Mark Scheme Patterns — Inference Using Normal and t-Distributions

标准给分结构

Single-Sample t-Test（6-7 marks）

Step	Marks	Typical wording
$H_0$ / $H_1$ correctly stated	B1	"$H_0: \mu = 16$, H_1: \mu < 16"
Correct test statistic formula	M1	"Use of $t = (\bar{x} - \mu)/\sqrt{s^2/n}$"
Correct t value (numerical)	A1	"$t = -2.771$"
Correct critical value / p-value	B1	"$t_{0.05}(9) = 1.833$"
Valid comparison	M1	"Compare $
Correct conclusion in context	A1	"Reject $H_0$ — sufficient evidence..."

Two-Sample Pooled t-Test（8-9 marks）

Step	Marks	Typical wording
$H_0$ / $H_1$ correctly stated	B1	"$H_0: \mu_1 = \mu_2$"
$s_p^2$ correctly calculated	M1	"Pooled variance formula used"
$s_p^2$ numerical value correct	A1	"$s_p^2 = 7.10$"
t statistic (formula + calc)	M1	"Substitution into t formula"
t value correct	A1	"$t = 2.630$"
df stated	B1	"df = 20"
Critical value	B1	"$t_{0.025}(20) = 2.086$"
Comparison + conclusion	M1 A1	"Since t > t_{\text{crit}}, reject $H_0$"

Paired t-Test（7-8 marks）

Step	Marks	Typical wording
Differences $d_i$ calculated	B1	"$d = [5, 2, 7, \dots]$"
$\bar{d}$ and $s_d^2$	M1 A1	"$\bar{d} = 4.25$, $s_d^2 = 4.50$"
t statistic formula + value	M1 A1	"$t = 5.667$"
df + critical value	B1	"$t_{0.05}(7) = 1.895$"
Comparison + conclusion	M1 A1	"Reject $H_0$ — mean has increased"

Confidence Interval（4-6 marks）

Step	Marks	Typical wording
Correct formula identified	B1	"$\bar{x} \pm t \cdot \sqrt{s^2/n}$"
Standard error correctly computed	M1 A1	"SE $= \sqrt{9.0/15} = 0.775$"
Correct $t$ or $z$ value from tables	B1	"$t_{0.025}(14) = 2.145$"
Interval endpoints (3 s.f.)	A1	"CI: $(40.3, 43.7)$"

常见扣分点

Mistake	Consequence
Wrong degrees of freedom	Lose B1 (usually method still marked)
Using $z$ instead of $t$	Lose M1 for wrong formula
Confusing one-tail / two-tail critical value	Lose B1
Not stating normality assumption	Lose B1 (when explicitly required)
Arithmetic error in $s_p^2$	Lose A1 (but can still get M1)
Wrong conclusion despite correct calculation	Lose A1

常见 MS 标记模式

• "Use of $t = (\bar{x} - \mu)/\sqrt{s^2/n}$" → M1 (method mark)
• "Correct substitution" → M1 if formula applied correctly
• "$t = ...$" → A1 (accuracy mark, follows a M1)
• "Critical value $t_{0.05}(11) = 1.796$" → B1 (independent mark)
• "|t| > t_{\text{crit}} \Rightarrow reject $H_0$" → M1 for comparison, A1 for conclusion
• "Assumption: normally distributed" → B1
• "3 s.f." → final A1 requires correct rounding

不同 paper 的分数分布参考

Paper	Q	Type	Marks
S20 Q4	4	Paired t-test + One-sample t	9
S20 Q5	5	CI (difference of means)	7
W20 Q1	1	One-sample t-test	6
W20 Q4	4	Paired t-test	8
S21 Q1	1	z-test	6
S21 Q4	4	Two-sample pooled t + CI	9
W21 Q1	1	One-sample t-test	6
W21 Q4	4	Two-sample pooled t	8
S22 Q1	1	One-sample t-test	6
S22 Q5	5	Two-sample z-test	7
W22 Q1	1	One-sample t-test	6
W22 Q6	6	CI (difference)	8
S23 Q1	1	One-sample t-test	6
S23 Q2	2	CI + two-sample test	9
W23 Q1	1	Paired t-test	7
W23 Q3	3	Pooled t + CI	10
S24 Q1	1	One-sample t-test	6
S24 Q6	6	Two-sample + CI	9
W24 Q6	6	CI + hypothesis test	8
S25 Q1	1	Paired t-test	7
S25 Q4	4	CI (two-sample)	7
W25 Q1	1	One-sample t-test	6
W25 Q3	3	CI + pooled t	9
W25 Q6	6	Two-sample z-test	7