Chi-squared with algebraic frequencies

A question is this type if and only if observed or expected frequencies are given algebraically (in terms of variables) and require manipulation or finding constraints.

5 questions · Standard +0.8

5.06b Fit prescribed distribution: chi-squared test
Sort by: Default | Easiest first | Hardest first
OCR Further Statistics AS 2018 June Q8
9 marks Challenging +1.2
8 The table shows the results of a random sample drawn from a population which is thought to have the distribution \(\mathrm { U } ( 20 )\).
Range\(1 \leqslant x \leqslant 8\)\(9 \leqslant x \leqslant 12\)\(13 \leqslant x \leqslant 20\)
Observed frequency12\(y\)\(28 - y\)
Find the range of values of \(y\) for which the data are not consistent with the distribution at the \(5 \%\) significance level. \section*{END OF QUESTION PAPER}
Edexcel S3 2022 June Q7
11 marks Challenging +1.2
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.
Number on die123456
Observed frequency\(x + 6\)\(x - 8\)\(x + 8\)\(x - 5\)\(x + 4\)\(x - 5\)
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.
OCR Further Statistics AS Specimen Q7
4 marks Standard +0.3
The discrete random variable \(X\) is equally likely to take values 0, 1 and 2. \(3N\) observations of \(X\) are obtained, and the observed frequencies corresponding to \(X = 0\), \(X = 1\) and \(X = 2\) are given in the following table.
\(x\)012
Observed frequency\(N - 1\)\(N - 1\)\(N + 2\)
The test statistic for a chi-squared goodness of fit test for the data is 0.3. Find the value of \(N\). [4]
OCR FS1 AS 2021 June Q4
9 marks Challenging +1.2
The table shows the results of a random sample drawn from a population which is thought to have the distribution \(U(20)\).
Range\(1 \leq x \leq 8\)\(9 \leq x \leq 12\)\(13 \leq x \leq 20\)
Observed frequency12\(y\)\(28 - y\)
Find the range of values of \(y\) for which the data are not consistent with the distribution at the \(5\%\) significance level. [9]
OCR FS1 AS 2017 Specimen Q7
4 marks Standard +0.3
The discrete random variable \(X\) is equally likely to take values 0, 1 and 2. \(3N\) observations of \(X\) are obtained, and the observed frequencies corresponding to \(X = 0\), \(X = 1\) and \(X = 2\) are given in the following table.
\(x\)012
Observed frequency\(N - 1\)\(N - 1\)\(N + 2\)
The test statistic for a chi-squared goodness of fit test for the data is 0.3. Find the value of \(N\). [4]