One-Sample Z-Test
Hypothesis testing is a fundamental aspect of Lean Six Sigma methodologies, which are employed to improve process efficiency and quality by eliminating defects and variability. Among the various statistical tests used in hypothesis testing, the One-Sample Z-Test holds a special place, especially when dealing with parametric tests for normal data. This article delves into the One-Sample Z-Test, outlining its purpose, application, and how it fits into the broader spectrum of Lean Six Sigma initiatives.
Introduction to One-Sample Z-Test
The One-Sample Z-Test is a parametric statistical method used to determine whether the mean of a sample differs significantly from a known or hypothesized population mean. This test is applicable when the data follows a normal distribution, and the population standard deviation is known. It's particularly useful in Lean Six Sigma projects for validating improvements or changes in processes against established benchmarks or standards.
When to Use the One-Sample Z-Test
The One-Sample Z-Test is most appropriate in situations where:
The sample size is large (n > 30) as per the Central Limit Theorem, which states that the sampling distribution of the mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.
The population standard deviation (σ) is known.
The data is quantitative and follows a normal distribution.
Steps for Conducting a One-Sample Z-Test
1.State the Hypotheses:
Null Hypothesis (H0): Assumes no difference between the sample mean and the population mean (μ0).
Alternative Hypothesis (Ha): Asserts a significant difference. This can be two-tailed, left-tailed, or right-tailed, depending on the situation.
2.Set the Significance Level (α): Commonly set at 0.05, this threshold defines the probability of rejecting the null hypothesis when it is actually true.
3.Calculate the Test Statistic: Use the formula
where:
xˉ is the sample mean.
μ0 is the hypothesized population mean.
σ is the population standard deviation.
n is the sample size.
5. Make a Decision: Compare the calculated Z-value with the critical value(s):
If the Z-value falls into the rejection region, reject the null hypothesis.
Otherwise, fail to reject the null hypothesis.
Application in Lean Six Sigma
In Lean Six Sigma projects, the One-Sample Z-Test can be applied in various scenarios, such as:
Evaluating the effectiveness of process improvements by comparing the mean output before and after implementing changes.
Verifying compliance with industry standards or benchmarks.
Assessing customer satisfaction levels against a target score.
Conclusion
The One-Sample Z-Test is a powerful tool in the Lean Six Sigma toolkit for hypothesis testing with normal data. It provides a statistical basis for making informed decisions about process improvements and quality control measures. By following a structured approach to hypothesis testing, Lean Six Sigma practitioners can substantiate their claims about process enhancements and drive meaningful change within their organizations.
Example One-Sample Z-Test
A pharmaceutical company claims that its new medication reduces blood pressure by an average of 8 mmHg with a population standard deviation (σ) of 2 mmHg. A health agency wants to verify this claim before recommending the medication for public use.
Objective:
To verify if the new medication effectively reduces blood pressure by the claimed average of 8 mmHg.
Methodology - One-Sample Z-Test:
Hypothesis Setup:
Null Hypothesis (H0): The mean reduction in blood pressure by the medication is equal to 8 mmHg. (µ = 8)
Alternative Hypothesis (H1): The mean reduction in blood pressure by the medication is not equal to 8 mmHg. (µ ≠ 8)
Data Collection:
The health agency conducts a study with a sample of 50 patients who have been prescribed the new medication. After a specified period, the mean reduction in blood pressure recorded in the sample is 7.5 mmHg.
Significance Level:
The agency decides on a significance level of 0.05 (α = 0.05), which means they are willing to accept a 5% chance of wrongly rejecting the null hypothesis.
Test Statistic Calculation:
The Z-Test formula is used to calculate the test statistic:
Where:
xˉ = sample mean = 7.5 mmHg
μ = hypothesized population mean = 8 mmHg
σ = population standard deviation = 2 mmHg
n = sample size = 50
Plugging the values in, the calculation gives:
Decision Rule:
The Z critical value for a two-tailed test at α = 0.05 is approximately ±1.96. Upon reviewing the table below, our objective is to locate the value corresponding to a significance level of 0.05. By adding the values from the intersecting row and column (1.9 + 0.06), we obtain 1.96, indicating the critical Z-value for our analysis.
To facilitate visualization, please refer to the chart below:
Since the calculated Z value of -1.77 does not exceed the critical value of ±1.96, the null hypothesis cannot be rejected.
Conclusion:
The health agency concludes that there is not enough statistical evidence to reject the null hypothesis at the 5% significance level. Therefore, it cannot be concluded that the medication's average blood pressure reduction differs from the claimed 8 mmHg based on this sample. This example illustrates how a One-Sample Z-Test helps in making informed decisions in real-life scenarios, such as verifying the efficacy of medical treatments.