Challenging +1.3 This question requires understanding of hypothesis testing terminology (Type II error), calculating critical regions for discrete distributions at 1% significance level, and then computing power under the alternative hypothesis. It combines multiple statistical concepts and requires careful probability calculations with Poisson distributions, going beyond routine hypothesis test questions but using standard S2 techniques.
The random variable \(R\) has the distribution Po\((\lambda)\). A significance test is carried out at the 1% level of the null hypothesis H\(_0\): \(\lambda = 11\) against H\(_1\): \(\lambda > 11\), based on a single observation of \(R\). Given that in fact the value of \(\lambda\) is 14, find the probability that the result of the test is incorrect, and give the technical name for such an incorrect outcome. You should show the values of any relevant probabilities. [6]
Critical region is \(\geq 20\). [CV 19 or 20 can imply first M1] [If only one probability shown, assume this is CR]
\(P(\geq 20) = 0.0093\)
A1
Probability 0.9907 or 0.0093 seen (even if CR is wrong) [from \(\lambda = 11\)]
\(P(\leq 19
\lambda = 14)\)
M1
\(= 0.9235\)
A1
Answer in range \([0.923, 0.924]\)
\(\text{Type II error}\)
B1
"Type II error" stated, allow "Type 2" (6 marks)
SC1: \(P(\leq 19) = 0.9907\) so CR is \(\geq 19\): M1A0A1M1A0B1 max 4/6
SC2: \(\lambda = 14\) used throughout, e.g. \(P(\geq 23) = 0.0093\): max B1
$P(\leq 19 | \lambda = 11) = 0.9907$ | M1 | Attempt to find critical region from $\lambda = 11$, allow even if tail wrong
$\text{so critical region} \geq 20$ | A1 | Critical region is $\geq 20$. [CV 19 or 20 can imply first M1] [If only one probability shown, assume this is CR]
$P(\geq 20) = 0.0093$ | A1 | Probability 0.9907 or 0.0093 seen (even if CR is wrong) [from $\lambda = 11$]
$P(\leq 19 | \lambda = 14)$ | M1 | Final P(not in CR | $\lambda = 14$), now be LH tail, i.e. 0.8826
$= 0.9235$ | A1 | Answer in range $[0.923, 0.924]$
$\text{Type II error}$ | B1 | "Type II error" stated, allow "Type 2" (6 marks)
| | SC1: $P(\leq 19) = 0.9907$ so CR is $\geq 19$: M1A0A1M1A0B1 max 4/6
| | SC2: $\lambda = 14$ used throughout, e.g. $P(\geq 23) = 0.0093$: max B1
The random variable $R$ has the distribution Po$(\lambda)$. A significance test is carried out at the 1% level of the null hypothesis H$_0$: $\lambda = 11$ against H$_1$: $\lambda > 11$, based on a single observation of $R$. Given that in fact the value of $\lambda$ is 14, find the probability that the result of the test is incorrect, and give the technical name for such an incorrect outcome. You should show the values of any relevant probabilities. [6]
\hfill \mbox{\textit{OCR S2 2016 Q9 [6]}}