Difference Between z-test and t-test

When it comes to statistical analysis, knowing which test to use can be the difference between accurate conclusions and flawed assumptions. The z-test and t-test are two commonly used methods to test hypotheses and make data-driven decisions. While both tests share similarities, they have distinct differences that are important to understand. In this article, we'll explore the primary differences between the z-test and t-test, helping you decide which test is best suited for your data analysis needs.

Statistics is essential for making informed decisions in various fields. Two popular methods to test hypotheses are z-test and t-test. It's crucial to understand their differences to choose the appropriate one. The following sections will explore the primary differences between these tests.

Table of Content

• Difference Between z-test and t-test
• What is z-test?
• What is t-test?
• Key Difference Between z-test and t-test

What is the Difference Between z-test and t-test?

Parameter	Z-test	T-test
Population Standard Deviation	Known	Unknown
Sample Size	Large (typically > 30)	Small (typically < 30)
Distribution	Normal distribution	Normal distribution
Test Statistic	Z-statistic	T-statistic
Degrees of Freedom	Not applicable	n - 1 (for one-sample t-test)
Critical Values	From the standard normal distribution	From the t-distribution
Hypothesis Testing	Tests if the population mean is equal to a hypothesized value	Tests if the population mean is equal to a hypothesized value
Underlying Assumptions	Data is normally distributed, and the population standard deviation is known	Data is normally distributed, and the population standard deviation is unknown
Robustness	Less robust to violations of normality assumption than the t-test	More robust to violations of normality assumption than the z-test
Confidence Interval Construction	Based on the standard normal distribution	Based on the t-distribution
Statistical Power	Generally higher power than the t-test when assumptions are met	Generally lower power than the z-test when assumptions are met
Effect Size Estimation	More accurate estimation of effect size when assumptions are met	Less accurate estimation of effect size when assumptions are violated
Applicability	Suitable for large sample sizes and known population standard deviation	Suitable for small sample sizes and unknown population standard deviation
Computational Complexity	Simpler calculations	More complex calculations due to the use of the t-distribution
Interpretation	Easier to interpret results based on the standard normal distribution	Slightly more complex interpretation due to the use of the t-distribution
Applications	Quality control, manufacturing processes, and other scenarios where population standard deviation is known	Research studies, experiments, and other scenarios where population standard deviation is unknown

What is z-test?

A z-test is a statistical test used to determine whether the mean of a population differs significantly from a hypothesized value or another population mean when the population standard deviation is known. The z-test assumes that the data is normally distributed and that the sample size is large (typically greater than 30).

Formula for Z-test

The formula for the z-test statistic is:

z = (x̄ - μ) / (σ / √n)

Where:

• x̄ is the sample mean
• μ is the hypothesized population mean
• σ is the known population standard deviation
• n is the sample size

Example of Z-test

Suppose a company claims that its products' average weight is 500 grams. You take a random sample of 50 products, and the sample mean weight is 510 grams. If the population standard deviation is known to be 20 grams, you can use a z-test to determine whether the company's claim is valid.

Given:

Sample mean (x̄) = 510 grams

Hypothesized population mean (μ) = 500 grams

Population standard deviation (σ) = 20 grams

Sample size (n) = 50

Substitute the values in the z-test formula:

z = (510 - 500) / (20 / √50)

z = 10 / (20 / 7.07) = 10 / 2.83

z = 3.53

Note: With a large sample size, the z-statistic follows a standard normal distribution. If the calculated z-value falls outside the critical region (e.g., beyond ±1.96 for a 95% confidence level), you can reject the null hypothesis and conclude that the population mean differs significantly from the hypothesized value.

What is a t-test?

A t-test is a statistical test used to determine whether the mean of a population differs significantly from a hypothesized value or another population mean when the population standard deviation is unknown. The t-test assumes that the data is normally distributed, and it is commonly used for small sample sizes (typically less than 30).

Formula for T-test

The formula for the one-sample t-test statistic is:

t = (x̄ - μ) / (s / √n)

Where:

• x̄ is the sample mean
• μ is the hypothesized population mean
• s is the sample standard deviation
• n is the sample size

Example of T-test

Suppose you want to test whether the average weight of a certain type of apple differs significantly from 150 grams. You take a random sample of 15 apples and find that the sample mean weight is 160 grams with a sample standard deviation of 10 grams.

Given:

Sample mean (x̄) = 160 grams

Hypothesized population mean (μ) = 150 grams

Sample standard deviation (s) = 10 grams

Sample size (n) = 15

Substitute the values in the t-test formula:

t = (160 - 150) / (10 / √15)

t = 10 / (10 / 3.87) = 10 / 2.58

t = 3.88

Note: With n - 1 = 15 - 1 = 14 degrees of freedom and a 95% confidence level, the critical t-value from the t-distribution table is approximately 2.145. Since the calculated t-value (3.88) is greater than the critical value, you can reject the null hypothesis and conclude that the population mean weight of the apples differs significantly from 150 grams.

Key Difference Between z-test and t-test

Here are the five most important key differences between the z-test and the t-test:

Population Standard Deviation Assumption:

• Z-test: The population standard deviation (σ) is assumed to be known.
• T-test: The population standard deviation (σ) is assumed to be unknown.

Sample Size Requirement:

• Z-test: Requires a large sample size, typically greater than 30.
• T-test: Can be used with small sample sizes, typically less than 30.

Underlying Distribution:

• Z-test: Assumes that the population is normally distributed.
• T-test: Assumes that the population is normally distributed, but it accounts for the uncertainty in estimating the population standard deviation from a small sample.

Test Statistic:

• Z-test: Uses the z-statistic, which is calculated based on the standard normal distribution.
• T-test: Uses the t-statistic, which is calculated based on the t-distribution with degrees of freedom (n - 1).

Critical Values:

• Z-test: Critical values are derived from the standard normal distribution.
• T-test: Critical values are derived from the t-distribution, which is more spread out than the standard normal distribution, especially for small degrees of freedom.

Conclusion

z-test and t-test are statistical methods to test the hypothesis, whether the assumed or given hypothesis is correct or not. In this article, we have briefly discussed what z-tests and t-tests are with the help of examples and the difference between z-tests and t-tests. Hope you will like the article.

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