📚 AQA Year 12 Statistics: Report Writing Framework and Model Paper | AQA Year 12 统计：论文写作框架与范文
Statistical investigations form a core part of AQA Year 12 Statistics. Whether you are tackling a data-based coursework task or constructing a formal examination response, clarity of structure, correct use of terminology and logical flow of reasoning are essential. This article presents a complete writing framework you can follow, illustrated with a fully worked model paper. Every section pairs an English explanation with its Chinese translation so you can master both the language and the statistical reasoning required for top marks.
统计调查是 AQA Year 12 统计课程的核心组成部分。无论你是在完成数据为基础的课业任务，还是构建正式的考试作答，清晰的结构、术语的正确使用以及推理的逻辑流都有决定性作用。本文提供一个你可以遵循的完整写作框架，并用一份完整范文加以说明。每个部分都配对了英文解释和中文翻译，帮助你同时掌握高分所需的语言表达与统计推理。
1. Defining the Research Question and Objectives | 确定研究问题与目标
Begin with a precise statement of what you intend to investigate. A focused research question prevents vague conclusions and helps you select appropriate methods. For our model paper, the question is: “Is there a significant linear relationship between the number of hours Year 12 students spend on social media per day and their nightly sleep duration?” The objectives are to collect paired data, describe the association, model it using linear regression, and perform a hypothesis test on the population correlation coefficient.
首先应精确说明你打算研究什么。一个明确的研究问题可以避免模糊的结论，也有助于你选择合适的统计方法。在本范文中，研究问题是：”Year 12 学生每日使用社交媒体的时长与其夜间睡眠时长之间是否存在显著的线性关系？” 目标包括收集配对数据、描述关联性、用线性回归建模，并对总体相关系数进行假设检验。
2. Data Collection and Sampling Method | 数据收集与抽样方法
A simple random sample of 30 Year 12 students was selected from a school population of 210. Each participant reported their average daily social media hours (X) and average nightly sleep hours (Y) over one week. The data were anonymised and collected via a standardised questionnaire. It is crucial to state the sampling method, the population from which the sample was drawn, and any steps taken to reduce bias.
从一所学校的 210 名 Year 12 学生中选取了一个包含 30 人的简单随机样本。每位参与者报告了自己在一周内日均社交媒体使用时长（X）和夜间睡眠时长（Y）。数据经过匿名化处理，通过标准化问卷收集。明确说明抽样方法、样本所来自的总体以及为减少偏差所采取的措施，这一点至关重要。
3. Descriptive Statistics and Summary Measures | 描述性统计与汇总指标
Summary statistics for the 30 paired observations are given below. The sample mean of X (social media hours) is x̄ = 3.24 hours, with a sample standard deviation sx = 1.52 hours. The sample mean of Y (sleep hours) is ȳ = 7.48 hours, with sy = 0.98 hours. The sample size n = 30. These measures give us a first look at central tendency and spread.
下面给出 30 组观测值的汇总统计量。X（社交媒体时长）的样本均值是 x̄ = 3.24 小时，样本标准差 sx = 1.52 小时。Y（睡眠时长）的样本均值是 ȳ = 7.48 小时，sy = 0.98 小时。样本容量 n = 30。这些指标让我们对数据的集中趋势和离散程度有了初步了解。
We also calculate the five-number summary for each variable to check for potential outliers. For X: minimum 0.8, Q₁ = 2.1, median = 3.3, Q₃ = 4.5, maximum 6.2. For Y: minimum 5.5, Q₁ = 6.8, median = 7.5, Q₃ = 8.1, maximum 9.3. No extreme outliers were flagged using the 1.5 × IQR rule.
我们还计算了每项变量的五数概括，以检查潜在的异常值。X：最小值 0.8，Q₁ = 2.1，中位数 = 3.3，Q₃ = 4.5，最大值 6.2。Y：最小值 5.5，Q₁ = 6.8，中位数 = 7.5，Q₃ = 8.1，最大值 9.3。使用 1.5 × IQR 规则未发现极端异常值。
4. Data Visualisation: Charts and Graphs | 数据可视化：图表
A scatterplot of Y against X clearly shows a downward trend: as social media hours increase, sleep hours appear to decrease. The scatter points roughly cluster around an imaginary straight line, suggesting a negative linear association. Also presented are side-by-side box plots, which confirm that the spread of sleep hours is fairly symmetric and that variability in X is larger than in Y.
Y 关于 X 的散点图呈现出明显的下降趋势：随着社交媒体使用时长增加，睡眠时长似乎减少。散点大致围绕一条假想直线分布，暗示存在负线性关联。同时还展示了并列箱线图，证实睡眠时长的分布较为对称，且 X 的变异性大于 Y。
| Social media (hours) | Sleep (hours) |
|---|---|
| 0.8 | 9.0 |
| 2.1 | 8.2 |
| 3.3 | 7.5 |
| 4.5 | 6.9 |
| 6.2 | 5.8 |
Table 1: A sample of five paired data points illustrating the negative trend. The full dataset of 30 points underpins all subsequent analysis.
表 1：五个配对数据点的样本，反映了负向趋势。完整的 30 个数据点支撑了后续的所有分析。
5. Probability Distributions and Model Assumptions | 概率分布与模型假设
Linear regression and Pearson’s correlation require assumptions of linearity, constant variance (homoscedasticity) and approximate bivariate normality. Examination of the scatterplot suggests the relationship is roughly linear. To check constant variance, a residual plot (residuals against fitted Y values) should show random scatter with no fan shape. In our model paper, the residual plot displayed no clear pattern, so the assumption of equal variances is deemed reasonable.
线性回归与皮尔逊相关系数要求满足线性性、等方差性（同方差性）以及近似的双变量正态性。通过检查散点图，可以看出关系大致呈线性。为检验等方差性，残差图（残差对应 Y 拟合值）应显示随机散布且无喇叭状。在我们的范文中，残差图未呈现明显模式，因此方差相等的假设被认为是合理的。
To assess approximate bivariate normality, we can check that histograms of both variables are unimodal and roughly symmetric. The histograms for X and Y meet these criteria well enough for a sample of size 30, so we proceed with the parametric test.
为评估近似的双变量正态性，可以检验两项变量的直方图是否呈单峰且大致对称。X 和 Y 的直方图在样本量为 30 的情况下较好地满足这些要求，因此我们继续使用参数检验。
6. Correlation and Regression Analysis | 相关与回归分析
We compute the Pearson product-moment correlation coefficient to quantify the strength of the linear relationship. Using the formula
r = [ Σ(xi − x̄)(yi − ȳ) ] / [ √( Σ(xi − x̄)² · Σ(yi − ȳ)² ) ]
we obtain r = −0.648. This indicates a moderately strong negative correlation. The coefficient of determination R² = r² = 0.420 means that 42.0% of the variation in sleep hours can be explained by the linear relationship with social media hours.
我们计算皮尔逊积矩相关系数，以量化线性关系的强度。利用公式 r = [ Σ(xi − x̄)(yi − ȳ) ] / [ √( Σ(xi − x̄)² · Σ(yi − ȳ)² ) ] 得到 r = −0.648。这表明存在中等强度的负相关。决定系数 R² = r² = 0.420，意味着睡眠时长变化的 42.0% 可以由与社交媒体时长的线性关系解释。
The equation of the least-squares regression line, with Y as the response variable, is:
Ŷ = 9.18 − 0.524 X
For every additional hour spent on social media, predicted sleep time decreases by about 0.52 hours, i.e. roughly 31 minutes. This model can be used for prediction within the observed range of X.
以 Y 为响应变量的最小二乘回归线方程为 Ŷ = 9.18 − 0.524 X。社交媒体使用时长每增加一小时，预测的睡眠时间减少约 0.52 小时，即大约 31 分钟。该模型可用于在 X 观测范围内的预测。
7. Hypothesis Testing: Setting Up the Test | 假设检验：设定检验
To determine if the observed sample correlation provides sufficient evidence of a linear relationship in the population, we conduct a two-tailed t-test for the population correlation coefficient ρ. The hypotheses are:
H₀: ρ = 0
H₁: ρ ≠ 0
We choose a significance level of α = 0.05. The test statistic is t = r × √(n − 2) / √(1 − r²), which follows a t-distribution with n − 2 = 28 degrees of freedom under H₀.
为判断样本相关是否提供了足够的证据表明总体中存在线性关系，我们对总体相关系数 ρ 进行双尾 t 检验。假设为：H₀: ρ = 0；H₁: ρ ≠ 0。我们选择显著性水平 α = 0.05。检验统计量为 t = r × √(n − 2) / √(1 − r²)，在 H₀ 下服从自由度为 n − 2 = 28 的 t 分布。
8. Performing the Test and Interpreting Results | 执行检验并解释结果
Substituting r = −0.648 and n = 30 into the formula:
t = −0.648 × √28 / √(1 − 0.648²)
We obtain t ≈ −4.38. Using statistical tables or technology, the two-tailed p-value is approximately 0.00015. Since p-value < 0.05, we reject H₀. There is statistically significant evidence at the 5% level to conclude that a non-zero linear correlation exists between social media hours and sleep hours in the population of Year 12 students.
将 r = −0.648 和 n = 30 代入公式：t = −0.648 × √28 / √(1 − 0.648²) 得到 t ≈ −4.38。查阅统计表或使用软件，双尾 p 值约为 0.00015。由于 p 值 < 0.05，我们拒绝 H₀。在 5% 显著性水平下，有统计显著证据表明，Year 12 学生总体中社交媒体时长与睡眠时长之间存在非零的线性相关。
9. Evaluation and Conclusion | 评估与结论
The analysis confirms a moderate negative relationship: more social media use is associated with less sleep. However, correlation does not imply causation. Confounding variables such as academic pressure, part-time jobs or screen brightness may influence both variables. Limitations include the relatively small sample from a single school, self-reported data prone to inaccuracy, and the snapshot nature of the measurements. A more robust study would use a larger, more diverse sample and objective measures of screen time and sleep. Despite these limitations, the report demonstrates a clear, repeatable statistical investigation framework suitable for AQA Year 12 Statistics.
分析证实了中等程度的负相关关系：更高的社交媒体使用量与更少的睡眠相关联。然而，相关性并不意味着因果性。诸如学业压力、兼职工作或屏幕亮度等混杂变量可能对两者均有影响。局限性包括：样本仅来自一所学校且较小、自我报告数据容易不准确、测量仅为截面性质。更严谨的研究会采用更大、更多样化的样本，并使用屏幕时间和睡眠的客观测量。尽管存在这些局限性，本报告展示了一套清晰、可重复的统计调查框架，适用于 AQA Year 12 统计课程。
10. Complete Model Report: from Question to Conclusion | 完整范文：从问题到结论
The model report presented below consolidates the previous sections into a single coherent statistical report. Each piece of English text is immediately followed by its Chinese translation so that you can observe how the final document reads.
以下范文将前面各节内容整合为一份连贯的统计报告。每段英文后面紧跟着中文翻译，以便你观察最终文档的表达方式。
“The purpose of this investigation is to determine whether a significant linear relationship exists between the number of hours Year 12 students spend on social media per day (X) and their average nightly sleep duration in hours (Y). A simple random sample of 30 students was drawn from a school population of 210. Data were collected through an anonymised questionnaire.”
“本调查旨在确定 Year 12 学生每日社交媒体使用时长（X）与其夜间平均睡眠时长（Y）之间是否存在显著的线性关系。从 210 名学生的学校总体中抽取了 30 人的简单随机样本，通过匿名问卷收集数据。”
“Summary statistics: x̄ = 3.24 h (sx = 1.52), ȳ = 7.48 h (sy = 0.98). A scatterplot revealed a negative linear trend. No outliers were detected using the 1.5×IQR criterion. The Pearson correlation coefficient was r = −0.648, with R² = 0.420. The regression equation is Ŷ = 9.18 − 0.524X.”
“汇总统计量：x̄ = 3.24 小时（sx = 1.52），ȳ = 7.48 小时（sy = 0.98）。散点图显示负线性趋势。使用 1.5×IQR 标准未发现异常值。皮尔逊相关系数 r = −0.648，R² = 0.420。回归方程为 Ŷ = 9.18 − 0.524X。”
“A two-tailed t-test for ρ was performed with H₀: ρ = 0, H₁: ρ ≠ 0, α = 0.05. The test statistic t = −4.38 with 28 d.f. gave a p-value ≈ 0.00015. Since p < 0.05, H₀ is rejected. There is significant evidence of a negative linear correlation between social media use and sleep duration in the population."
“对 ρ 进行双尾 t 检验，H₀: ρ = 0，H₁: ρ ≠ 0，α = 0.05。检验统计量 t = −4.38（自由度 28），p 值 ≈ 0.00015。由于 p < 0.05，拒绝 H₀。有显著证据表明总体中社交媒体使用与睡眠时长之间存在负线性相关。"
“The main limitations are the small, single-school sample, reliance on self-reported data and the observational design, which precludes causal conclusions. Future research should expand the sample and include additional variables.”
“主要局限性在于样本较小且仅来自一所学校、依赖自我报告数据，以及观察性设计不允许因果推断。未来的研究应扩大样本并纳入更多变量。”
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