Hypothesis Testing (Z-Test)

Hypothesis testing helps us determine if there's enough evidence to support a claim about a population.

Steps:

State the Hypotheses:
- Null Hypothesis $H_0$: No effect or difference (e.g., $\mu = \mu_0$).
- Alternative Hypothesis $H_1$: A significant effect or difference (e.g., $\mu \neq \mu_0$).
Select the Significance Level $\alpha$ (commonly 0.05).
Calculate the Z-statistic:

$$ Z = \frac{\bar{X} - \mu_0}{\frac{\sigma}{\sqrt{n}}} $$

where $\bar{X}$ is the sample mean, $\mu_0$ is the population mean under $H_0$, $\sigma$ is the population standard deviation, and $n$ is the sample size.
Compare the Z-statistic to Critical Value (or use the p-value approach):
- If $|Z|$ is greater than the critical value (1.96 for $\alpha = 0.05$ in a two-tailed test), reject $H_0$.
Alternatively, compute the $p$-value (the probability that the $Z$ statistic is at least extreme as the one computed)
1. Reject if the $p$ value is less than the significant level $\alpha$

Confidence Intervals

A confidence interval provides a range of plausible values for a population parameter based on a sample statistic.

Formula for a Confidence Interval for the Mean (known population standard deviation):

$$ CI = \bar{X} \pm Z_{\frac{\alpha}{2}} \cdot \frac{\sigma}{\sqrt{n}} $$

$\bar{X}$: Sample mean
$Z_{\frac{\alpha}{2}}$: Z-value corresponding to the desired confidence level (e.g., 1.96 for 95%)
$\sigma$: Population standard deviation
$n$: Sample size

A 95% confidence interval means that if we repeated the experiment many times, 95% of the calculated intervals would contain the true population mean.

Practice:

Review quiz 2