1 Variable Z Test Calculator

What is a 1 Variable Z Test?

The 1 Variable Z Test is a statistical method used to test hypotheses about a population mean when the population standard deviation is known. It's commonly used in quality control, manufacturing, and other fields where population parameters are well-established.

This test compares a sample mean to a hypothesized population mean to determine whether the difference is statistically significant. The test statistic follows a standard normal distribution (Z-distribution) under the null hypothesis.

Key Concepts

• Null Hypothesis (H₀): The sample mean is equal to the hypothesized population mean (μ₀)
• Alternative Hypothesis (H₁): The sample mean is not equal to the hypothesized population mean (two-tailed test)
• Test Statistic: Z = (x̄ - μ₀) / (σ/√n) where x̄ is the sample mean, μ₀ is the hypothesized population mean, σ is the population standard deviation, and n is the sample size
• Critical Value: The Z-value that corresponds to the chosen significance level (α)
• P-value: The probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true

When to Use

Use the 1 Variable Z Test when:

• The population standard deviation (σ) is known
• The sample size is large (n ≥ 30)
• You want to test whether a sample mean differs from a known population mean

How to Use This Calculator

1. Enter the hypothesized population mean (μ₀)
2. Enter the sample mean (x̄)
3. Enter the population standard deviation (σ)
4. Enter the sample size (n)
5. Select the significance level (α)
6. Click "Calculate" to perform the test

The calculator will display the Z-score, p-value, and decision about the null hypothesis.

Formula

The test statistic for the 1 Variable Z Test is calculated as:

Where:

• Z = Z-score
• x̄ = Sample mean
• μ₀ = Hypothesized population mean
• σ = Population standard deviation
• n = Sample size

The p-value is calculated based on the standard normal distribution and the calculated Z-score.

Worked Example

Suppose a manufacturer claims that the mean lifetime of their light bulbs is 1000 hours (μ₀ = 1000). A sample of 50 bulbs has a mean lifetime of 990 hours (x̄ = 990) with a known standard deviation of 50 hours (σ = 50).

Using the calculator:

1. Enter μ₀ = 1000
2. Enter x̄ = 990
3. Enter σ = 50
4. Enter n = 50
5. Select α = 0.05
6. Click "Calculate"

The calculator will show:

• Z-score = -2.00
• P-value = 0.0455
• Decision: Reject the null hypothesis (p < α)

This means there is statistically significant evidence (at the 5% significance level) that the mean lifetime of the bulbs is less than 1000 hours.

Interpreting Results

After running the test, you'll receive:

• Z-score: The standardized test statistic
• P-value: The probability of observing the data if the null hypothesis is true
• Decision: Whether to reject or fail to reject the null hypothesis

Decision Rules

• If p-value < α: Reject the null hypothesis (there is significant evidence against H₀)
• If p-value ≥ α: Fail to reject the null hypothesis (there is not enough evidence to reject H₀)