Hypothesis Testing: Z-Test

Setting up the hypothesis test

Step 1 — State hypotheses:

$H_0: \mu = 500$ (battery life is as claimed)

$H_a: \mu \neq 500$ (battery life differs from claim)

Step 2 — Significance level: $\alpha = 0.05$ (two-tailed).

Computing the test statistic

Step 3 — Calculate $z$:

$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} = \frac{490 - 500}{30 / \sqrt{36}} = \frac{-10}{5} = -2.0$

Making the decision

Step 4 — Find critical values. For two-tailed at $\alpha = 0.05$: $z_{\text{crit}} = \pm 1.96$.

Step 5 — Compare: $|z| = 2.0 > 1.96$. The test statistic falls in the rejection region.

Step 6 — p-value: $P(|Z| > 2.0) = 2 \times 0.0228 = 0.0456 < 0.05$.

$\boxed{\text{Reject } H_0. \text{ Evidence suggests battery life differs from 500 hours.}}$

Interpretation

At the 5% significance level, the sample mean of 490 is significantly different from the claimed 500. The company's claim is not supported by this data.

Try it in the visualization

The normal curve shows the rejection regions (red shaded tails). The test statistic is marked. If it falls in the red zone → reject $H_0$.