Difference between Z-Test and T-Test

Z-tests are used when the population variance is known and the sample size is large, while t-tests are used when the population variance is unknown and the sample size is small.

This article explains the differences between Z-tests and T-tests, detailing their purposes, assumptions, sample size requirements, and applications in statistical hypothesis testing.

Z-test is a statistical test used to determine whether there is a significant difference between sample and population means or between the means of two samples.

It is typically used when the sample size is large (generally n > 30) and the population standard deviation is known. The Z-test is based on the standard normal distribution (Z-distribution).

The Z-test compares the means of two populations with a large sample size (typically ≥30) and known population standard deviation. It assesses whether the difference between the means is statistically significant.

Types of Z-Test

There are two types of Z-test i.e.,

• One-Sample Z-Test
  • Compares the sample mean to a known population mean.
  • Used to determine if the sample comes from a population with a specific mean.
• Two-Sample Z-Test
  • Compares the means of two independent samples.
  • Used to determine if there is a significant difference between the two sample means.

Read More about Z-Score.

T-test is a statistical test used to determine whether there is a significant difference between the means of two groups.

It is particularly useful when the sample size is small (typically n < 30) and the population standard deviation is unknown. The T-test relies on the t-distribution, which is similar to the normal distribution but has heavier tails.

Types of T-Tests

There are three types of T-test i.e.,

• One-Sample T-Test
  • Compares the sample mean to a known value (usually a population mean).
  • Used to determine if the sample comes from a population with a specific mean.
• Two-Sample T-Test (Independent T-Test)
  • Compares the means of two independent samples.
  • Used to determine if there is a significant difference between the means of two groups.
• Paired Sample T-Test (Dependent T-Test)
  • Compares means from the same group at different times (e.g., before and after a treatment) or from matched pairs.
  • Used to determine if there is a significant difference between paired observations.

Read More about T-test.

Some of the common difference between Z-test and T-test are:

In conclusion, the Z-test and T-test are both valuable tools for comparing population means, with the Z-test suited for large samples with known standard deviations, and the T-test for smaller samples with unknown deviations. Their applications vary based on sample size and data characteristics.

Read More,

• Statistics
• Z-Score Table
• Normal Distribution

What is the main difference between a Z-test and a T-test?

• Z-Test: Used when the sample size is large (n > 30) and the population standard deviation is known.
• T-Test: Used when the sample size is small (n < 30) and the population standard deviation is unknown.

When should I use a Z-test instead of a T-test?

Use a Z-test when you have a large sample size and the population standard deviation is known. It’s often used for hypothesis testing about means when these conditions are met.

When should I use a T-test instead of a Z-test?

Use a T-test when the sample size is small and the population standard deviation is unknown. It’s also used when comparing the means of two samples or paired observations.

How do I interpret the results of a Z-test or T-test?

Compare the test statistic (Z or t) to the critical value from the Z or t distribution table, or compare the P-value to your significance level (e.g., 0.05). If the test statistic exceeds the critical value or the P-value is less than the significance level, reject the null hypothesis.

Can I use a Z-test for small samples?

Generally, no. Z-tests are not recommended for small samples because the Z distribution assumes a large sample size for the Central Limit Theorem to hold. For small samples, use a T-test.