Chi-squared tests

David carries out an experiment with 4 identical dice, each with faces numbered 1 to 6. He rolls the 4 dice and counts the number of dice showing an even number on the uppermost face. He repeats this 150 times. The results are summarised in the table below.

David defines the random variable C as the number of dice showing an even number on the uppermost face when the four dice are thrown. David claims that C ~ B(4, 0.5)

Stating your hypotheses clearly and using a 1% level of significance, test David’s claim. Show your working clearly. John claims that C ~ B(4, p)

Calculate an estimate of the value of p from the summary of the results of David’s experiment. Show your working clearly. John decides to test his claim. He calculates expected frequencies using the results of David’s experiment and obtains the following table.

Calculate, to 2 decimal places, the value of d and the value of e

State suitable hypotheses to test John’s claim. John obtained a test statistic of 16.9 and carries out a test at the 1% level of significance.

State what conclusion John should make about his claim.

The pointer shown in Figure 1 is spun so that it comes to rest between 0 and 360 degrees. Linda claims that it is equally likely to come to rest at any point between 0 and 360 degrees. She spins the pointer 100 times and her results are summarised in the table below. She calculates expected frequencies for some of the possible outcomes and these are also given in the table below.

Find the values of the missing expected frequencies a, b and c.

Stating your hypotheses clearly and using a 5% level of significance, test whether or not Linda’s claim is supported by these data.

A psychologist carries out a survey of the perceived body weight of 150 randomly chosen people. He asks them if they think they are underweight, about right or overweight. His results are summarised in the table below.

The psychologist calculates two of the expected frequencies, to 2 decimal places, for a test of independence between perceived body weight and gender. These results are shown in the table below.

Complete the table of expected frequencies shown above.

Test, at the 10% level of significance, whether or not perceived body weight is independent of gender. State your hypotheses clearly.

The psychologist now combines the male and female data to test whether or not body weight types are chosen equally.

Find the smallest significance level, from the tables in the formula booklet, for which there is evidence of a preference.

A new drug to vaccinate against influenza was given to 110 randomly chosen volunteers. The volunteers were given the drug in one of 3 different concentrations, A, B and C, and then were monitored to see if they caught influenza. The results are shown in the table

Test, at the 10% level of significance, whether or not there is an association between catching influenza and the concentration of the new drug. State your hypotheses and show your working clearly. You should state your expected frequencies to 2 decimal places.

An airport manager carries out a survey of families and their luggage. Each family is allowed to check in a maximum of 4 suitcases. She observes 50 families at the check-in desk and counts the total number of suitcases each family checks in. The data are summarised in the table below.

The manager claims that the data can be modelled by a binomial distribution with p = 0.3

Test the manager’s claim at the 5% level of significance. State your hypotheses clearly. Show your working clearly and give your expected frequencies to 2 decimal places.

The manager also carries out a survey of the time taken by passengers to check in. She records the number of passengers that check in during each of 100 five-minute intervals. The manager makes a new claim that these data can be modelled by a Poisson distribution. She calculates the expected frequencies given in the table below.

Find the value of r and the value of s giving your answers to 2 decimal places.

Stating your hypotheses clearly, use a 1% level of significance to test the manager’s new claim.

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The sketch in Figure 1 represents a target which consists of 4 regions formed from 4 concentric circles of radii 4 cm, 7 cm, 9 cm and 10 cm. The regions are coloured as labelled in Figure 1. A random sample of 100 children each choose a point on the target and their results are summarised in the table below.

Caitland is trying to model the distribution of the points chosen by the children. She defines the random variable D to be the distance, in cm, of a point from the centre of the target and assumes D ~ U[0, 10].

Stating your hypotheses clearly and using a 1% level of significance, test whether or not U[0, 10] is a suitable model for these data.

Question 6 continues on page 18

Question 6 continues on page 18

Henry claims that the points are randomly distributed over the target and the probability of a point being in any particular region is proportional to the area of that region. He calculates expected frequencies and obtains the following table.

Find the value of r and the value of s.

Henry obtained a test statistic of 6.188 and no groups were pooled.

State what conclusion Henry should make about his claim.

Phoebe believes that the children chose the region of the target according to colour. She believes that boys and girls would favour different colours and splits the original data by gender to obtain the following table.

State suitable hypotheses to test Phoebe’s belief.

Phoebe calculated the following expected frequencies to carry out a suitable test.

Show how the value of 25.35 was obtained.

Phoebe carried out the test using 2 degrees of freedom and a 10% level of significance. She obtained a test statistic of 1.411

Explain clearly why Phoebe used 2 degrees of freedom.

Stating your critical value clearly, determine whether or not these data support Phoebe’s belief.

A doctor takes a random sample of 100 patients and measures their intake of saturated fats in their food and the level of cholesterol in their blood. The results are summarised in the table below.

Using a 5% level of significance, test whether or not there is an association between cholesterol level and intake of saturated fats. State your hypotheses and show your working clearly.

Customers at a post office are timed to see how long they wait until being served at the counter. A random sample of 50 customers is chosen and their waiting times, x minutes, are summarised in Table 1.

Show that an estimate of = 5.49 and an estimate of s²_x = 6.88

The post office manager believes that the customers’ waiting times can be modelled by a normal distribution. Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2.

Find the value of a and the value of b.

Test, at the 5% level of significance, the manager’s belief. State your hypotheses clearly.

John thinks that a person’s eye colour is related to their hair colour. He takes a random sample of 600 people and records their eye and hair colours. The results are shown in

John carries out a ² test in order to test whether eye colour and hair colour are related. He calculates the expected frequencies shown in Table 2.

Show how the value 47 in Table 2 has been calculated.

Write down the number of degrees of freedom John should use in this ² test.

Given that the value of the ² statistic is 20.6, to 3 significant figures,

find the smallest value of for which the null hypothesis will be rejected at the % level of significance.

Use the data from Table 1 to test at the 5% level of significance whether or not the proportions of people in the population with black, brown, red and blonde hair are in the ratio 2:6:1:3 State your hypotheses clearly.

Two breeds of chicken are surveyed to measure their egg yield. The results are shown in the table below.

Showing each stage of your working clearly, test, at the 5% significance level, whether or not there is an association between egg yield and breed of chicken. State your hypotheses clearly. (10)

A total of 100 random samples of 6 items are selected from a production line in a factory and the number of defective items in each sample is recorded. The results are summarised in the table below.

Show that the mean number of defective items per sample is 2.91 (2) A factory manager suggests that the data can be modelled by a binomial distribution with

shown in the table below.

Calculate the value of a and the value of b giving your answers to 2 decimal places.(4)

Test, at the 5% level, whether or not the binomial distribution is a suitable model for the number of defective items in samples of 6 items. State your hypotheses clearly. (8)

A factory manufactures batches of an electronic component. Each component is manufactured in one of three shifts. A component may have one of two types of defect, D₁ or D₂, at the end of the manufacturing process. A production manager believes that the type of defect is dependent upon the shift that manufactured the component. He examines 200 randomly selected defective components and classifies them by defect type and shift. The results are shown in the table below.

Stating your hypotheses, test, at the 10% level of significance, whether or not there is evidence to support the manager’s belief. Show your working clearly.

The number of hurricanes per year in a particular region was recorded over 80 years. The results are summarised in Table 1 below.

Write down two assumptions that will support modelling the number of hurricanes per year by a Poisson distribution.

Show that the mean number of hurricanes per year from Table 1 is 4.4875

Use the answer in part (b) to calculate the expected frequencies r and s given in Table 2 below to 2 decimal places.

Test, at the 5% level of significance, whether or not the data can be modelled by a Poisson distribution. State your hypotheses clearly.

The number of goals scored by a football team is recorded for 100 games. The results are summarised in Table 1 below.

Calculate the mean number of goals scored per game.

The manager claimed that the number of goals scored per match follows a Poisson distribution. He used the answer in part (a) to calculate the expected frequencies given in

Find the value of r and the value of s giving your answers to 3 decimal places.

Stating your hypotheses clearly, use a 5% level of significance to test the manager’s claim.