Moderate -0.3 This is a straightforward one-tailed binomial hypothesis test with clear parameters (n=20, x=19, p=0.82, α=0.10). Students must state the assumption of independence, set up H₀: p=0.82 vs H₁: p>0.82, and calculate P(X≥19). The calculation is routine and the question structure is standard for S2, making it slightly easier than average but still requiring proper statistical methodology.
2 A hockey player found that she scored a goal on \(82 \%\) of her penalty shots. After attending a coaching course, she scored a goal on 19 out of 20 penalty shots. Making an assumption that should be stated, test at the 10\% significance level whether she has improved.
\(20 \times 0.82^{19} \times 0.18 + 0.82^{20} = 0.102\) (3 sf); No evidence that improved
M1, A1, B1f [5]
For use of \(\text{Bin}(20, 0.82)\) and either \(P(19)\) and/or \(P(20)\) attempted; valid comparison with 0.05 if \(H_1\ p \neq 0.82\); correct conclusion ft numerical errors in 0.102 only
Normal approx'n note: B1 B1 (\(\mu = 16.4\) acceptable), then \(CR = 1.222\) from \(\frac{18.5 - 20 \times 0.82}{\sqrt{20 \times 0.82 \times (1-0.82)}}\); comp \(z = 1.282\); No evidence that improved SC 1; Same scheme for proportions
## Question 2:
| Assume shots independent OR prob of scoring constant | B1 | In context |
| $H_0: P(\text{score}) = 0.82$; $H_1: P(\text{score}) > 0.82$ | B1 | Both, allow $p$ |
| $20 \times 0.82^{19} \times 0.18 + 0.82^{20} = 0.102$ (3 sf); No evidence that improved | M1, A1, B1f [5] | For use of $\text{Bin}(20, 0.82)$ and either $P(19)$ and/or $P(20)$ attempted; valid comparison with 0.05 if $H_1\ p \neq 0.82$; correct conclusion ft numerical errors in 0.102 only |
Normal approx'n note: B1 B1 ($\mu = 16.4$ acceptable), then $CR = 1.222$ from $\frac{18.5 - 20 \times 0.82}{\sqrt{20 \times 0.82 \times (1-0.82)}}$; comp $z = 1.282$; No evidence that improved SC 1; Same scheme for proportions
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2 A hockey player found that she scored a goal on $82 \%$ of her penalty shots. After attending a coaching course, she scored a goal on 19 out of 20 penalty shots. Making an assumption that should be stated, test at the 10\% significance level whether she has improved.
\hfill \mbox{\textit{CAIE S2 2013 Q2 [5]}}