One Sample Z-Test Calculator

Sample Mean (X̄):

Population Mean (μ):

Population Standard Deviation (σ):

Sample Size (n):

Calculate Z-Score

 

About One Sample Z-Test Calculator (Formula)

A One Sample Z-Test is a statistical test used to determine whether the mean of a single sample differs significantly from a known population mean or a hypothesized mean. This test is commonly used in hypothesis testing when you have a single set of data points and want to determine if it’s representative of a larger population or if there’s a significant difference between the sample and the population.

Here’s the formula for a One Sample Z-Test:

Z = (X̄ – μ) / (σ / √(n))

Where:

• Z is the Z-statistic.
• X̄ (pronounced as “X-bar”) is the sample mean.
• μ (pronounced as “mu”) is the population mean (the known mean or the hypothesized mean).
• σ (pronounced as “sigma”) is the population standard deviation (if known).
• n is the sample size.

The steps to perform a One Sample Z-Test are as follows:

1. Formulate the null hypothesis (H0) and the alternative hypothesis (H1 or Ha).
   • H0: The sample mean is equal to the population mean (μ).
   • Ha: The sample mean is not equal to the population mean (μ), indicating a two-tailed test. Alternatively, you can use a one-tailed test if you have a specific direction in mind (greater than or less than).
2. Collect your sample data and calculate the sample mean (X̄) and, if possible, the population standard deviation (σ).
3. Determine the significance level (α), which represents the probability of making a Type I error (rejecting the null hypothesis when it is true). Common choices for α include 0.05 and 0.01.
4. Calculate the Z-statistic using the formula mentioned above.
5. Compare the calculated Z-statistic to the critical Z-value(s) from the standard normal distribution table or use a statistical calculator. The critical value(s) correspond to your chosen significance level (α) and the type of test (two-tailed or one-tailed).
6. Make a decision:
   • If |Z| > critical value: Reject the null hypothesis (H0) in favor of the alternative hypothesis (Ha).
   • If |Z| ≤ critical value: Fail to reject the null hypothesis (H0).
7. Draw a conclusion based on your decision and report the results.

This test helps you determine whether the observed difference between your sample mean and the population mean is statistically significant or if it could have occurred due to random sampling variation.

Keep in mind that for practical purposes, it’s often recommended to use statistical software or calculators to perform One Sample Z-Tests because they can handle the calculations and critical value lookup efficiently.