🧪 Z-Test

Last Updated: Jan 2026

A Z-Test is a hypothesis test used to determine whether a sample mean is significantly different from a population mean when the population variance is known and the sample size is large.

🗣 Hinglish Tip: Z-Test = jab sample bada ho aur population ka $\sigma$ pata ho, tab mean compare karte hain

When to Use Z-Test?

Use Z-Test when all conditions are satisfied:

• Sample size: $n \ge 30$
• Population standard deviation $\sigma$ is known
• Data is normally distributed or CLT applies
• Random & independent samples

Types of Z-Test

1. One-Sample Z-Test → Compare sample mean with population mean
2. Two-Sample Z-Test → Compare means of two large samples
3. Z-Test for Proportion → Compare proportions

👉 In this tutorial, we cover One-Sample Z-Test (most common)

Z-Test Notation (Math Standard)

• Population mean → $\mu$
• Sample mean → $\bar{x}$
• Population standard deviation → $\sigma$
• Sample size → $n$
• Significance level → $\alpha$
• Z statistic → $Z$

Z-Test Formula

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

Decision Rule (Z-Table Method)

For $\alpha = 0.05$:

Example

A company claims that the average battery life is 100 hours. A sample of 50 batteries is tested and the following is observed:

• Sample mean: $\bar{x} = 98$
• Population standard deviation: $\sigma = 5$
• Significance level: $\alpha = 0.05$

Test whether the company's claim is correct.

Step 1: State the Hypotheses

Since we are checking difference, this is a two-tailed test.

$H_0: \mu = 100
\\[0.5em]
H_1: \mu \ne 100$

Step 2: Identify Test Type

• $n = 50 \ge 30$ $\checkmark$
• $\sigma$ known $\checkmark$

👉 Use Z-Test

Step 3: Compute Standard Error

$\frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{50}}
\\[0.5em]
= \frac{5}{7.07}
\approx 0.707$

Step 4: Calculate Z-Statistic

$Z = \frac{98 - 100}{0.707}
\\[0.5em]
Z = \frac{-2}{0.707}
\\[0.5em]
Z \approx -2.83$

Step 5: Find Critical Value

From Z-table ($\alpha = 0.05$, two-tailed):

$\text{Critical Z} = \pm 1.96$

Step 6: Decision

Since:

$|Z| = 2.83 > 1.96$

👉 Reject $H_0$

Step 7: Conclusion

There is sufficient evidence to conclude that the average battery life is NOT equal to 100 hours.

🗣 Hinglish Tip: Z value limit cross kar gaya → claim reject

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