2 Sample T Test Or Z Test

Choosing Between a Z-Test and a T-Test: A Practical Guide with Examples

Determining whether to use a Z-test or a t-test is a crucial step in hypothesis testing. Both tests assess whether there's a significant difference between a sample mean and a population mean (or between two sample means), but they differ in their assumptions and applications. This article will clarify these differences, providing two sample scenarios illustrating each test. We'll focus on understanding the context in which each test is most appropriate and demonstrating the process step-by-step.

Understanding the Differences: Z-Test vs. T-Test

The core difference lies in the knowledge of the population standard deviation.

• Z-test: Uses a Z-statistic and assumes you know the population standard deviation (σ). This is rare in real-world scenarios. Z-tests are often used for large sample sizes where the sample standard deviation is a good approximation of the population standard deviation.

• T-test: Uses a t-statistic and estimates the population standard deviation from the sample standard deviation (s). This is the more common approach, particularly with smaller sample sizes. The t-distribution accounts for the uncertainty introduced by estimating the population standard deviation.

Example 1: Z-Test – Population Standard Deviation Known

Let's say a company claims their light bulbs have an average lifespan of 1000 hours (µ = 1000). Through rigorous testing, they've established the population standard deviation is 50 hours (σ = 50). A consumer advocacy group tests a sample of 100 bulbs (n = 100) and finds a mean lifespan of 980 hours (x̄ = 980). They want to test if this significantly differs from the company's claim.

Hypothesis:

• Null hypothesis (H₀): µ = 1000 (The company's claim is true)
• Alternative hypothesis (H₁): µ ≠ 1000 (The company's claim is false)

Calculation:

1. Calculate the Z-statistic: Z = (x̄ - µ) / (σ / √n) = (980 - 1000) / (50 / √100) = -4

2. Determine the p-value: Using a Z-table or statistical software, find the probability of observing a Z-statistic as extreme as -4 (or 4) under the null hypothesis. This p-value will be very small, indicating strong evidence against the null hypothesis.

3. Conclusion: Since the p-value is very low (typically below a significance level of 0.05), we reject the null hypothesis. There is significant evidence to suggest the average lifespan of the bulbs is different from the company's claim.

Example 2: T-Test – Population Standard Deviation Unknown

A researcher wants to compare the average height of men and women. They sample 30 men (n₁ = 30) and find a mean height of 175 cm (x̄₁ = 175) with a sample standard deviation of 5 cm (s₁ = 5). They also sample 35 women (n₂ = 35) and find a mean height of 165 cm (x̄₂ = 165) with a sample standard deviation of 4 cm (s₂ = 4). They want to know if there's a significant difference in average height. This requires an independent samples t-test.

Hypothesis:

• Null hypothesis (H₀): µ₁ = µ₂ (There is no difference in average height)
• Alternative hypothesis (H₁): µ₁ ≠ µ₂ (There is a difference in average height)

Calculation:

This involves calculating the t-statistic, considering the degrees of freedom (df), and using a t-table or statistical software to find the p-value. The precise calculation requires a formula which is beyond the scope of this simple explanation, but statistical software packages readily perform this calculation.

Conclusion: Based on the calculated t-statistic and p-value, the researcher would either reject or fail to reject the null hypothesis, concluding whether a statistically significant difference exists between the average heights of men and women.

Choosing the Right Test:

Remember, the choice hinges on whether you know the population standard deviation. If you know it, use a Z-test. If you don't (the more realistic scenario), use a t-test. Always consider the sample size; for very large samples, the difference between Z and t-tests becomes negligible. Using statistical software simplifies the calculations and p-value determination significantly.