Standard +0.3 This is a straightforward one-tailed binomial hypothesis test with clearly stated hypotheses (p=0.65 vs p>0.65), standard significance level, and small sample size allowing direct calculation from tables. It requires routine application of the binomial test procedure with no conceptual complications, making it slightly easier than average.
2 Jill shoots arrows at a target. Last week, \(65 \%\) of her shots hit the target. This week Jill claims that she has improved. Out of her first 20 shots this week, she hits the target with 18 shots. Assuming shots are independent, test Jill's claim at the \(1 \%\) significance level.
Allow one end error. Allow p/q mix. Allow \((1-)\) for M mark
\(= 0.0121\) (3 sf)
A1
A mark recovered following valid comparison
Compare 0.01; There is no evidence (at the 1% level) that she has improved
M1 A1\(\checkmark\) [5]
For valid comparison; She has probably not improved. No contradictions. (SR Use of Normal M0, but M1A1 for valid comparison could be awarded)
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0$: $P(\text{hit target}) = 0.65$; $H_1$: $P(\text{hit target}) > 0.65$ | **B1** | Allow $p = 0.65$; allow $p > 0.65$ |
| $^{20}C_2 \times 0.35^2 \times 0.65^{18} + 19 \times 0.35 \times 0.65^{19} + 0.65^{20}$ | **M1** | Allow one end error. Allow p/q mix. Allow $(1-)$ for M mark |
| $= 0.0121$ (3 sf) | **A1** | A mark recovered following valid comparison |
| Compare 0.01; There is no evidence (at the 1% level) that she has improved | **M1 A1$\checkmark$** [5] | For valid comparison; She has probably not improved. No contradictions. (SR Use of Normal M0, but M1A1 for valid comparison could be awarded) |
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2 Jill shoots arrows at a target. Last week, $65 \%$ of her shots hit the target. This week Jill claims that she has improved. Out of her first 20 shots this week, she hits the target with 18 shots. Assuming shots are independent, test Jill's claim at the $1 \%$ significance level.
\hfill \mbox{\textit{CAIE S2 2016 Q2 [5]}}