Z-Test Calculator
This Z-test calculator computes data for both one-sample and two-sample Z-tests. It also provides a diagram to show the position of the Z-score and the acceptance/rejection regions. When making a two-sample Z-test calculation, the population mean difference, d, represents the difference between the population means of sample one and sample two, which is μ1-μ2. To use this calculator, simply select the type of calculation from the tab, enter the values, and click the 'Calculate' button.
The Z-test is a statistical procedure used to determine whether there is a significant difference between means, either between a sample mean and a known population mean (one-sample Z-test) or between the means of two independent samples (two-sample Z-test). It assumes that the data is normally distributed and is particularly useful when the sample sizes are large (>30) and the population standard deviations are known. When analyzing data to make informed decisions, statistical hypothesis tests are indispensable tools used to determine if evidence exists to reject a prevailing assumption or theory, known as the null hypothesis. The Z-test is one of these tests.
One-Sample Z-Test
The one-sample Z-test is used when you want to compare the mean of a single sample to a known population mean to see if there is a significant difference. This is particularly common in quality control and other scenarios where the standard deviation of the population is known.
Hypotheses
- Null Hypothesis (H0): The sample mean is equal to the population mean (x̅=μ).
- Alternative Hypothesis (H1): The sample mean is not equal to the population mean (x̅≠μ). This can also be one-tailed (x̅>μ or x̅<μ) depending on the direction of interest.
Formula
The formula for the Z-statistic in a one-sample Z-test is:
| Z = |
|
where:
- x̅ is the sample mean
- μ is the population mean
- σ is the population standard deviation
- n is the sample size
Example: Suppose a school administrator knows the national average score for a standardized test is 500 with a standard deviation of 50. A sample of 100 students from a new teaching program scores an average of 520. To determine if this program significantly differs from the national average:
| Z = |
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= |
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= 4 |
This Z-value would then be compared against a critical value from the Z-distribution table typically at a 0.05 significance level. The critical value for a 0.05 significance level is approximately ±1.96. The Z-value of 4 is greater than 1.96. Therefore, the null hypothesis is rejected and the score of this program is considered significantly different from the national average at the 0.05 significance level.
Two-Sample Z-Test
The two-sample Z-test (or independent samples Z-test) compares the means from two independent groups to determine if there is a statistically significant difference between them.
Hypotheses
- Null Hypothesis (H0): The two population means have a difference of d (μ1-μ2=d). If d is 0, the null hypothesis states that the two population means are equal (μ1=μ2).
- Alternative Hypothesis (H1): The difference between two population means is not d (μ1-μ2≠d), which can also be directional (μ1-μ2>d or μ1-μ2<d). If d is 0, the alternative hypothesis becomes μ1≠μ2 , or μ1>μ2 or μ1<μ2 if it is directional.
Formula
The formula for calculating the Z-statistic in a two-sample Z-test is:
| Z = |
|
where:
- x̅1 and x̅2 are the sample means of groups 1 and 2, respectively
- μ1 and μ2 are the population means, with μ1 - μ2 = d. d is often hypothesized to be zero under the null hypothesis.
- σ1 and σ2 are the population standard deviations
- n1 and n2 are the sample sizes of the two groups
Example: Consider two groups of employees from different branches of a company undergoing training. Group A has 50 employees with an average score of 80 and a standard deviation of 10, and Group B has 50 employees with an average score of 75 and a standard deviation of 12. To test if there's a significant difference:
| Z = |
|
= |
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= 2.26 |
This Z-value is then compared to the critical Z-values to assess significance. The critical value of a 0.05 significance level is around ±1.95. The Z-value of 2.26 is more than 1.95. Therefore, the two group has significant difference at 0.05 significance level.
Significance Level
The significance level (α) is a critical concept in hypothesis testing. It represents the probability threshold below which the null hypothesis will be rejected. Common levels are 0.05 (5%) or 0.01 (1%). The choice of α affects the Z-critical value, which is used to determine whether to reject the null hypothesis based on the computed Z-score.
- Critical Value: This is a point on the Z-distribution that the test statistic must exceed to reject the null hypothesis. For instance, at a 5% significance level in a two-tailed test, the critical values are approximately ±1.96. The significance level (probability) and critical value (Z-score) can be converted with each other the Z-distribution table or use our Z/P converter.
Using the above examples, if the computed Z-scores exceed the respective critical values, the null hypotheses in each case would be rejected, indicating a statistically significant difference as per the alternative hypotheses. These examples demonstrate how the Z-test is applied in different scenarios to test hypotheses concerning population means.