Chi-squared with algebraic frequencies

7 The following table shows observed frequencies, where \(x\) is an integer, from an experiment to test whether or not a six-sided die is biased. A goodness of fit test is conducted to determine if there is evidence that the die is biased.

1. Write down suitable null and alternative hypotheses for this test. It is found that the null hypothesis is not rejected at the \(5 \%\) significance level.
2. Hence
   1. find the minimum value of \(x\)
   2. determine the minimum number of times the die was rolled.

1. Charlie carried out a survey on the main type of investment people have.

The contingency table below shows the results of a survey of a random sample of people.

1. Find an expression, in terms of \(a , b , c\) and \(d\), for the difference between the observed and the expected value \(( O - E )\) for the group whose main type of investment is Bonds and are aged 45-75
   Express your answer as a single fraction in its simplest form. Given that \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 9.62\) for this information,
2. test, at the \(5 \%\) level of significance, whether or not there is evidence of an association between the age of a person and the main type of investment they have. You should state your hypotheses, critical value and conclusion clearly. You may assume that no cells need to be combined.

7 The discrete random variable \(X\) is equally likely to take values 0,1 and 2 . \(3 N\) observations of \(X\) are obtained, and the observed frequencies corresponding to \(X = 0 , X = 1\) and \(X = 2\) are given in the following table. The test statistic for a chi-squared goodness of fit test for the data is 0.3 . Find the value of \(N\).