Test statistic given, complete the test

4 The table summarises the usual method of travelling to school for 200 randomly selected pupils from primary and secondary schools in a city.

1. Write down null and alternative hypotheses for a test to examine whether there is any association between method of travel and type of school.
2. Calculate the expected frequency for primary school bus users. Calculate also the corresponding contribution to the test statistic for the usual \(\chi ^ { 2 }\) test.
3. Given that the value of the test statistic for the usual \(\chi ^ { 2 }\) test is 42.64 , carry out the test at the \(5 \%\) level of significance, stating your conclusion clearly. The mean travel time for pupils who travel by bus is known to be 18.3 minutes. A survey is carried out to determine whether the mean travel time to school by car is different from 18.3 minutes. In the survey, 20 pupils who travel by car are selected at random. Their mean travel time is found to be 22.4 minutes.
4. Assuming that car travel times are Normally distributed with standard deviation 8.0 minutes, carry out a test at the \(10 \%\) level, stating your hypotheses and conclusion clearly.
5. Comment on the suggestion that pupils should use a bus if they want to get to school quickly.

5. A Policy Unit wished to find out whether attitudes to the European Union varied with age. It conducted a survey asking 200 individuals to which of three age groups they belonged and whether they regarded themselves as generally pro-Europe or Eurosceptic. The results are shown in the table below.

1. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether attitudes to Europe are associated with age.
   (11 marks)
   The survey also asked people if they voted at the last election. When the above test was repeated using only the results from those who had voted a value of 4.872 was calculated for \(\sum \frac { ( O - E ) ^ { 2 } } { E }\). No classes were combined.
2. Find if this value leads to a different result.

1. Sharma believes that each computer game he sells appeals equally to all age ranges.

To investigate this, he takes a random sample of 100 people who play these games and asks them which of the games \(A , B\) or \(C\) they prefer.
The results are summarised in the table below.

1. Write down hypotheses for a suitable test to assess Sharma's belief.
2. For the test, calculate the expected frequency for
   1. those players aged under 20 who prefer game \(C\)
   2. those players aged between 20 and 30 who prefer game \(A\)
3. State the degrees of freedom of the test statistic for this test. Sharma correctly calculates the test statistic for this test to be 11.542 (to 3 decimal places).
4. Using a \(5 \%\) significance level, and stating your critical value, comment on Sharma's belief.

A random sample of the invoices, for books purchased by the customers of a large bookshop, was classified by book cover (hardback, paperback) and type of book (novel, textbook, general interest). As part of the analysis of these invoices, an approximate \(\chi^2\) statistic was calculated and found to be 11.09. Assuming that there was no need to amalgamate any of the classifications, carry out an appropriate test to determine whether or not there was any association between book cover and type of book. State your hypotheses clearly and use a 5% level of significance. [6]

In a test of association of two factors, \(A\) and \(B\), a \(2 \times 2\) contingency table yielded \(5.63\) for the value of \(\chi^2\) with Yates' correction.

1. State the null hypothesis and alternative hypothesis for the test. [1]
2. State how Yates' correction is applied, and whether it increases or decreases the value of \(\chi^2\). [2]
3. Carry out the test at the \(2\frac{1}{2}\%\) significance level. [3]

A restaurant has asked Sylvia to conduct a \(\chi^2\) test for association between meal ordered and age of customer.

1. State the hypotheses that Sylvia should use for her test. [1 mark]
2. Sylvia correctly calculates her value of the test statistic to be 44.1 She uses a 5% level of significance and the degrees of freedom for the test is 30 Sylvia accepts the null hypothesis. Explain whether or not Sylvia was correct to accept the null hypothesis. [4 marks]
3. State in context the correct conclusion to Sylvia's test. [1 mark]