Hypothesis Testing for the Normal Distribution

This lesson covers hypothesis testing for a population mean using the normal distribution as required by the Edexcel A-Level Mathematics specification (9MA0), Paper 3 Section A -- Statistics. You need to test whether a sample provides evidence of a change in the population mean, using a test statistic and critical values.

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When to Use a Normal Distribution Test

Use this test when:

• The population is normally distributed (or large sample -- Central Limit Theorem).
• The population standard deviation (sigma) is known.
• You are testing a claim about the population mean (mu).

The test statistic

Z = (X-bar - mu0) / (sigma / sqrt(n))

where X-bar is the sample mean, mu0 is the hypothesised mean, sigma is the known population SD, and n is the sample size.

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Setting Up the Hypotheses

• H0: mu = mu0
• H1: mu < mu0 (one-tailed, lower)
• H1: mu > mu0 (one-tailed, upper)
• H1: mu ≠ mu0 (two-tailed)

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The Standard Error

Standard error = sigma / sqrt(n)

As n increases, the standard error decreases -- larger samples give more precise estimates.

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