Statstics

z-test: Definition, Assumptions, Advantages, Disadvantages, and Real-Life Example

Published on August 25, 2023
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Table of Contents

1 Definition:
2 Mathematical Notation:
3 Assumptions:
4 Advantages:
5 Disadvantages:

Definition:

The z-test is a statistical hypothesis test used to assess whether the means of two populations are significantly different when the population standard deviation is known. It’s particularly suitable for large sample sizes. The test statistic follows a standard normal distribution (z-distribution), and it’s calculated by comparing the difference between sample means to the expected sampling variation.

Mathematical Notation:

Let’s consider two independent samples, Sample 1 with mean $\bar{x}_1$ and Sample 2 with mean $\bar{x}_2$ , and a known population standard deviation $\sigma$ . The z-test statistic is calculated as:

$z = \frac{(\bar{x}_1 - \bar{x}_2)}{(\sigma / \sqrt{n})}$

Where:
– $\bar{x}_1$ and $\bar{x}_2$ are the sample means.
– $\sigma$ is the known population standard deviation.
– $n$ is the sample size.

Assumptions:

1. The samples are random and independent.
2. The data within each sample are normally distributed or the sample size is sufficiently large for the Central Limit Theorem to apply.
3. The population standard deviation $\sigma$ is known.

Advantages:

– Simple and straightforward to calculate.
– Provides a clear test statistic (z-value) for comparing means.
– Suitable for large sample sizes.
– Offers a direct way to determine statistical significance.

Disadvantages:

– Requires knowledge of the population standard deviation, which is often unavailable.
– Limited to situations where the assumptions of normality and known population standard deviation are met.

Example
Imagine a social science study examining the impact of a new teaching method on student test scores. Researchers want to determine if the average test scores of students who underwent the new teaching method are significantly different from those who followed the traditional method.

In a random sample of 100 students who received the new teaching method, their average test score ( $\bar{x}_1$ ) was 85, with a known population standard deviation ( $\sigma$ ) of 10. On the other hand, a sample of 120 students who underwent the traditional teaching method had an average score ( $\bar{x}_2$ ) of 80.

Using the z-test formula:

$z = \frac{(85 - 80)}{(10 / \sqrt{100})} = 2.5$

Assuming a significance level ( $\alpha$ ) of 0.05, we can compare the z-value to the critical z-score for a two-tailed test (z-critical ≈ 1.96). Since 2.5 > 1.96, we reject the null hypothesis, concluding that there is a significant difference in test scores between the two teaching methods.

In this example, the z-test allowed researchers to determine the statistical significance of the difference in test scores between the two teaching methods, providing valuable insights for educational decision-making.

In summary, the z-test is a useful tool in statistics for comparing means when the population standard deviation is known. It’s applicable in various fields, including social sciences, economics, and more, but its assumptions and limitations should be carefully considered when interpreting results.