Find critical region for F-test

A question is this type if and only if it asks the student to determine the set of values of a sample variance (or F-statistic) that would lead to rejection of the null hypothesis in an F-test, expressed as an inequality or interval.

3 questions · Standard +0.6

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Edexcel FS2 2022 June Q5
8 marks Standard +0.8
  1. The concentration of an air pollutant is measured in micrograms \(/ \mathrm { m } ^ { 3 }\)
Samples of air were taken at two different sites and the concentration of this particular air pollutant was recorded. For Site \(A\) the summary statistics are shown below.
\cline { 2 - 3 } \multicolumn{1}{c|}{}number of samples\(S _ { A } ^ { 2 }\)
Site \(A\)136.39
For Site \(B\) there were 9 samples of air taken.
A test of the hypothesis \(\mathrm { H } _ { 0 } : \sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 }\) against the hypothesis \(\mathrm { H } _ { 1 } : \sigma _ { A } ^ { 2 } \neq \sigma _ { B } ^ { 2 }\) is carried out using a \(2 \%\) level of significance.
  1. State a necessary assumption required to carry out the test. Given that the assumption in part (a) holds,
  2. find the set of values of \(s _ { B } ^ { 2 }\) that would lead to the null hypothesis being rejected,
  3. find a 99\% confidence interval for the variance of the concentration of the air pollutant at Site A.
Edexcel S4 Q1
2 marks Standard +0.3
Find the value of the constant \(a\) such that $$\text{P}(a < F_{8,10} < 3.07) = 0.94$$ [2]
Edexcel S4 2002 June Q1
3 marks Standard +0.8
The random variable \(X\) has an \(F\) distribution with 10 and 12 degrees of freedom. Find \(a\) and \(b\) such that \(\text{P}(a < X < b) = 0.90\). [3]