Further Hypothesis Testing

This lesson extends the hypothesis testing framework to include tests for the mean of a normal distribution and the correlation coefficient. These are key topics at A-Level and bring together the concepts of the normal distribution, sampling, and statistical inference.

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Hypothesis Test for the Mean of a Normal Distribution

If $X \sim N(\mu, \sigma^2)$X∼N(μ,σ2) and $\sigma^2$σ2 is known, we can test hypotheses about $\mu$μ using the sample mean $\bar{X}$Xˉ.

Distribution of the Sample Mean

If $X \sim N(\mu, \sigma^2)$X∼N(μ,σ2), then the sample mean of a random sample of size $n$n follows:

$\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$Xˉ∼N(μ,nσ2​)

The standard error of the mean is:

$\text{SE} = \frac{\sigma}{\sqrt{n}}$SE=n​σ​

Test Procedure

1. State hypotheses:

   • $H_0: \mu = \mu_0$H0​:μ=μ0​
   • $H_1: \mu > \mu_0$H1​:μ>μ0​ or $\mu < \mu_0$μ<μ0​ or $\mu \neq \mu_0$μ=μ0​

2. Assume $H_0$H0​ is true: $\bar{X} \sim N\left(\mu_0, \frac{\sigma^2}{n}\right)$Xˉ∼N(μ0​,nσ2​)

3. Calculate the test statistic: $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$z=σ/n​xˉ−μ0​​

4. Compare with the critical value or find the p-value.

5. Make a decision and state the conclusion in context.