Hypothesis Testing (Further)

This lesson extends hypothesis testing to cover the z-test, t-test, tests for proportions, and tests for the difference between two means. These are the core inference procedures in Further Statistics 2.

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Framework for Hypothesis Testing

Every hypothesis test follows the same logical structure:

Step	Action
1	State $H_0$H0​ (null hypothesis) and $H_1$H1​ (alternative hypothesis)
2	Choose significance level $\alpha$α
3	Calculate the test statistic
4	Find the critical value or p-value
5	Make a decision: reject or do not reject $H_0$H0​
6	State the conclusion in context

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z-Test for the Mean ($\sigma$σ Known)

$H_0: \mu = \mu_0$H0​:μ=μ0​. Test statistic:

$Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$Z=σ/n​xˉ−μ0​​

Compare with $z_{\alpha}$zα​ (one-tailed) or $z_{\alpha/2}$zα/2​ (two-tailed).

Worked Example

A manufacturer claims mean weight is 500 g ($\sigma = 10$σ=10). A sample of 36 gives $\bar{x} = 497$xˉ=497. Test at 5% (two-tailed).

$Z = \frac{497 - 500}{10/\sqrt{36}} = \frac{-3}{1.667} = -1.80$Z=10/36​497−500​=1.667−3​=−1.80

Critical value: $z_{0.025} = \pm 1.960$z0.025​=±1.960. Since $|Z| = 1.80 < 1.96$∣Z∣=1.80<1.96, do not reject $H_0$H0​.

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