TOPIC 01 • SAMPLING SYSTEM
Sampling
Sampling is the process of selecting observations from a population so that information about the population can be obtained without studying every member.
Population and Sample
The population is the complete set of individuals or objects being studied. A sample is a subset selected from that population.
Simple Random Sampling
In simple random sampling, every possible sample of the required size has an equal chance of being selected.
Members of a population can be numbered and random numbers can then be used to select the required sample.
Stratified Sampling
The population is divided into strata according to a relevant characteristic. A sample is then taken from each stratum, usually in proportion to the size of that stratum.
Systematic Sampling
Individuals are selected at regular intervals from an ordered sampling frame, usually after choosing a suitable starting point.
Quota Sampling
The investigator sets quotas for different groups and continues selecting individuals until the required quotas are filled.
Comparing Sampling Methods
- Simple random sampling can reduce selection bias but requires a suitable sampling frame.
- Stratified sampling can ensure important groups are represented appropriately.
- Systematic sampling can be straightforward to administer.
- Quota sampling can be convenient but may introduce interviewer or selection bias.
TOPIC 02 • DISTRIBUTION NETWORK
Combinations of Random Variables
S3 extends the Normal distribution to combinations of independent Normal random variables.
Linear Combinations
Suppose X and Y are independent Normal random variables:
Then a linear combination of X and Y is also Normally distributed.
Variances are added after multiplying them by the squares of the coefficients. The standard deviations themselves are not added.
WORKED EXAMPLE
X ~ N(20, 4²) and Y ~ N(15, 3²), independently. Find the distribution of 2X − Y.
Mean:
Variance:
Therefore:
TOPIC 03 • ESTIMATION ENGINE
Estimators and Standard Error
An estimator is a statistic used to estimate an unknown population parameter.
Estimator
The sample mean is commonly used as an estimator of the population mean μ.
Bias
An estimator is unbiased if its expected value is equal to the population parameter that it estimates.
Variance of the Sample Mean
For a random sample of n independent observations from a population with variance σ²:
Standard Error
Increasing the sample size reduces the standard error and therefore generally makes the sample mean a more precise estimator of μ.
Estimator of Population Variance
For a random sample, the usual unbiased estimator of the population variance uses n − 1 in the denominator.
TOPIC 04 • CONFIDENCE FIELD
Confidence Intervals
A confidence interval provides a range of plausible values for an unknown population parameter.
Normal Mean with Known Variance
If the population is Normal with known variance σ², an interval estimate for μ can be constructed using the sampling distribution of X̄.
Here z is the appropriate percentage point of the standard Normal distribution for the required confidence level.
Interpreting a Confidence Interval
A confidence interval should be interpreted in terms of the method used to construct it. For example, a 95% confidence procedure produces intervals that contain the true parameter in approximately 95% of repeated samples.
Width of an Interval
A larger sample size gives a smaller standard error and therefore usually produces a narrower confidence interval.
TOPIC 05 • HYPOTHESIS ENGINE
Hypothesis Tests for a Normal Mean
Hypothesis testing provides a formal method for deciding whether sample evidence is inconsistent with a proposed population parameter.
Null and Alternative Hypotheses
The statement assumed to be true unless the sample provides sufficient evidence against it.
Alternative hypothesis H₁:The competing statement supported when the null hypothesis is rejected.
Test Statistic
When the population variance is known, the sample mean can be standardised using:
One-Tailed and Two-Tailed Tests
- A one-tailed test investigates whether a parameter is greater than or less than a specified value.
- A two-tailed test investigates whether a parameter differs from a specified value in either direction.
Significance Level
The significance level is the probability of rejecting H₀ when H₀ is actually true.
Conclusion
A statistical conclusion should be stated clearly in the context of the question.
Do not say that H₀ has been proved true. A test either provides sufficient evidence to reject H₀ or does not provide sufficient evidence to reject H₀.
TOPIC 06 • CENTRAL LIMIT FIELD
Central Limit Theorem
The Central Limit Theorem allows the distribution of a sample mean to be approximated by a Normal distribution for sufficiently large samples, even when the underlying population is not Normal.
Sampling Distribution of the Mean
For a population with mean μ and variance σ², a sample mean based on n independent observations has:
For a Normal population, the sampling distribution of X̄ is exactly Normal. For a sufficiently large sample from a non-Normal population, the Central Limit Theorem provides a Normal approximation.
Standardising the Sample Mean
Always identify the population mean, population variance, sample size and required probability before standardising.
TOPIC 07 • COMPARISON MATRIX
Comparing Two Population Means
S3 also considers inference about the difference between two population means using independent samples.
Difference Between Sample Means
If two independent samples are taken from Normal populations with known variances:
Standardising the Difference
This can be used to construct confidence intervals or perform hypothesis tests about the difference between two population means.
Large-sample methods can also be used when population variances are unknown, using appropriate estimates of variability.
TOPIC 08 • RELATIONSHIP NETWORK
Spearman's Rank Correlation
Spearman's rank correlation coefficient measures the strength and direction of a monotonic relationship between two variables using their ranks.
Rank Differences
For each observation, calculate the difference d between its two ranks and then calculate d².
Interpreting rₛ
- A value close to +1 indicates strong positive association.
- A value close to −1 indicates strong negative association.
- A value close to 0 indicates little evidence of a monotonic relationship.
Ties
When tied values occur, appropriate ranks must be assigned before calculating the correlation coefficient.
Limitations
Correlation does not establish causation. Spearman's coefficient also describes association rather than proving that one variable causes changes in another.
TOPIC 09 • CORRELATION TEST NODE
Testing for Zero Correlation
A hypothesis test can be used to investigate whether there is evidence of a relationship between two variables.
Hypotheses
Depending on the wording of the question, a one-tailed alternative may be appropriate.
Critical Values
For Spearman's rank correlation, critical values can be used to determine whether the observed rank correlation is sufficiently extreme to reject the null hypothesis.
Compare the calculated statistic with the appropriate critical value for the chosen significance level and alternative hypothesis.
TOPIC 10 • CHI-SQUARED ANALYSIS
Goodness of Fit Tests
A chi-squared goodness-of-fit test can be used to investigate whether observed frequencies are consistent with a proposed probability model.
Observed and Expected Frequencies
The observed frequencies are the actual data. Expected frequencies are calculated from the model assumed under the null hypothesis.
Hypotheses
The data are consistent with the proposed distribution.
H₁:The data are not consistent with the proposed distribution.
Degrees of Freedom
The number of degrees of freedom depends on the number of categories and the number of parameters estimated from the data.
Expected Frequencies
Expected frequencies should generally be sufficiently large for the chi-squared approximation to be appropriate. Where necessary, categories may need to be combined according to the conditions of the test.
TOPIC 11 • ASSOCIATION MATRIX
Contingency Tables
Contingency tables are used to investigate whether two categorical variables are associated.
Observed Frequencies
The observed frequencies are arranged in rows and columns according to the two categorical variables.
Expected Frequency
Chi-Squared Statistic
Degrees of Freedom
Hypotheses
The two categorical variables are independent.
H₁:The two categorical variables are associated.
Conclusion
The final conclusion should be written in context and should refer to the evidence for or against independence.
INFERENCE SIMULATION • PRACTICE
S3 Mixed Practice
These questions combine different S3 ideas rather than separating practice by chapter.
Show answer
Proportion = 180 / 480.
Required sample = (180 / 480) × 60 = 22.5.
In practice, the allocation must be adjusted to whole students while maintaining the required total sample.
Show answer
Mean = 2(12) + 3(8) = 48.
Variance = 2²(3²) + 3²(2²) = 36 + 36 = 72.
Therefore:
2X + 3Y ~ N(48, 72).
Show answer
SE(X̄) = σ / √n
= 12 / √36 = 2.
Show answer
An estimator is unbiased if its expected value is equal to the population parameter that it estimates.
Show answer
Var(X̄) = σ² / n.
Show answer
Z = (51 − 50) / (10 / √100) = 1.
Show answer
It allows the sampling distribution of the sample mean to be approximated by a Normal distribution.
Show answer
rₛ = 1 − 6Σd² / [n(n² − 1)].
Show answer
df = 5 − 1 − 2 = 2.
Show answer
df = (4 − 1)(3 − 1) = 6.
Show answer
Expected frequency = (80 × 50) / 200 = 20.
Show answer
Statistical significance means that the sample provides sufficient evidence against the null hypothesis at the chosen significance level. Practical importance concerns whether the size of the observed effect is meaningful in the real-world context.
INFERENCE DATABASE • QUICK REVIEW
S3 Key Ideas
- Identify the population parameter being investigated.
- State H₀ and H₁ clearly.
- Choose the correct one-tailed or two-tailed test.
- Calculate the appropriate test statistic.
- Use the correct significance level or critical value.
- State the statistical decision.
- Finish with a conclusion in context.