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Year 12 Maths

Statistical Inference

Understand the normal distribution, construct confidence intervals, and apply hypothesis testing to draw conclusions from data.

The Normal Distribution

The normal distribution (bell curve) is the most important probability distribution in statistics. It is symmetric about the mean and described by two parameters: the mean (μ) and standard deviation (σ).

Notation

X ~ N(μ, σ2)

X follows a normal distribution with mean μ and variance σ2.

The 68-95-99.7 Rule (Empirical Rule)

68%

within 1σ of μ

95%

within 2σ of μ

99.7%

within 3σ of μ

Z-scores standardise any normal distribution: z = (x − μ) / σ. A z-score tells you how many standard deviations a value is from the mean.

Confidence Intervals

A confidence interval gives a range of plausible values for a population parameter, based on sample data. A 95% CI means we are 95% confident the true parameter lies within this range.

95% Confidence Interval for a Mean

x̄ ± 1.96 × σ √n

x̄ = sample mean, σ = population standard deviation, n = sample size.

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Example: A sample of 100 students has mean test score 72, σ = 10. Find a 95% CI.

Margin of error: 1.96 × 10/√100 = 1.96 × 1 = 1.96

95% CI: 72 ± 1.96 = (70.04, 73.96)

Hypothesis Testing

Hypothesis testing uses sample data to test a claim about a population parameter. The process has clear steps.

Null Hypothesis (H0)

The claim we assume true unless evidence says otherwise. Usually “no effect” or “no difference.”

Alternative Hypothesis (H1)

What we suspect is true. Can be one-tailed (< or >) or two-tailed (≠).

Steps for Hypothesis Testing

  1. State H0 and H1
  2. Choose significance level (α, typically 0.05)
  3. Calculate the test statistic (z-score)
  4. Find the p-value or compare to critical value
  5. Make a decision: reject H0 if p-value < α

Important: “Not rejecting H0” does not prove H0 is true. It means there is insufficient evidence to reject it.

Knowledge Check

Test your understanding of statistical inference concepts.

Question 1

In a normal distribution, approximately what percentage of data falls within one standard deviation of the mean?

Question 2

A value has a z-score of −2. What does this mean?

Question 3

X ~ N(50, 25). What is the standard deviation?

Question 4

Calculate the z-score for x = 85 when μ = 70 and σ = 10.

Question 5

What does a 95% confidence interval mean?

Question 6

In hypothesis testing, what is the null hypothesis (H0)?

Question 7

If the significance level is α = 0.05 and the p-value is 0.03, what is the decision?

Question 8

A sample of 64 has mean 50, σ = 8. What is the 95% confidence interval for the population mean?

Question 9

What happens to the width of a confidence interval when the sample size increases?

Question 10

A company claims their batteries last 500 hours. A sample of 36 batteries has mean 490 hours with σ = 30. Test at α = 0.05 whether the mean is less than 500. What is the z-score?

Key Concepts Summary