Mathematical Methods
Random Variables, Normal Distribution, and Confidence Intervals
Year 11 · Year 12
- ✓By the end of this lesson students will be able to define and classify discrete and continuous random variables.
- ✓By the end of this lesson students will be able to describe the properties of the normal distribution and calculate probabilities using it.
- ✓By the end of this lesson students will be able to apply the Central Limit Theorem to understand the sampling distribution of the sample mean.
- ✓By the end of this lesson students will be able to construct and interpret confidence intervals for population means (with known standard deviation or large sample size) and population proportions.
- ✓By the end of this lesson students will be able to use appropriate technology (e.g., graphics calculator) to solve problems involving the normal distribution and confidence intervals.
Key concepts
A random variable is a variable whose value is a numerical outcome of a random phenomenon. Random variables can be classified as either discrete or continuous.\n\n* **Discrete Random Variable**: A variable that can take on a finite or countably infinite number of values. These values are often integers and result from counting. Examples include the number of heads in three coin tosses, or the number of cars passing a point in an hour.\n* **Continuous Random Variable**: A variable that can take on any value within a given range. These values are typically measurements. Examples include the height of a student, the time taken to complete a task, or the temperature of a room.
The normal distribution is a continuous probability distribution that is symmetric about its mean, forming a bell-shaped curve. It is one of the most important distributions in statistics due to the Central Limit Theorem. It is characterised by two parameters: the mean (μ) and the standard deviation (σ). We denote a normal distribution as N(μ, σ²).\n\nKey properties:\n* Symmetric about the mean (μ), so the mean, median, and mode are all equal.\n* The total area under the curve is 1.\n* Approximately 68% of the data falls within one standard deviation of the mean (μ ± σ).\n* Approximately 95% of the data falls within two standard deviations of the mean (μ ± 2σ).\n* Approximately 99.7% of the data falls within three standard deviations of the mean (μ ± 3σ).
The standard normal distribution is a special case of the normal distribution where the mean (μ) is 0 and the standard deviation (σ) is 1. Any normal random variable X can be transformed into a standard normal random variable Z using the Z-score formula. This standardisation allows us to compare values from different normal distributions and use standard normal tables or calculators to find probabilities.
The sampling distribution of the sample mean (X̄) is the probability distribution of all possible sample means that could be drawn from a population of a given size. The Central Limit Theorem (CLT) is fundamental here:\n\n* If the population itself is normally distributed with mean μ and standard deviation σ, then the sampling distribution of X̄ will also be normally distributed with mean E(X̄) = μ and standard deviation SD(X̄) = σ/√n, regardless of sample size n.\n* If the population is not normally distributed, but the sample size n is sufficiently large (typically n ≥ 30), then the sampling distribution of X̄ will be approximately normally distributed with mean E(X̄) = μ and standard deviation SD(X̄) = σ/√n. The standard deviation of the sample mean, σ/√n, is often called the standard error of the mean.
A confidence interval (CI) provides a range of plausible values for an unknown population parameter (like the mean μ) based on sample data. A 95% confidence interval, for example, means that if we were to take many samples and construct a CI from each, approximately 95% of these intervals would contain the true population mean.\n\nFor a population mean μ, when the population standard deviation σ is known (or when the sample size n is large enough for the CLT to apply and the sample standard deviation 's' can be used as an estimate for σ, allowing z-scores), the confidence interval is calculated as:\n\nCI = Sample Mean ± Margin of Error\nMargin of Error = Critical Z-value × Standard Error
Similar to the mean, a confidence interval for a population proportion (p) provides a range of plausible values for the true proportion of a characteristic in a population. This is used when dealing with categorical data (e.g., proportion of people who prefer a certain brand).\n\nFor a population proportion p, the confidence interval is calculated as:\n\nCI = Sample Proportion ± Margin of Error\nMargin of Error = Critical Z-value × Standard Error of Proportion
Key facts to remember
- 1Random variables are numerical outcomes of random phenomena, classified as discrete (countable) or continuous (measurable).
- 2The normal distribution is symmetric, bell-shaped, and defined by its mean (μ) and standard deviation (σ).
- 3The Z-score (Z = (X - μ) / σ) standardises any normal variable to the standard normal distribution (μ=0, σ=1).
- 4The Central Limit Theorem states that for large sample sizes, the sampling distribution of the sample mean is approximately normal, regardless of the population distribution.
- 5The standard error of the mean is σ/√n, and the standard error of the proportion is √(p̂(1-p̂)/n).
- 6A confidence interval provides a range of plausible values for a population parameter, with a specified level of confidence.
- 7The margin of error in a confidence interval is calculated as Critical Z-value × Standard Error.
Worked examples
Example 1
The scores on a standardised maths test are normally distributed with a mean of 65 and a standard deviation of 8. What is the probability that a randomly selected student scores between 60 and 70?
Answer
P(60 < X < 70) ≈ 0.4680
Always draw a sketch of the normal curve to visualise the area you are calculating.
Example 2
A random sample of 100 Year 12 students is taken, and their average height is found to be 170 cm. Assume the population standard deviation of heights for Year 12 students is 8 cm. Construct a 95% confidence interval for the true mean height of all Year 12 students.
Answer
The 95% confidence interval for the true mean height of all Year 12 students is (168.43 cm, 171.57 cm) (rounded to two decimal places).
Interpretation: We are 95% confident that the true mean height of all Year 12 students lies between 168.43 cm and 171.57 cm.
Example 3
In a survey of 250 voters, 130 stated they would vote for Party A. Construct a 90% confidence interval for the true proportion of voters who would vote for Party A.
Answer
The 90% confidence interval for the true proportion of voters who would vote for Party A is (0.468, 0.572).
Ensure that n*p̂ ≥ 5 and n*(1-p̂) ≥ 5 for the normal approximation to be valid. Here, 250*0.52 = 130 and 250*0.48 = 120, both are ≥ 5.
Common mistakes
- ✗Confusing discrete and continuous random variables, especially when determining which probability distribution to use.
- ✗Incorrectly calculating Z-scores or misinterpreting the area under the normal curve (e.g., finding P(X > x) instead of P(X < x)).
- ✗Not understanding that a confidence interval estimates the population parameter, not individual data points or future sample values.
- ✗Using the sample standard deviation (s) instead of the population standard deviation (σ) when σ is known, or failing to use the correct standard error formula for means vs. proportions.
- ✗Rounding intermediate calculations too early, leading to inaccuracies in the final confidence interval.
Exam tips
- ★Always draw a sketch of the normal distribution curve for probability questions to visualise the area you need to find.
- ★Clearly state the parameters (μ, σ, n) and the critical Z-value (z*) used in confidence interval calculations.
- ★Show all steps in your working, including the formula used, substitution of values, and the final calculation, especially when using a calculator.
- ★Interpret confidence intervals in the context of the problem, explaining what the interval means in plain language.
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