at the 10% significance level to determine whether the heights of the surveyed giraffes fit this normal distribution.

Complete the table below. Give each probability correct to four decimal places and each expected frequency correct to two decimal places.

Write down the null and alternative hypotheses.

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

Find the $p$-value for this test.

State whether the results of the test support the null hypothesis. Give a reason for your answer.

A cat shelter looks after 328 cats. As part of the health check of each new cat, its weight, $W$ pounds, is measured. The weights are summarised in the table below.

The weights of cats in the general cat population can be modelled by $W~\mathrm{N}(10.4,0.9^2)$. The shelter wants to find out whether the weights of the cats in its care fit this distribution.

A ${\chi }^2$ goodness of fit test is carried out at the 10% significance level.

Write down the null and alternative hypotheses.

Complete the table of expected frequencies below, if the shelter's cats follow the weight distribution of the general cat population. Give each expected frequency correct to two decimal places.

Find the $p$-value for this test.

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

A football coach wants to know whether the number of hours a team spends training in the week before a match affects the result of the match.

Data are collected for 100 matches. For each match, the number of hours spent training in the week before the match, $h$, is recorded with the result: win, lose or draw. Some of the data are shown in the table below.

Complete the table of observed data.

A ${\chi }^2$ test for independence is carried out at the 10% significance level.

State the null hypothesis.

Find the expected number of losses for a team that trains for at least 16 hours but fewer than 24 hours in the week before a match, if training time and match result are independent.

Find the number of degrees of freedom.

Find the ${\chi }^2$ statistic for this data.

The critical value for this test is 7.779.

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

A spinner used on a game show is divided into 6 equal sections coloured red, yellow, green, blue, purple and orange. There are reports that the spinner is not fair, so an experiment is carried out: the spinner is spun 720 times.

Write down the number of times the spinner would be expected to land on red if it is fair.

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

Write down the name of a statistical test that can be used to compare the observed results with the expected results, to determine whether the spinner is fair.

Write down the null and alternative hypotheses.

Carry out the test using your GDC, and find the $p$-value.

The test is carried out at the 1% significance level.

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

Comment on the suitability of the 1% significance level for this test. Justify your answer.

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

Explain how the two $p$-values can be used to compare the fairness of the two spinners.

The technology company Flooglesoft has recently lost some key employees to its main competitor. In response, the company wants to find out whether the monthly wages of its employees can be modelled by a normal distribution with standard deviation $400. The monthly wages, $w$ dollars, of Flooglesoft's employees are summarised in the table below.

Given that $\sum w=2124000$, find the unbiased estimate of the population mean monthly wage.

A ${\chi }^2$ goodness of fit test is to be carried out at the 10% significance level to test whether the monthly wages can be modelled by a normal distribution with standard deviation $400.

State the null and alternative hypotheses.

The table below shows the expected frequencies for the data, using the mean found in part (a).

Find the values of $a$, $b$ and $c$. Give your answers correct to 4 decimal places.

Explain why the test has 5 degrees of freedom.

The critical value for the test is 9.236.

Find the ${\chi }^2$ statistic, and state the conclusion of the test. Give a reason for your answer.

Leander works for a manufacturing company. Each shift lasts four hours, and during each shift she records the number of times a machine breaks down. The data from 160 shifts are shown in the table below.

Leander believes that the number of breakdowns during a four-hour shift can be modelled by a Poisson distribution. She decides to test her belief using a ${\chi }^2$ goodness of fit test at the 5% significance level.

Write down suitable null and alternative hypotheses to test Leander's belief.

Calculate an estimate of the mean number of breakdowns during a four-hour shift.

The table below shows the expected frequencies, using the mean found in part (b).

Find the values of $a$, $b$ and $c$. Give your answers correct to 3 decimal places.

Write down the number of degrees of freedom.

Find the $p$-value for the test, and determine whether the test supports Leander's belief. Justify your answer.

An IB student is investigating the study habits of students at her school. She wants to find out whether the time of day that a student prefers to study is independent of their year group. Years 12 and 13 at the school are the same size. She selects 20 students at random from Year 12 and 20 students at random from Year 13, and asks each of them to choose their preferred time of study: morning, afternoon or evening.

State the type of sampling used in her investigation.

She carries out a ${\chi }^2$ test for independence at the 5% significance level.

State the null and alternative hypotheses.

The results of the survey are listed below.

Complete the contingency table of observed frequencies below.

Show that, if preferred time of study and year group are independent, the expected number of Year 12 students who prefer to study in the morning is 6.5.

The critical value for the test is 5.991.

Find the ${\chi }^2$ statistic, and state the conclusion of the test in the context of the question. Give a reason for your answer.

A study has found that the top 20% of phone users spend more than 4.5 hours a day on their phones. A health charity wants to investigate whether phone usage is independent of age group. A survey of 85 people records each person's age group (child, teenager or adult) and the mean number of hours, $h$, that they spend on their phone each day.

The results are shown in the table below.

Complete the table of expected frequencies below. Give each value correct to two decimal places.

A ${\chi }^2$ test for independence is carried out at the 5% significance level, using the critical value from the table below.

Carry out the test. You should state the null and alternative hypotheses, the number of degrees of freedom, the critical value and the ${\chi }^2$ statistic, and the conclusion of the test in context.

State two steps that could be taken to make the conclusion of the test more valid.

Melanie is playing an online game with two digital four-sided dice, each numbered 1 to 4. In each turn, the pair of dice is rolled together three times. High points are scored for rolling doubles, that is, when both dice land on the same number.

The table below shows the number of doubles scored in each turn when Melanie plays 400 turns.

Melanie suspects that the number of doubles in a turn can be modelled by a binomial distribution with the probability of rolling a double equal to 0.25. A ${\chi }^2$ goodness of fit test is carried out at the 5% significance level to test Melanie's suspicion.

Carry out the test to show that the number of doubles cannot be modelled by the binomial distribution $\mathrm{B}(3,0.25)$.

Melanie still suspects that the number of doubles can be modelled by a binomial distribution, but with a different probability of rolling a double.

Assuming that the rolls of the dice are independent, use the table of results to estimate the probability of rolling a double.

Melanie now tests her new suspicion with a ${\chi }^2$ goodness of fit test at the 5% significance level, using the probability found in part (b).

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

(ii) Carry out the test, and state the conclusion of the test. Give a reason for your answer.

About 1.6 million sets of twins are born each year worldwide. A doctor models the sexes of the two babies in a set of twins as independent, with each baby equally likely to be male or female.

Use this model to find the probability that both babies in a randomly chosen set of twins are male.

Using the doctor's model, complete the probability table for the number of male babies in a set of twins.

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

Complete the table of expected frequencies below.

The numbers of male babies in the sets of twins born in this hospital during the year are shown in the table below.

A ${\chi }^2$ goodness of fit test is carried out at the 5% significance level. The critical value for the test is 5.991.

Find the ${\chi }^2$ statistic, and state the conclusion of the test. Give a reason for your answer.

Suggest a reason why the doctor's model does not fit the data.

A class of 40 students take part in a science lesson on reaction times. A computer program measures the time taken, in milliseconds, for each student to press a button with their dominant hand after a green light is shown.

One student carries out a ${\chi }^2$ goodness of fit test at the 10% significance level, to test whether the class's reaction times follow a normal distribution. The $p$-value is 0.130.

State the conclusion of her test. Justify your answer.

A second class of 40 students repeats the experiment using their non-dominant hand. A ${\chi }^2$ goodness of fit test, also at the 10% significance level, gives a $p$-value of 0.0854.

Compare the results of the two tests. Justify your answer.

The mean reaction times of the two classes are then compared using a two-tailed, two-sample $t$-test at the 10% significance level. The $p$-value for this test is 0.0667.

Explain what can be concluded from this test.

Gordon takes the bus to work at the same time each day, and he records the time, $t$ minutes, of each journey. His data are shown in the table below.

Gordon wants to use a ${\chi }^2$ goodness of fit test at the 10% significance level to test whether the journey times can be modelled by a normal distribution.

State the null and alternative hypotheses.

The mean journey time of the sample is 25.3 minutes and the standard deviation of the sample is 7.38 minutes.

Write down the unbiased estimate of the population mean journey time, and find the unbiased estimate of the population variance.

Using the unbiased estimates from part (b), complete the table of expected frequencies below. Give each expected frequency correct to 3 decimal places.

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

The critical value for the test is 4.605.

Find the ${\chi }^2$ statistic, and state the conclusion of the test. Give a reason for your answer.