One-Sample Z Tests for Means
So far, we have learned the concepts behind hypothesis testing, including null and alternative hypotheses and possible errors.
Now we move from concepts to actual statistical testing. The one-sample Z test is one of the simplest and most important hypothesis tests.
What Is a One-Sample Z Test?
A one-sample Z test is used to determine whether the mean of a population is significantly different from a known or claimed value.
It compares:
- The sample mean
- The population mean (claimed value)
This test is used when the population standard deviation is known and the sample size is sufficiently large.
When Can We Use a Z Test?
A one-sample Z test is appropriate when:
- The population standard deviation is known
- The sample size is large (usually n ≥ 30)
- The sample is randomly selected
If these conditions are not met, other tests (like the t-test) are more appropriate.
Setting Up the Hypotheses
Before performing the test, we clearly define the hypotheses.
| Hypothesis | Meaning |
|---|---|
| H₀ (Null) | Population mean equals the claimed value |
| H₁ (Alternative) | Population mean differs from the claimed value |
Depending on the problem, the alternative hypothesis can be:
- Two-tailed (≠)
- Left-tailed (<)
- Right-tailed (>)
The Z Test Statistic
The Z test statistic measures how far the sample mean is from the population mean, in standard deviation units.
The formula is:
Z = ( x̄ − μ ) ÷ ( σ / √n )
- x̄ = sample mean
- μ = population mean (claimed)
- σ = population standard deviation
- n = sample size
Deep Numerical Example (Step-by-Step)
A company claims that the average weight of its product is 500 grams.
A random sample of 36 products is taken.
- Sample mean (x̄) = 492 grams
- Population standard deviation (σ) = 12 grams
- Significance level (α) = 0.05
Step 1: State the Hypotheses
H₀: μ = 500
H₁: μ ≠ 500
Step 2: Calculate the Z Value
First calculate the standard error:
σ / √n = 12 / √36 = 12 / 6 = 2
Now calculate Z:
Z = (492 − 500) ÷ 2 = −8 ÷ 2 = −4
Step 3: Determine the Critical Value
For a two-tailed test at α = 0.05:
- Critical Z values = ±1.96
Step 4: Make the Decision
The calculated Z value is −4.
Since −4 lies beyond −1.96, it falls in the rejection region.
Decision: Reject the null hypothesis
Interpretation in Plain English
There is strong statistical evidence that the average product weight is different from 500 grams.
The company’s claim is not supported by the data.
Using the P-Value Approach
Instead of critical values, we can use the p-value.
The p-value represents the probability of observing a result as extreme as the sample, assuming H₀ is true.
Decision rule:
- If p-value ≤ α → Reject H₀
- If p-value > α → Fail to reject H₀
Common Mistakes to Avoid
- Using a Z test when σ is unknown
- Confusing sample mean with population mean
- Ignoring test direction (one-tailed vs two-tailed)
- Claiming H₀ is proven true
Quick Check
When do we reject the null hypothesis using the Z-value method?
Practice Quiz
Question 1:
What happens if |Z| < critical value?
Question 2:
What does the Z statistic measure?
Question 3:
Is a Z test valid for small samples when σ is known?
Mini Practice
A factory claims its average daily output is 200 units.
- x̄ = 195
- σ = 10
- n = 25
- α = 0.05
Perform a one-sample Z test and decide whether to reject H₀.
What’s Next
In the next lesson, we will study One-Sample Tests for Proportions, which apply similar logic to percentages instead of means.