When to Use Z Test Vs. T Test - A Simple Guide

Navigating the world of data science is like solving a grand puzzle. Hypothesis testing is the key, but choosing the right hypothesis test can be challenging. Decode the Z-test vs. T-test comparison with this blog, and get ready to conquer hypothesis testing and supercharge your data science journey!

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Suppose that you are a data scientist trying to determine whether a new drug lowers blood pressure or if a website redesign boosts user engagement. How do you know if the results are just random chance or something real? Welcome to the world of hypothesis testing, where you put your ideas against the odds. In the world of data science, numbers tell stories. But how can you trust those stories to be true? That's where statistical methods like the Z-test and T-test come in, offering you the power to decipher data's secrets. This blog will present a detailed Z-test vs. T-test comparison, highlight their key differences, and guide you on when to use each. So, if you are ready to turn data into wisdom, let us discover the incredible power of these tests and master the art of hypothesis testing and statistics in data science.

Z-Test vs. T-Test - The Differences, Formula and Examples

First, we will discuss the two tests and understand what they are before moving on to the comparison between them.

What is Z-Test?

The Z-test is a fundamental statistical method used for hypothesis testing and making inferences about population parameters. This hypothesis test allows you to determine whether a sample statistic (usually the sample mean) significantly differs from a known population parameter (usually the population mean). The Z-test is useful when you have access to a large dataset and possess knowledge of the population standard deviation, which is essential for accurate statistical calculations.

The Z-test comes in several variations, including the one sample Z-test and the two sample Z-test, each suited to different scenarios

1. One Sample Z-Test

The one sample Z-test compares whether the sample mean (X) is significantly different from a known population mean (μ) when the population standard deviation (σ) is known.

One sample Z-test formula-

Where,

• Z is the Z-score.

• X is the sample mean.

• μ is the population mean.

• σ is the population standard deviation.

• n is the sample size.

2. Two Sample Z-Test

The two sample Z-test is a statistical test that determines whether there is a significant difference between the means of two independent samples when the population standard deviations are known.

Two sample Z-test formula-

Where,

• Z is the Z-score.

• X1 and X2 are the sample means of the two independent sample groups.

• μ1 and μ2 are the population means of the two groups.

• σ1 and σ2 are the population standard deviations of the two groups.

• n1 and n2 are sample sizes of two groups.

The Z-test is used when you meet two critical criteria-

1. Large Sample Size- The Z-test is most reliable when your sample size is sufficiently large, typically n≥30. With a large sample, the sample mean follows a normal distribution, allowing you to use the Z-test confidently.

2. Known Population Standard Deviation (σ)- You should know the population standard deviation. This knowledge is often obtained from historical data or prior research.

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What is T-Test ?

A T-test, similar to the Z-test, is a statistical test that assesses whether there is a significant difference between the means of two groups. It is useful when dealing with small sample sizes or situations where you do not know the population’s standard deviation. The T-test comes in several variations, including the one-sample T-test and the two-sample T-test, each suited to different scenarios.

1. One Sample T-Test

The one-sample T-test determines whether a sample mean differs significantly from a known population mean.

One-sample T-test formula-

where

• t is the T statistics.

• X is the sample mean.

• μ is the population mean (hypothesized mean).

• s is the sample standard deviation.

• n is the sample size.

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2. Two Sample T-Test

The two-sample T-test is used when comparing the means of two independent samples to determine if they significantly differ.

Two-sample T-test formula-

Where

• t is the T statistics.

• x1 and x2 are the sample means of the two independent sample groups.

• Sp is the pooled standard deviation.

• n1 and n2 are sample sizes of two groups.

The T-test is an ideal choice in the following scenarios-

• Small Sample Size- When dealing with small samples (typically n<30), the T-test is more reliable than the Z-test because it accounts for the additional uncertainty caused by a smaller dataset.

• Unknown Population Standard Deviation- If you do not have information about the population standard deviation, the T-test is ideal as it estimates the standard error from the sample data.

Expert Definition Of The T-Test

Javier Fernandez, Data Scientist at Vodafone, defines the T-test in one of his articles as follows-

Now that you have grasped the basic knowledge of these hypothesis testing procedures, let us understand how you can perform each of them successfully for any given data point.

How To Perform Z-Test vs. T-Test?

This section will mainly discuss how to perform these parametric tests step by step using different real-world example use cases.

How To Perform The Z-Test?

Let us understand the key steps involved in performing a Z-test with the help of an example scenario. Suppose you work for an e-commerce company and want to determine if the average time customers take to complete an online purchase has changed from the previously recorded average of 8 minutes.

1. Step 1- State Your Null And Alternative Hypotheses

Define your null hypothesis (H0) and alternative hypothesis (H1) based on your research question. The null hypothesis typically states no significant difference between the sample and population mean. 

In this example scenario, 

• Null Hypothesis (H0)- The average time to complete an online purchase is still 8 minutes, i.e., μ=8.

• Alternative Hypothesis (H1)- The average time to complete an online purchase has changed, i.e., μ is not equal to 8.

2. Step 2- Collect Data

The next step is to gather a sufficiently large sample from your population of interest. You must ensure that the data is randomly selected and unbiased.

In this case, you must randomly sample 50 recent online purchases and record the time it took for each purchase to be completed.

3. Step 3- Calculate The Z-Score

This step involves using the one-sample Z-test formula by plugging in the values for your sample mean (X), the population mean (μ), population standard deviation (σ), and sample size (n).

In this case, let us assume the sample mean (X) is 7.5 minutes, the population standard deviation (σ) is 1.2 minutes, and the sample size (n) is 50.

Z = (7.5 - 8) / (1.2 / sqrt(50)) ~ -2.88

4. Step 4- Determine The Critical Value

Based on your desired level of significance (α) and the test's distribution (usually the standard normal distribution), find the critical Z-value(s) from a Z-table or calculator.

In this example, you must choose a significance level (e.g., α=0.05) and look up the critical Z-value(s) from a Z-table or calculator. For α=0.05, the critical Z-values are approximately -1.96 and 1.96.

5. Step 5- Compare The Z-Score And Critical Value

If the absolute value of the Z-score exceeds the critical value, you reject the null hypothesis (H0). Otherwise, you accept it.

The calculated Z-score in this example (-2.88) is less than -1.96 (the critical value for α=0.05). Hence, you must reject the null hypothesis (H0).

6. Step 6- Draw Conclusion

The final step involves interpreting the results in the context of your research question. If you reject the null hypothesis, it indicates that the sample and population mean significantly differ.

In this case, there is strong evidence to suggest that the average time to complete an online purchase has changed from the previously recorded average of 8 minutes based on this sample.

How To Perform The T-Test?

Let us understand the key steps involved in performing a T-test with the help of an example scenario. Suppose you work for an e-commerce company and want to determine if a new website design has significantly increased the average time users spend on the website compared to the old design.

1. Step 1- State Your Null And Alternative Hypotheses

Define your null hypothesis (H0) and alternative hypothesis (H1)  based on your research question. The null hypothesis typically states no significant difference between the sample and population mean. 

In this example scenario, 

• Null Hypothesis (H0)- The average time users spend on the website with the new design is the same as the old design, i.e., μ new =μ old.

• Alternative Hypothesis (H1)- The average time users spend on the website with the new design is greater than the old design, i.e., μ new >μ old.

2. Step 2- Collect Data

The second step is to gather your sample data, ensuring that it meets the assumptions of the T-test (e.g., random sampling, independence).

For this example, you must record the time spent by a random sample of 30 users on the website with the new design and another random sample of 30 users on the website with the old design.

3. Step 3- Calculate The T-Statistic

Now, it's time to use the appropriate T-test formula for your scenario and plug in the values for your sample mean (X), the hypothesized population mean (μ), sample standard deviation (s), and sample size (n).

In this case, you will use the two-sample T-test formula. Let us assume that for the new design, the sample mean (X new) is 5 minutes, the sample standard deviation (

s new) is 1.2 minutes, and the sample size (n new) is 30. For the old design, let us assume X old is 4.5 minutes, s old is 1.1 minutes, and n old is 30.

t = (5 - 4.5) / [sqrt{(1.2^2/30)+(1.1^2/30)}] ~ 2.27

4. Step 4- Determine The Critical Value

Based on your chosen level of significance (α) and degrees of freedom, you must find the critical T-value(s) from a T-table or calculator.

In this example, you must choose a significance level (e.g., α=0.05) and find the critical T-value(s) from a T-table or calculator. For α=0.05 and 58 degrees of freedom (assuming equal sample sizes in both groups), the critical T-value is approximately 1.671.

5. Step 5- Compare The T-Statistic And Critical Value

If the absolute value of the T-statistic is higher than the critical value, you reject the null hypothesis (H0). Otherwise, you accept it.

In this example, the calculated T-statistic (2.27) is greater than 1.671 (the critical value for α=0.05). Hence, you must reject the null hypothesis.

6. Step 6- Draw Conclusion

The last step is to interpret the results in the context of your research question. If you reject the null hypothesis, it suggests that the sample and population mean differ.

Based on the sample of the above T-test example, there is strong evidence to suggest that the new website design has caused a significant increase in the average time users spend on the website compared to the old design.

Let us discuss the significant differences between the two parametric tests to know which is better.

T-Test vs. Z-Test- Key Differences

The table below shows the key differences between the two statistical methods, Z-test and T-test, to help you better understand these tests.

Expert Opinion on Key Differences Between T-Test Vs. Z-Test

Udit Saini, Principal Applied Data Scientist at UnitedHealth Group, discusses some of the main differences between these two tests as follows-

Are you still confused about when to use Z-test vs. T-test? Let us find out in the following section which test is better suited in which case.

When To Use T-Test vs. Z-Test?

Selecting the right statistical test is crucial for drawing valid conclusions from your data. The decision to use either a Z-test or a T-test hinges on several factors, including the size of your sample and your knowledge of the population standard deviation. Let us understand when do you use Z-test vs. T-test-

1. Sample Size

The Z-test is most reliable when you have a large sample size. This is typically defined as a sample size greater than or equal to 30. The Central Limit Theorem states that the sample mean tends to have a normal distribution for sufficiently large samples. Therefore, you can confidently use the Z-test to analyze your data.

The distribution of the sample mean may not closely resemble a normal distribution when working with small to moderate sample sizes.  In such cases, using the T-test is more suitable. The T-test accounts for the additional uncertainty associated with smaller datasets.

2. Population Standard Deviation Knowledge

Before using the Z-test, you must know the population standard deviation (σ). Historically collected data points, past research, or established facts are commonly used to gather this information. The Z-test gives more reliable results when the population standard deviation is known.

The T-test is ideal if you lack the population standard deviation (σ) knowledge. In many real-world scenarios, when you won't have access to this population parameter, the T-test is the right choice as it estimates the standard error from the sample data itself.

3. Types Of Hypotheses

The Z-test is mainly used for testing hypotheses related to sample variance in population means. It lets you determine whether the sample mean significantly differs from a known population mean.

Like the Z-test, the T-test allows you to test hypotheses related to population means. It is mainly useful when the sample size is insufficient to rely on the Z-test.

4. Comparing Two Samples

The Z-test is not specifically designed for comparing two independent samples. It is mainly used for single-sample hypothesis testing.

The T-test is a suitable option if your analysis involves comparing the means of two independent samples to see if there is a significant difference between them. This situation often occurs in A/B testing, clinical trials, and comparative studies.

5. Assumptions

The Z-test assumes that the data follows a normal distribution. While this assumption holds well for large samples, it may not be as accurate for smaller samples.

Like the Z-test, the T-test assumes that the data follows a normal distribution. However, it is more robust to deviations from normality, especially when dealing with smaller samples. It is crucial to assess and validate this assumption when using T tests.

Decode When to Use Z-Test Vs. T-Test with ProjectPro

Ready to discover the power of statistics in your data science journey? The question of when to use a Z-test or T-test isn't just an academic one. It's the key to making data-driven decisions in the real world. Now, how will you step into the shoes of a data scientist, armed with the knowledge of these tests? The answer lies in practical experience. Dive into Data Science projects with ProjectPro. Hands-on practice isn't just about grasping theory; it's about building effective data analysis solutions leveraging the potential of Z-tests, T-tests, and more. Elevate your data science game, bring insights to life, and shape your career with these industry-level projects from the ProjectPro repository. Explore, learn, and create!

FAQs on Z-Test vs. T-Test

Should I use a T-test or a Z-test?

You should use a Z-test when you have a large sample size (typically 30 or more) and know the population standard deviation. You should opt for T tests when dealing with small to moderate sample sizes (typically <30) or when the population standard deviation remains unknown. Your choice depends on these key factors for accurate and meaningful hypothesis testing.

What are Z tests used for?

Z-tests are primarily used to assess whether a sample mean significantly differs from a known population mean when you have a large sample size (typically 30 or more) and know the population standard deviation. They are used in hypothesis testing to draw conclusions about population variance between means in various fields, including quality control, finance, and manufacturing, where large datasets and known parameters are available for analysis.

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