SQA Statistics Case Studies: Practical Walkthrough | SQA 统计：案例分析实战演练

📚 SQA Statistics Case Studies: Practical Walkthrough | SQA 统计：案例分析实战演练

In SQA Statistics, theory comes alive through real datasets. This article walks you through a complete case study – from formulating a question to interpreting results – so you can see how each statistical technique fits together. We will investigate the relationship between weekly study hours and final exam scores among Year 11 students. Along the way, you will practice descriptive statistics, probability, confidence intervals, hypothesis testing, correlation, and regression, exactly as required for your assessments.

在 SQA 统计中，理论通过真实数据变得鲜活。本文带你走完一个完整案例——从提出研究问题到解释结果——让你看清每项统计技术如何衔接。我们将调查 Year 11 学生每周学习时长与期末考试成绩之间的关系。沿途你将练习描述性统计、概率、置信区间、假设检验、相关与回归，完全贴合考试要求。

1. The Case Study Question | 案例研究问题

We begin with a clear research question: "Is there a significant positive relationship between the number of hours a Year 11 student studies per week and their final exam score out of 100?" A random sample of 30 students from a large secondary school provided their self-reported weekly study hours and their exam results. The data are stored in a table; each row represents a student.

我们从清晰的研究问题开始：“一名 Year 11 学生每周学习小时数与百分制期末考试成绩之间是否存在显著的正相关？”从一所大型中学随机抽取的 30 名学生提供了他们自报的每周学习时长和考试成绩。数据存储在一个表格中；每一行代表一名学生。

2. Data Collection and Variable Types | 数据收集与变量类型

Study hours are a continuous numerical variable (ratio scale), measured to one decimal place. Exam scores are also continuous numerical. Both are quantitative. No categorical variable is used in this core analysis, but later we might split students into 'high' and 'low' study groups for a two-sample test. The data were collected via an anonymous online questionnaire, minimising response bias. Random sampling ensures the results can be generalised to the Year 11 population at that school.

学习小时数是一个连续型数值变量（比率尺度），精确到一位小数。考试分数同样是连续型数值。两者都是定量变量。当前核心分析中没有使用分类变量，但后续我们可能把学生分为“高学习时长”和“低学习时长”组进行双样本检验。数据通过匿名在线问卷收集，最大限度减少了回答偏差。随机抽样确保结果可推广至该校 Year 11 总体。

3. Descriptive Statistics: Centre and Spread | 描述性统计：集中趋势与离散程度

.4. The median study time is 7.8 hours, and the median score is 74. Since the mean and median are close for both variables, the distributions are roughly symmetric. The interquartile range (IQR) for study hours is 10.1 – 6.3 = 3.8 hours, confirming moderate variability.

在这 30 名学生中，平均学习时间为 8.2 小时，标准差为 3.1 小时。平均考试分数为 72.5，标准差为 12.4。学习时间中位数为 7.8 小时，分数中位数为 74。由于两个变量的均值与中位数接近，分布大致对称。学习小时数的四分位距（IQR）为 10.1 – 6.3 = 3.8 小时，也证实了适中的变异性。

4. Visualising the Data | 数据可视化

ape, while the histogram of exam scores appears bell-shaped. Boxplots confirm no outliers beyond 1.5 × IQR. Always check for outliers before calculating correlation, as they can distort the relationship.

考试分数对学习小时数的散点图显示出明显的向上趋势。学习小时数的直方图表现为单峰、轻微右偏的形状，而考试分数的直方图则呈钟形。箱线图确认没有超出 1.5 × IQR 的异常值。在计算相关之前，务必检查异常值，因为它们可能扭曲关系。

5. Probability Foundations | 概率基础

Suppose the probability that a randomly selected student studies more than 10 hours per week is 0.3, and the probability of scoring above 80 is 0.4. If studying more than 10 hours and scoring above 80 are not independent, the conditional probability P(score > 80 | study > 10) might be higher than 0.4. Using a two-way table from our sample, we can estimate such joint and conditional probabilities – a key skill for SQA Statistics.

假设随机挑选一名学生每周学习超过 10 小时的概率是 0.3，得分超过 80 的概率是 0.4。如果学习超过 10 小时和得分超过 80 并非相互独立，则条件概率 P(score > 80 | study > 10) 可能高于 0.4。利用样本中的双向表，我们可以估计这类联合概率和条件概率——这是 SQA 统计的关键技能。

6. Sampling Distributions and the Central Limit Theorem | 抽样分布与中心极限定理

We treat our sample mean of 72.5 as an estimate of the population mean. If we took many random samples of size 30, the sampling distribution of the sample mean would be approximately normal with mean μ and standard error σ/√n. Here, we estimate σ by s = 12.4, so the standard error is 12.4 / √30 population distribution is somewhat skewed, thanks to the Central Limit Theorem (n ≥ 30).

我们将样本均值 72.5 视为总体均值的估计。如果重复抽取许多大小为 30 的随机样本，样本均值的抽样分布将近似正态，其均值为 μ，标准误为 σ/√n。这里我们用 s = 12.4 估计 σ，因此标准误为 12.4 / √30 ≈ 2.26。即使总体分布略偏，由于中心极限定理（n ≥ 30），这一正态性依然成立。

7. Constructing a Confidence Interval | 构建置信区间

A 95% confidence interval for the true mean exam score is given by x̄ ± t* × (s/√n). With n=30, degrees of freedom = 29, the t-critical value for 95% pproximately 2.045. Thus, 72.5 ± 2.045 × 2.26 gives an interval of (67.9, 77.1). We are 95% confident that the average score of all Year 11 students falls within this range.

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真实考试平均分的 95% 置信区间由公式 x̄ ± t* × (s/√n) 给出。n=30，自由度为 29，95% 对应的 t 临界值约为 2.045。因此 72.5 ± 2.045 × 2.26 得到的区间为 (67.9, 77.1)。我们可以 95% 地确信所有 Year 11 学生的平均分数在这一范围内。

8. Hypothesis Test for a Mean | 均值的假设检验

ional average of 70. Set H₀: μ = 70, H₁: μ ≠ 70 (two-tailed). The test statistic t = (72.5 – e 0.278 > 0.05, we fail to reject H₀. There is insufficient evidence to say this school's mean score differs from the national average.

我们想检验总体平均考试分数是否与全国平均分 70 不同。设 H₀: μ = 70，H₁: μ ≠ 70（双尾）。检验统计量 t = (72.5 – 70) / 2.26 = 1.106。自由度 29 时，p 值 ≈ 0.278。因为 0.278 > 0.05，无法拒绝 H₀。没有充分证据表明该校平均分与全国平均分存在差异。

9. Correlation and Its Test | 相关系数及其检验

s tend to increase. To test significance, H₀: ρ = 0, H₁: ρ ≠ 0. The test statistic t = r √[(n-2)/(1-r²)] = 0.72 × √28 / 0.4816 h df=28, p < 0.001. We reject H₀ and conclude a significant linear relationship exists.

学习小时数与考试分数之间的皮尔逊相关系数为 r = 0.72。正 r 表明随着学习时间增加，考试分数也倾向增加。为检验显著性，设 H₀: ρ = 0，H₁: ρ ≠ 0。检验统计量 t = r √[(n-2)/(1-r²)] = 0.72 × √28 / 0.4816 ≈ 5.49。自由度为 28 时，p < 0.001。我们拒绝 H₀，得出存在显著线性关系的结论。

10. Simple Linear Regression | 简单线性回归

redicted exam score increases by about 2.72 points. The R² value is 0.72² = 0.518, so 51.8% ed by study hours. Always check residual plots to confirm linearity and constant variance.

最小二乘回归线为：分数 = 50.2 + 2.72 × (学习小时数)。斜率 2.72 意味着每周多学习一小时，预测考试分数提高约 2.72 分。R² 值为 0.72² = 0.518，因此考试分数的变异中有 51.8% 可由学习小时数解释。始终要检查残差图以确认线性和等方差性。

11. Chi-Squared Test for Independence | 卡方检验的独立性应用

To extend the analysis, we categorise study hours as Low (<7), Medium (7–10), High (>10) and exam scores as Fail (<60), Pass (60– =0.05, we reject the null hypothesis of independence; study category and result category are associated.

为扩展分析，我们将学习小时数分为低 (<7)、中 (7–10)、高 (>10)，考试分数分为不及格 (<60)、及格 (60–80)、优秀 (>80)。3×3 表格中的观测频数显示出关联。卡方检验得到 χ² = 14.7，自由度 4，p = 0.005。在 α=0.05 水平下，我们拒绝独立性的原假设；学习时间类别与成绩类别存在关联。

Observed frequency table | 观测频数表

Study / Result	Fail	Pass	Distinction	Total
Low	6	4	1	11
Medium	2	7	3	12
High	0	2	5	7
Total	8	13	9	30

12. Reporting Findings and Exam Tips | 报告结果与考试技巧

text, stating the test used, the p-value, and a non-technical interpretation. For example: " .001). This suggests that increased study time is associated with higher exam performance." ons (normality, linearity, independence) before each test and mention if any are violated.

在 SQA 考试中，你必须写出清晰的上下文结论，陈述所用检验、p 值以及非技术性的解释。例如：“皮尔逊相关检验显示学习小时数与考试分数之间存在显著的正线性关系（r=0.72，p<0.001）。这表明增加学习时间与更高的考试表现相关。”每次检验前都应检查假设条件（正态性、线性、独立性），如有违背要提及。

Published by TutorHao | Statistics Revision Series | aleveler.com

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