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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t-Test for the Mean ($\sigma$σ Unknown)

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

$T = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \sim t_{n-1}$T=s/n​xˉ−μ0​​∼tn−1​

Worked Example

A sample of 15 batteries has mean lifetime 48.5 hours and $s = 3.2$s=3.2. The manufacturer claims $\mu = 50$μ=50. Test at 5% (one-tailed: $H_1: \mu < 50$H1​:μ<50).

$T = \frac{48.5 - 50}{3.2/\sqrt{15}} = \frac{-1.5}{0.826} = -1.816$T=3.2/15​48.5−50​=0.826−1.5​=−1.816

Critical value: $t_{14, 0.05} = 1.761$t14,0.05​=1.761. Since $|T| = 1.816 > 1.761$∣T∣=1.816>1.761, reject $H_0$H0​. There is evidence that the mean lifetime is less than 50 hours.

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Test for a Proportion

For a large sample, test $H_0: p = p_0$H0​:p=p0​ using:

$Z = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}$Z=p0​(1−p0​)/n​p^​−p0​​

Note: Under $H_0$H0​, the standard error uses $p_0$p0​ (not $\hat{p}$p^​).

Worked Example

A coin is tossed 200 times and lands heads 115 times. Test whether the coin is biased at the 5% level (two-tailed).

$H_0: p = 0.5$H0​:p=0.5, $H_1: p \neq 0.5$H1​:p=0.5.

$Z = \frac{0.575 - 0.5}{\sqrt{0.5 \times 0.5/200}} = \frac{0.075}{0.03536} = 2.121$Z=0.5×0.5/200​0.575−0.5​=0.035360.075​=2.121

Since $2.121 > 1.960$2.121>1.960, reject $H_0$H0​. There is significant evidence that the coin is biased.

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Test for the Difference of Two Means

Known Variances (z-test)