Standard +0.3 This is a straightforward application of the sign test with standard two-tailed hypothesis testing at a given significance level. Students need to recognize the binomial distribution (n=30, p=0.5) and find critical values where P(X≤k) ≤ 0.025 in each tail, which is routine bookwork for S4. The question requires no novel insight, just methodical application of a standard procedure.
1 A meteorologist claims that the median daily rainfall in London is 2.2 mm . A single sample sign test is to be used to test the claim, using the following hypotheses:
\(\mathrm { H } _ { 0 }\) : a sample comes from a population with median 2.2,
\(\mathrm { H } _ { 1 }\) : the sample does not come from a population with median 2.2.
30 randomly selected observations of daily rainfall in London are compared with 2.2, and given a '+' sign if greater than 2.2 and a '-' sign if less than 2.2. (You may assume that no data values are exactly equal to 2.2.) The test is to be carried out at the \(5 \%\) level of significance. Let the number of ' + ' signs be \(k\). Find, in terms of \(k\), the critical region for the test showing the values of any relevant probabilities.
Use of tables, even if 1 tail. Allow even if incorrect e.g. look for 0.0025.
\(F(20) = 0.9786\), \(F(19) = 0.9506\)
M1
Look for 2.5% at both ends. Can be implied by attempt at \(k \geq 21\). May be by symmetry. N(15,7.5) M1M1A1A0 max.
\(k \leq 9\)
A1
\(k \geq 21\)
A1
[4]
# Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $F(9) = 0.0214$, $F(10) = 0.0494$ | M1 | Use of tables, even if 1 tail. Allow even if incorrect e.g. look for 0.0025. |
| $F(20) = 0.9786$, $F(19) = 0.9506$ | M1 | Look for 2.5% at both ends. Can be implied by attempt at $k \geq 21$. May be by symmetry. N(15,7.5) M1M1A1A0 max. |
| $k \leq 9$ | A1 | |
| $k \geq 21$ | A1 | |
| | **[4]** | |
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1 A meteorologist claims that the median daily rainfall in London is 2.2 mm . A single sample sign test is to be used to test the claim, using the following hypotheses:\\
$\mathrm { H } _ { 0 }$ : a sample comes from a population with median 2.2,\\
$\mathrm { H } _ { 1 }$ : the sample does not come from a population with median 2.2.\\
30 randomly selected observations of daily rainfall in London are compared with 2.2, and given a '+' sign if greater than 2.2 and a '-' sign if less than 2.2. (You may assume that no data values are exactly equal to 2.2.) The test is to be carried out at the $5 \%$ level of significance. Let the number of ' + ' signs be $k$. Find, in terms of $k$, the critical region for the test showing the values of any relevant probabilities.
\hfill \mbox{\textit{OCR S4 2017 Q1 [4]}}