Section A 📊 → Sampling, Hypothesis, Probabilities, Presentation of Data

Discusses different ways to collect data and assesses the reliability and validity of both primary and secondary data sources in order for one to achieve a 9 on his/her/their AQA GCSE Statistics Exam.

Data Representation

Involves interpreting graphs and understanding frequency tables.

Probability

Includes basic probability calculations and the use of tree and Venn diagrams.

Sampling

Encompasses different sampling methods (random, stratified, systematic) and considerations for bias and sample size calculations.

Statistical Measures

Involves understanding range, mean, median, mode, interquartile range, and standard deviation.

Hypothesis Testing

Focuses on formulating null and alternative hypotheses, conducting tests, and interpreting significance levels.

Scatter Graphs and Correlation

Includes drawing scatter graphs, identifying correlation, and interpreting correlation coefficients.

Time Series Analysis

Involves interpreting time series data, recognizing trends, patterns, and making predictions.

Critical Evaluation

Focuses on evaluating statistical methods, identifying limitations, improvements, and drawing conclusions.

Practical Applications

Applying statistical concepts to real-life scenarios, problem-solving, and effectively communicating findings.

t-Test

$$t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$$

Sample Mean

$$\bar{x}$$

Population Mean

$$\mu$$

Sample Standard Deviation

$${s}$$

Sample Size

$${n}$$

Chi-Square Test

$$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$$

Observed Frequency in Category ${i}$

${O_i}$

Expected Frequency in Category ${i}$

${E_i}$

Type I Error

A false positive, meaning that you falsely reject a true null hypothesis.

Type II Error

• A false negative, meaning that you fail to reject a false null hypothesis.

• In real life scenarios, Type II Errors commonly result in more serious/dangerous errors than Type I.

You can never __________ a __________ hypothesis.

Accept; Null.

Null Hypothesis

$H_0:$ $\text{No effect or difference}$

Alternative hypothesis

$\newline H_1: \text{There is an effect or difference}$

Significance Level

$\alpha = 0.05\ (5\%)$

Low p-value (< .05)

→ strong evidence against null evidence indicating a significant effect of observations

High p-value (≥ .05)

→ weak/insufficient evidence against null hypothesis indicating observations are likely a coincidence/random chance

Random Sampling

Every member of the population has an equal chance of being selected

Systematic Sampling

Selecting every nth member from a list

Stratified Sampling

Population divided into subgroups, sample taken from each subgroup

Cluster Sampling

Population divided into random groups (clusters), then the clusters to collect data from are randomly selected

Quota Sampling

Non-probability, balanced method in which the population is divided into groups (quotas) based on categories (i.e., age, gender, etc.) to ensure each quota has an equal size.

Probability of an Event Formula

$$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$

Complementary Events Formula:

$$P(\text{not } E) = 1 - P(E)$$

Range

• Difference between the highest and lowest values

• Formula: $\text{Range} = \text{Max} - \text{Min}$

Standard Deviation

• Square root of the variance

• Formula: $\sigma = \sqrt{\frac{\sum{(x - \bar{x})^2}}{n}}$

Mean

• Sum of all values divided by the number of values

• Formula: Mean = Total/Number of Values

Median

• Middle value when data is ordered

• If even number of values, median is the average of the two central values