Example Of One Sample Z Test

One Sample Z-Test: A Comprehensive Guide with Examples

The one-sample z-test is a powerful statistical tool used to determine whether a sample mean significantly differs from a known population mean. Understanding its application is crucial for researchers across various fields, from healthcare and social sciences to engineering and finance. This comprehensive guide will delve into the intricacies of the one-sample z-test, explaining its assumptions, calculations, interpretation, and providing illustrative examples to solidify your understanding.

Understanding the One-Sample Z-Test

The core principle behind the one-sample z-test lies in comparing a sample's mean to a pre-determined population mean. This comparison isn't just about checking for a difference; it assesses whether the observed difference is statistically significant, meaning it's unlikely to have occurred by random chance alone. The test utilizes the z-statistic, a standardized score that measures the distance between the sample mean and the population mean in terms of standard errors.

When to Use a One-Sample Z-Test:

A one-sample z-test is appropriate when the following conditions are met:

• Known Population Standard Deviation: You know the population standard deviation (σ). This is a crucial differentiator from the t-test, which is used when the population standard deviation is unknown.
• Random Sampling: The sample data must be obtained through a random sampling method to ensure the sample is representative of the population.
• Large Sample Size (or Normally Distributed Population): While the central limit theorem states that larger samples (generally n ≥ 30) tend toward a normal distribution regardless of the population distribution, using a z-test with smaller samples requires the population to be approximately normally distributed. Severe departures from normality can invalidate the z-test's results.
• Independent Observations: Each observation in the sample should be independent of the others. This means that the value of one observation does not influence the value of another.

The Z-Statistic Formula and its Components

The z-statistic is calculated using the following formula:

z = (x̄ - μ) / (σ / √n)

Where:

• x̄: The sample mean. This is the average of your sample data.
• μ: The population mean. This is the known mean of the population you're comparing your sample to.
• σ: The population standard deviation. This is a measure of the spread or variability of the population data.
• n: The sample size. This is the number of observations in your sample.

Let's break down each component:

• (x̄ - μ): This represents the difference between the sample mean and the population mean. This difference is the core of our hypothesis test. A large difference suggests a potential significant divergence.
• (σ / √n): This is the standard error of the mean. The standard error measures the variability of the sample mean. A smaller standard error indicates a more precise estimate of the population mean based on the sample. Notice that the standard error decreases as the sample size (n) increases. This reflects the increased precision with larger samples.

Hypothesis Testing with the One-Sample Z-Test

The one-sample z-test is a hypothesis test, meaning we formulate a null hypothesis (H₀) and an alternative hypothesis (Hₐ).

Typical Hypotheses:

• Null Hypothesis (H₀): There is no significant difference between the sample mean and the population mean. This is often stated as: x̄ = μ
• Alternative Hypothesis (Hₐ): There is a significant difference between the sample mean and the population mean. This can be:
  • Two-tailed: x̄ ≠ μ (The sample mean is different from the population mean – this tests for differences in either direction).
  • One-tailed (left-tailed): x̄ < μ (The sample mean is significantly less than the population mean).
  • One-tailed (right-tailed): x̄ > μ (The sample mean is significantly greater than the population mean).

Choosing a Significance Level (Alpha)

Before conducting the test, you need to choose a significance level (α), typically set at 0.05 (5%). This represents the probability of rejecting the null hypothesis when it is actually true (Type I error). A lower alpha value indicates a stricter threshold for rejecting the null hypothesis.

Calculating the p-value

The p-value is the probability of observing a sample mean as extreme as, or more extreme than, the one obtained, assuming the null hypothesis is true. A small p-value (typically less than α) suggests strong evidence against the null hypothesis, leading to its rejection.

Making a Decision: Reject or Fail to Reject the Null Hypothesis

Based on the calculated p-value and the chosen significance level, you make a decision:

• If p-value ≤ α: Reject the null hypothesis. There is sufficient evidence to conclude that there is a significant difference between the sample mean and the population mean.
• If p-value > α: Fail to reject the null hypothesis. There is not enough evidence to conclude that there is a significant difference between the sample mean and the population mean.

Example 1: Testing the Average Height of Students

Let's say a researcher wants to determine if the average height of students at a particular university is different from the national average. The national average height (μ) is known to be 68 inches, with a standard deviation (σ) of 4 inches. The researcher randomly selects a sample of 100 students (n=100) and calculates their average height (x̄) as 69 inches. Let's perform a two-tailed z-test at a significance level of α = 0.05.

1. State the Hypotheses:

   • H₀: x̄ = 68 inches
   • Hₐ: x̄ ≠ 68 inches

2. Calculate the z-statistic: z = (69 - 68) / (4 / √100) = 2.5

3. Find the p-value: Using a z-table or statistical software, the p-value for a two-tailed test with z = 2.5 is approximately 0.0124.

4. Make a Decision: Since the p-value (0.0124) is less than α (0.05), we reject the null hypothesis.

5. Conclusion: There is sufficient evidence to conclude that the average height of students at this university is significantly different from the national average.

Example 2: Assessing Employee Satisfaction

A company wants to evaluate employee satisfaction. They know the average satisfaction score across all similar companies (μ) is 7.5 (on a scale of 1-10), with a standard deviation (σ) of 1.2. They randomly survey 50 employees (n=50) and find the average satisfaction score (x̄) to be 7. Let's perform a one-tailed (left-tailed) z-test at a significance level of α = 0.01 to see if employee satisfaction is significantly lower than the industry average.

1. State the Hypotheses:

   • H₀: x̄ ≥ 7.5
   • Hₐ: x̄ < 7.5

2. Calculate the z-statistic: z = (7 - 7.5) / (1.2 / √50) ≈ -2.95

3. Find the p-value: For a left-tailed test with z ≈ -2.95, the p-value is approximately 0.0016.

4. Make a Decision: Since the p-value (0.0016) is less than α (0.01), we reject the null hypothesis.

5. Conclusion: There is sufficient evidence to conclude that employee satisfaction at this company is significantly lower than the industry average.

Interpreting the Results and Limitations

The results of a one-sample z-test should be interpreted carefully within the context of the study. A statistically significant result doesn't necessarily imply practical significance. The effect size should be considered along with the statistical significance to fully understand the implications.

Limitations of the One-Sample Z-Test:

• Dependence on known population standard deviation: This is often the biggest limitation. The population standard deviation is rarely known in real-world scenarios.
• Assumption of normality (for small samples): Violation of the normality assumption can lead to inaccurate results.
• Sensitivity to outliers: Extreme values in the sample can disproportionately influence the results.
• Sample representativeness: The validity of the results hinges on the sample being truly representative of the population.

Alternatives to the One-Sample Z-Test

When the population standard deviation is unknown, the one-sample t-test is the appropriate alternative. The t-test uses the sample standard deviation as an estimate of the population standard deviation, which makes it more versatile for real-world applications.

Conclusion

The one-sample z-test is a valuable tool for comparing a sample mean to a known population mean. Understanding its assumptions, calculations, and interpretations is crucial for conducting and interpreting statistical analyses correctly. While it has limitations, particularly the requirement of a known population standard deviation, its simplicity and ease of understanding make it a foundational concept in statistics. Remember to carefully consider the context, limitations, and potential alternatives when applying this powerful statistical technique.