Z-Tests for a Population Mean

Scope Label

Core 9758. This branch covers one-tailed and two-tailed $Z$-tests for a single population mean. It is the main procedural branch for Hypothesis Testing.

Use it with Hypothesis Testing Logic and Decision Methods for the conceptual language.

When a $Z$-Test Is Used

The core $Z$-test applies when testing a claim about one population mean $\mu$ and the population standard deviation $\sigma$ is known.

The sample statistic is $\bar{X}$.

Under $H_0:\mu ={\mu }_0$, the test statistic is

$$Z=\frac{\bar{X}-{\mu }_0}{\sigma /\sqrt{n}}.$$

For observed data, use

$$z_{\text{obs}}=\frac{\bar{x}-{\mu }_0}{\sigma /\sqrt{n}}.$$

The denominator is the standard error of the sample mean. This links directly to Sampling Distribution of the Sample Mean.

Distributional Justification

The $Z$-test is justified in two common cases:

• if the population is normal, then $\bar{X}$ is exactly normal;
• if the sample size is large, then $\bar{X}$ is approximately normal by the central limit theorem.

In both cases, under $H_0$,

$$Z\sim N(0,1)$$

exactly or approximately.

Standard Workflow

For a $Z$-test:

1. Define $\mu$ clearly in context.
2. State $H_0$ and $H_1$.
3. Identify the tail direction.
4. State or compute the test statistic.
5. Use either the critical-region method or the p-value method.
6. Write a contextual conclusion.

Caption: A core $Z$-test follows a fixed chain from context and hypotheses to a statistic, decision method, and conclusion.

Two Routes to the Same $Z$ Statistic

The formula for the $Z$ statistic is the same in the two common justifications, but the reason it is valid is different.

Situation	Distribution of $\bar{X}$	Comment
Population normal, $\sigma$ known	exact normal	works for any $n$
Population not necessarily normal, $\sigma$ known, $n$ large	approximately normal	uses the central limit theorem

In both cases, under $H_0:\mu ={\mu }_0$,

$$Z=\frac{\bar{X}-{\mu }_0}{\sigma /\sqrt{n}}$$

is standard normal exactly or approximately.

Do not use this core $Z$-test formula if the population standard deviation is not given and the question is not clearly using a large-sample approximation.

Critical-Region Method

The critical-region method compares the observed statistic with critical values.

For common $Z$-tests:

Test	Alternative	Critical-region idea
Lower-tailed	$H_1:\mu <{\mu }_0$	reject for sufficiently negative $z$
Upper-tailed	$H_1:\mu >{\mu }_0$	reject for sufficiently positive $z$
Two-tailed	$H_1:\mu \ne {\mu }_0$	reject for sufficiently large $

The critical values come from the standard normal distribution.

p-Value Method

The p-value is computed under $H_0$.

For $z_{\text{obs}}$:

• lower-tailed test: $P(Z\le z_{\text{obs}})$;
• upper-tailed test: $P(Z\ge z_{\text{obs}})$;
• two-tailed test: probability of being at least as extreme in either direction.

Reject $H_0$ if

$$\text{p-value}<\alpha .$$

Caption: The critical-region and p-value methods lead to the same decision when applied to the same test.

Worked Example 1: Lower-Tailed $Z$-Test

A manufacturer claims that the mean mass of a component is $50$ g. The population standard deviation is known to be $4$ g. A random sample of $36$ components has mean mass $48.8$ g.

Test at the $5\%$ level whether the mean mass has decreased.

Step 1: Define the parameter

Let $\mu$ be the true mean mass of the components.

Step 2: State the hypotheses

$$H_0:\mu =50,H_1:\mu <50.$$

This is a lower-tailed test.

Step 3: Compute the statistic

$$z_{\text{obs}}=\frac{48.8-50}{4/\sqrt{36}}=\frac{-1.2}{2/3}=-\mathrm{1.8.}$$

Critical-region method

For a lower-tailed $5\%$ test, the critical value is approximately $-1.645$.

Since

$$-1.8<-1.645,$$

the observed statistic lies in the critical region. Reject $H_0$.

Conclusion: there is sufficient evidence at the $5\%$ level that the mean mass has decreased.

p-Value method

The p-value is

$$P(Z<-1.8)\approx \mathrm{0.0359.}$$

Since

$$0.0359<0.05,$$

reject $H_0$.

The conclusion is the same.

Worked Example 2: Two-Tailed $Z$-Test

A machine is claimed to fill packets with mean mass $500$ g. The population standard deviation is known to be $6$ g. A random sample of $25$ packets has mean mass $502.8$ g.

Test at the $5\%$ level whether the machine is incorrectly calibrated.

Let $\mu$ be the true mean fill mass.

$$H_0:\mu =500,H_1:\mu \ne 500.$$

This is a two-tailed test because “incorrectly calibrated” means too low or too high.

The observed statistic is

$$z_{\text{obs}}=\frac{502.8-500}{6/\sqrt{25}}=\frac{2.8}{1.2}=2.333\dots .$$

For a two-tailed $5\%$ test, the critical values are approximately

$$z=\pm \mathrm{1.96.}$$

Since

$$2.333\dots >1.96,$$

reject $H_0$.

Conclusion: there is sufficient evidence at the $5\%$ level that the machine is incorrectly calibrated.

Worked Example 3: Insufficient Evidence

A company claims that a process has mean completion time $40$ minutes. The population standard deviation is $8$ minutes. A sample of $64$ jobs has mean time $38.6$ minutes.

Test at the $1\%$ level whether the mean completion time is less than $40$ minutes.

$$H_0:\mu =40,H_1:\mu <40.$$

The observed statistic is

$$z_{\text{obs}}=\frac{38.6-40}{8/\sqrt{64}}=\frac{-1.4}{1}=-\mathrm{1.4.}$$

For a lower-tailed $1\%$ test, the critical value is approximately $-2.326$.

Since

$$-1.4>-2.326,$$

the statistic is not in the critical region. Do not reject $H_0$.

Conclusion: there is insufficient evidence at the $1\%$ level that the mean completion time is less than $40$ minutes.

Calculator Discipline

A graphing calculator can be used to:

• compute tail probabilities;
• compute p-values;
• find normal critical values.

However, the calculator does not decide:

• the correct $H_1$;
• the test direction;
• whether to use $\sigma$ or $\sigma /\sqrt{n}$;
• how to write the conclusion.

Those decisions must come from the context and the sampling distribution.

When reading calculator output, identify:

• the test statistic, usually reported as $z$;
• the p-value;
• whether the tail choice entered into the calculator matches $H_1$.

The calculator may give a number, but it does not know whether the question asks for an increase, decrease, or any change. That decision must be made before using the calculator.

Common Pitfalls

• Using $X$ instead of $\bar{X}$ in the test statistic.
• Forgetting the denominator $\sigma /\sqrt{n}$.
• Using a one-tailed critical value for a two-tailed test.
• Failing to state $\mu$ in context.
• Writing “accept $H_0$” when the correct conclusion is “do not reject $H_0$”.
• Reporting the decision without a contextual conclusion.

Revision Checklist

• Can you state when a $Z$-test for one mean is appropriate?
• Can you compute $z_{\text{obs}}$ correctly?
• Can you choose the correct critical region from $H_1$?
• Can you calculate and interpret a p-value?
• Can you write a conclusion in context?