Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.
Recommended Resources
Free eBook: Top Programming Languages For A Data Scientist
Normality Test in Minitab: Minitab with Statistics
Machine Learning Career Guide: A Playbook to Becoming a Machine Learning Engineer
PMP, PMI, PMBOK, CAPM, PgMP, PfMP, ACP, PBA, RMP, SP, and OPM3 are registered marks of the Project Management Institute, Inc.
Hypothesis Testing Calculator
$H_o$:
$H_a$:
μ
≠
μ₀
$n$
=
$\bar{x}$
=
=
$\text{Test Statistic: }$
=
$\text{Degrees of Freedom: } $
$df$
=
$ \text{Level of Significance: } $
$\alpha$
=
Type II Error
$H_o$:
$\mu$
$H_a$:
$\mu$
≠
$\mu_0$
$n$
=
σ
=
$\mu$
=
$\text{Level of Significance: }$
$\alpha$
=
The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.
$\sigma$ Known
$\sigma$ Unknown
Test Statistic
$ z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{{\color{Black} n}}} $
$ t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}} $
Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.
Lower Tail Test
Upper Tail Test
Two-Tailed Test
$H_0 \colon \mu \geq \mu_0$
$H_0 \colon \mu \leq \mu_0$
$H_0 \colon \mu = \mu_0$
$H_a \colon \mu
$H_a \colon \mu \neq \mu_0$
In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.
To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.
In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.
To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.
Lower Tail Test
Upper Tail Test
Two-Tailed Test
If $z \leq -z_\alpha$, reject $H_0$.
If $z \geq z_\alpha$, reject $H_0$.
If $z \leq -z_{\alpha/2}$ or $z \geq z_{\alpha/2}$, reject $H_0$.
If $t \leq -t_\alpha$, reject $H_0$.
If $t \geq t_\alpha$, reject $H_0$.
If $t \leq -t_{\alpha/2}$ or $t \geq t_{\alpha/2}$, reject $H_0$.
When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.
Condition
$H_0$ True
$H_a$ True
Conclusion
Accept $H_0$
Correct
Type II Error
Reject $H_0$
Type I Error
Correct
Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.
Skip to secondary menu
Skip to main content
Skip to primary sidebar
Statistics By Jim
Making statistics intuitive
Z Test: Uses, Formula & Examples
By Jim Frost Leave a Comment
What is a Z Test?
Use a Z test when you need to compare group means. Use the 1-sample analysis to determine whether a population mean is different from a hypothesized value. Or use the 2-sample version to determine whether two population means differ.
A Z test is a form of inferential statistics . It uses samples to draw conclusions about populations.
For example, use Z tests to assess the following:
One sample : Do students in an honors program have an average IQ score different than a hypothesized value of 100?
Two sample : Do two IQ boosting programs have different mean scores?
In this post, learn about when to use a Z test vs T test. Then we’ll review the Z test’s hypotheses, assumptions, interpretation, and formula. Finally, we’ll use the formula in a worked example.
Related post : Difference between Descriptive and Inferential Statistics
Z test vs T test
Z tests and t tests are similar. They both assess the means of one or two groups, have similar assumptions, and allow you to draw the same conclusions about population means.
However, there is one critical difference.
Z tests require you to know the population standard deviation, while t tests use a sample estimate of the standard deviation. Learn more about Population Parameters vs. Sample Statistics .
In practice, analysts rarely use Z tests because it’s rare that they’ll know the population standard deviation. It’s even rarer that they’ll know it and yet need to assess an unknown population mean!
A Z test is often the first hypothesis test students learn because its results are easier to calculate by hand and it builds on the standard normal distribution that they probably already understand. Additionally, students don’t need to know about the degrees of freedom .
Z and T test results converge as the sample size approaches infinity. Indeed, for sample sizes greater than 30, the differences between the two analyses become small.
William Sealy Gosset developed the t test specifically to account for the additional uncertainty associated with smaller samples. Conversely, Z tests are too sensitive to mean differences in smaller samples and can produce statistically significant results incorrectly (i.e., false positives).
When to use a T Test vs Z Test
Let’s put a button on it.
When you know the population standard deviation, use a Z test.
When you have a sample estimate of the standard deviation, which will be the vast majority of the time, the best statistical practice is to use a t test regardless of the sample size.
However, the difference between the two analyses becomes trivial when the sample size exceeds 30.
Learn more about a T-Test Overview: How to Use & Examples and How T-Tests Work .
Z Test Hypotheses
This analysis uses sample data to evaluate hypotheses that refer to population means (µ). The hypotheses depend on whether you’re assessing one or two samples.
One-Sample Z Test Hypotheses
Null hypothesis (H 0 ): The population mean equals a hypothesized value (µ = µ 0 ).
Alternative hypothesis (H A ): The population mean DOES NOT equal a hypothesized value (µ ≠ µ 0 ).
When the p-value is less or equal to your significance level (e.g., 0.05), reject the null hypothesis. The difference between your sample mean and the hypothesized value is statistically significant. Your sample data support the notion that the population mean does not equal the hypothesized value.
Related posts : Null Hypothesis: Definition, Rejecting & Examples and Understanding Significance Levels
Two-Sample Z Test Hypotheses
Null hypothesis (H 0 ): Two population means are equal (µ 1 = µ 2 ).
Alternative hypothesis (H A ): Two population means are not equal (µ 1 ≠ µ 2 ).
Again, when the p-value is less than or equal to your significance level, reject the null hypothesis. The difference between the two means is statistically significant. Your sample data support the idea that the two population means are different.
These hypotheses are for two-sided analyses. You can use one-sided, directional hypotheses instead. Learn more in my post, One-Tailed and Two-Tailed Hypothesis Tests Explained .
Related posts : How to Interpret P Values and Statistical Significance
Z Test Assumptions
For reliable results, your data should satisfy the following assumptions:
You have a random sample
Drawing a random sample from your target population helps ensure that the sample represents the population. Representative samples are crucial for accurately inferring population properties. The Z test results won’t be valid if your data do not reflect the population.
Related posts : Random Sampling and Representative Samples
Continuous data
Z tests require continuous data . Continuous variables can assume any numeric value, and the scale can be divided meaningfully into smaller increments, such as fractional and decimal values. For example, weight, height, and temperature are continuous.
Other analyses can assess additional data types. For more information, read Comparing Hypothesis Tests for Continuous, Binary, and Count Data .
Your sample data follow a normal distribution, or you have a large sample size
All Z tests assume your data follow a normal distribution . However, due to the central limit theorem, you can ignore this assumption when your sample is large enough.
The following sample size guidelines indicate when normality becomes less of a concern:
One-Sample : 20 or more observations.
Two-Sample : At least 15 in each group.
Related posts : Central Limit Theorem and Skewed Distributions
Independent samples
For the two-sample analysis, the groups must contain different sets of items. This analysis compares two distinct samples.
Related post : Independent and Dependent Samples
Population standard deviation is known
As I mention in the Z test vs T test section, use a Z test when you know the population standard deviation. However, when n > 30, the difference between the analyses becomes trivial.
Related post : Standard Deviations
Z Test Formula
These Z test formulas allow you to calculate the test statistic. Use the Z statistic to determine statistical significance by comparing it to the appropriate critical values and use it to find p-values.
The correct formula depends on whether you’re performing a one- or two-sample analysis. Both formulas require sample means (x̅) and sample sizes (n) from your sample. Additionally, you specify the population standard deviation (σ) or variance (σ 2 ), which does not come from your sample.
I present a worked example using the Z test formula at the end of this post.
Learn more about Z-Scores and Test Statistics .
One Sample Z Test Formula
The one sample Z test formula is a ratio.
The numerator is the difference between your sample mean and a hypothesized value for the population mean (µ 0 ). This value is often a strawman argument that you hope to disprove.
The denominator is the standard error of the mean. It represents the uncertainty in how well the sample mean estimates the population mean.
Learn more about the Standard Error of the Mean .
Two Sample Z Test Formula
The two sample Z test formula is also a ratio.
The numerator is the difference between your two sample means.
The denominator calculates the pooled standard error of the mean by combining both samples. In this Z test formula, enter the population variances (σ 2 ) for each sample.
Z Test Critical Values
As I mentioned in the Z vs T test section, a Z test does not use degrees of freedom. It evaluates Z-scores in the context of the standard normal distribution. Unlike the t-distribution , the standard normal distribution doesn’t change shape as the sample size changes. Consequently, the critical values don’t change with the sample size.
To find the critical value for a Z test, you need to know the significance level and whether it is one- or two-tailed.
0.01
Two-Tailed
±2.576
0.01
Left Tail
–2.326
0.01
Right Tail
+2.326
0.05
Two-Tailed
±1.960
0.05
Left Tail
+1.650
0.05
Right Tail
–1.650
Learn more about Critical Values: Definition, Finding & Calculator .
Z Test Worked Example
Let’s close this post by calculating the results for a Z test by hand!
Suppose we randomly sampled subjects from an honors program. We want to determine whether their mean IQ score differs from the general population. The general population’s IQ scores are defined as having a mean of 100 and a standard deviation of 15.
We’ll determine whether the difference between our sample mean and the hypothesized population mean of 100 is statistically significant.
Specifically, we’ll use a two-tailed analysis with a significance level of 0.05. Looking at the table above, you’ll see that this Z test has critical values of ± 1.960. Our results are statistically significant if our Z statistic is below –1.960 or above +1.960.
The hypotheses are the following:
Null (H 0 ): µ = 100
Alternative (H A ): µ ≠ 100
Entering Our Results into the Formula
Here are the values from our study that we need to enter into the Z test formula:
IQ score sample mean (x̅): 107
Sample size (n): 25
Hypothesized population mean (µ 0 ): 100
Population standard deviation (σ): 15
The Z-score is 2.333. This value is greater than the critical value of 1.960, making the results statistically significant. Below is a graphical representation of our Z test results showing how the Z statistic falls within the critical region.
We can reject the null and conclude that the mean IQ score for the population of honors students does not equal 100. Based on the sample mean of 107, we know their mean IQ score is higher.
Now let’s find the p-value. We could use technology to do that, such as an online calculator. However, let’s go old school and use a Z table.
To find the p-value that corresponds to a Z-score from a two-tailed analysis, we need to find the negative value of our Z-score (even when it’s positive) and double it.
In the truncated Z-table below, I highlight the cell corresponding to a Z-score of -2.33.
The cell value of 0.00990 represents the area or probability to the left of the Z-score -2.33. We need to double it to include the area > +2.33 to obtain the p-value for a two-tailed analysis.
P-value = 0.00990 * 2 = 0.0198
That p-value is an approximation because it uses a Z-score of 2.33 rather than 2.333. Using an online calculator, the p-value for our Z test is a more precise 0.0196. This p-value is less than our significance level of 0.05, which reconfirms the statistically significant results.
See my full Z-table , which explains how to use it to solve other types of problems.
Share this:
Reader Interactions
Comments and questions cancel reply.
7.4.1 - Hypothesis Testing
Five step hypothesis testing procedure.
In the remaining lessons, we will use the following five step hypothesis testing procedure. This is slightly different from the five step procedure that we used when conducting randomization tests.
Check assumptions and write hypotheses. The assumptions will vary depending on the test. In this lesson we'll be confirming that the sampling distribution is approximately normal by visually examining the randomization distribution. In later lessons you'll learn more objective assumptions. The null and alternative hypotheses will always be written in terms of population parameters; the null hypothesis will always contain the equality (i.e., \(=\)).
Calculate the test statistic. Here, we'll be using the formula below for the general form of the test statistic.
Determine the p-value. The p-value is the area under the standard normal distribution that is more extreme than the test statistic in the direction of the alternative hypothesis.
Make a decision. If \(p \leq \alpha\) reject the null hypothesis. If \(p>\alpha\) fail to reject the null hypothesis.
State a "real world" conclusion. Based on your decision in step 4, write a conclusion in terms of the original research question.
General Form of a Test Statistic
When using a standard normal distribution (i.e., z distribution), the test statistic is the standardized value that is the boundary of the p-value. Recall the formula for a z score: \(z=\frac{x-\overline x}{s}\). The formula for a test statistic will be similar. When conducting a hypothesis test the sampling distribution will be centered on the null parameter and the standard deviation is known as the standard error.
This formula puts our observed sample statistic on a standard scale (e.g., z distribution). A z score tells us where a score lies on a normal distribution in standard deviation units. The test statistic tells us where our sample statistic falls on the sampling distribution in standard error units.
7.4.1.1 - Video Example: Mean Body Temperature
Research question: Is the mean body temperature in the population different from 98.6° Fahrenheit?
7.4.1.2 - Video Example: Correlation Between Printer Price and PPM
Research question: Is there a positive correlation in the population between the price of an ink jet printer and how many pages per minute (ppm) it prints?
7.4.1.3 - Example: Proportion NFL Coin Toss Wins
Research question: Is the proportion of NFL overtime coin tosses that are won different from 0.50?
StatKey was used to construct a randomization distribution:
Step 1: Check assumptions and write hypotheses
From the given StatKey output, the randomization distribution is approximately normal.
The p value will be the area on the z distribution that is more extreme than the test statistic of 2.542, in the direction of the alternative hypothesis. This is a two-tailed test:
The p value is the area in the left and right tails combined: \(p=0.0055110+0.0055110=0.011022\)
Step 4: Make a decision
The p value (0.011022) is less than the standard 0.05 alpha level, therefore we reject the null hypothesis.
Step 5: State a "real world" conclusion
There is evidence that the proportion of all NFL overtime coin tosses that are won is different from 0.50
7.4.1.4 - Example: Proportion of Women Students
Research question : Are more than 50% of all World Campus STAT 200 students women?
Data were collected from a representative sample of 501 World Campus STAT 200 students. In that sample, 284 students were women and 217 were not women.
StatKey was used to construct a sampling distribution using randomization methods:
Because this randomization distribution is approximately normal, we can find the p value by computing a standardized test statistic and using the z distribution.
The assumption here is that the sampling distribution is approximately normal. From the given StatKey output, the randomization distribution is approximately normal.
We can find the p value by constructing a standard normal distribution and finding the area under the curve that is more extreme than our observed test statistic of 3.045, in the direction of the alternative hypothesis. In other words, \(P(z>3.045)\):
Our p value is 0.0011634
4. Make a decision
Our p value is less than or equal to the standard 0.05 alpha level, therefore we reject the null hypothesis.
5. State a "real world" conclusion
There is evidence that the proportion of all World Campus STAT 200 students who are women is greater than 0.50.
7.4.1.5 - Example: Mean Quiz Score
Research question: Is the mean quiz score different from 14 in the population?
\(H_0\colon \mu = 14\)
\(H_a\colon \mu \ne 14\)
The sample statistic is the mean in the original sample, 13.746 points. The null parameter is 14 points. And, the standard error, 0.142, can be found on the StatKey output.
The p value will be the area on the z distribution that is more extreme than the test statistic of -1.789, in the direction of the alternative hypothesis:
This was a two-tailed test. The p value is the area in the left and right tails combined: \(p=0.0368074+0.0368074=0.0736148\)
The p value (0.0736148) is greater than the standard 0.05 alpha level, therefore we fail to reject the null hypothesis.
There is not enough evidence to state that the mean quiz score in the population is different from 14 points.
7.4.1.6 - Example: Difference in Mean Commute Times
Research question: Do the mean commute times in Atlanta and St. Louis differ in the population?
From the given StatKey output, the randomization distribution is approximately normal.
The observed sample statistic is \(\overline x _1 - \overline x _2 = 7.14\). The null parameter is 0. And, the standard error, from the StatKey output, is 1.136.
\(test\;statistic=\dfrac{7.14-0}{1.136}=6.285\)
The p value will be the area on the z distribution that is more extreme than the test statistic of 6.285, in the direction of the alternative hypothesis:
This was a two-tailed test. The area in the two tailed combined is 0.000000. Theoretically, the p value cannot be 0 because there is always some chance that a Type I error was committed. This p value would be written as p < 0.001.
The p value is smaller than the standard 0.05 alpha level, therefore we reject the null hypothesis.
There is evidence that the mean commute times in Atlanta and St. Louis are different in the population.
Hypothesis Testing Formula
Hypothesis Testing Formula (Table of Contents)
What is the hypothesis testing formula.
Before deep diving into hypothesis testing, we need to understand the hypothesis in the first place. In simple language, an idea is an educated and informed guess about anything around you, which can be tested by experiment or observation.
For example, A new mobile variant will be accepted by people; new medicine might work or not, etc. So a hypothesis test is a statistical tool for testing the hypothesis we will make and whether that statement is full or not. We select a sample from the data set and test a hypothesis statement by determining the likelihood that a sample statistics. So If your results from that test are not significant, it means that the hypothesis is not valid.
Formula For Hypothesis Testing:
Start Your Free Investment Banking Course
Download Corporate Valuation, Investment Banking, Accounting, CFA Calculator & others
The z-test gives hypothesis testing. The formula for Z – Test is given as follows:
X – Sample Mean
U – Population Mean
SD – Standard Deviation
n – Sample size
But this is not as simple as it seems. To correctly perform the hypothesis test, you need to follow specific steps:
Step 1: First and foremost, to perform a hypothesis test, we must define the null and alternative hypotheses. An example of the null and alternate hypothesis is given by:
H0 (null hypothesis): Mean value > 0
For this, Alternate Hypothesis (Ha): Mean < 0
Step 2: Next thing we have to do is that we need to find out the level of significance. Generally, its value is 0.05 or 0.01
Step 3: Find the z-test value, also called test statistic, as stated in the above formula.
Step 4: Find the z score from the z table given the significance level and mean .
Step 5: Compare these two values, and if the test statistic is greater than the z score, reject the null hypothesis. You cannot reject the null hypothesis if the test statistic is less than the z score.
Examples of Hypothesis Testing Formula (With Excel Template)
Let’s take an example to understand the calculation of the Hypothesis Testing formula in a better manner.
Hypothesis Testing Formula – Example #1
Suppose you have been given the following parameters, and you have to find the Z value and state if you accept the null hypothesis or not:
Null hypothesis H0: Population Mean = 30
Alternate hypothesis Ha: Population Mean ≠ 30
Z – Test is calculated using the formula given below
Z = (X – U) / (SD / √n)
Z – Test = ( 27 – 30 ) / (20 / SQRT(10))
Z – Test = -0.474
Level of significance = 0.05
This is a Two tail test, so the probability lies on both sides of the distribution. So 0.025 on each side, and we will look at this value on the z table.
Source: https://www.z-table.com/
Since the significance level is 0.025 on each side, we need to find 0.025 in the z table. Once we see that value from the table, we must extract the z value.
If you see here, on the left side, the values of z are given, and in the top row, decimal places are given. So from that, we can say that 0.025 will give a z value of -1.96
So Z – Score = -1.96
We can reject the null hypothesis since the Z Test > Z Score.
Hypothesis Testing Formula – Example #2
Let’s say you are a school principal; you are claiming that the students in your school are above average intelligence. An analyst wants to double-check your claim and use hypothesis testing. He measures the IQ of all the students in the school and then takes a sample of 20 students. The following are the data points:
Z – Test = (112 – 110)/ (15 / SQRT(20))
Z – Test = 3.58
Null Hypothesis: Since population mean = 100,
H0 : Mean = 100
Ha: Mean > 100
Level of Significance = 0.05
Since the significance level is 0.05, we must find 1 – 0.05 = 0.95 in the z table. Once we find that value from the table, we must extract the z value.
Z – Table:
If you see here, on the left side, the values of z are given, and in the top row, decimal places are given. So from that, we can say that 0.95 lies between 1.64 to 1.65, mid-point of 1.645.
So Z Score = 1.645
Since the Z Test > Z Score, we can reject the null hypothesis and say students’ intelligence is above average.
Explanation
Everyone should remember that No hypothesis test is 100% correct, and there is always a chance of making an error. There is 2 type of errors that can arise in hypothesis testing: type I and type II.
Type 1: When the null hypothesis is true but rejected in the model. The level of significance gives the probability of this. So if the significance level is 0.05, there is a 5% chance that you will reject the true null.
Type 2: When the null hypothesis is not true but not rejected in the model. The probability of this is given the power of the test. Large sample size can help reduce the probability of this type of error, providing greater confidence in the model.
Relevance and Uses of Hypothesis Testing Formula
As discussed above, the hypothesis test helps the analyst test the statistical sample and, in the end, will either accept or reject the null hypothesis. The test assists in determining the accuracy of the formed hypothesis. If unexpected results occur, it may necessitate the formulation of a new hypothesis, which can then be tested. There are steps for any hypothesis test. The first step is to state the hypothesis, both the null and alternate hypothesis.
The next step is determining all the relevant parameters like mean, standard deviation , level of significance, etc., which helps determine the z-test value . The third step determines the z score from the z table, and for this step, we need to see if it is a two-tail or single-tail test and accordingly extract the z score. The fourth and final step is to compare the results and then, based on that, either accept or reject the null hypothesis.
Hypothesis Testing Formula Calculator
You can use the following Hypothesis Testing Calculator
X
U
SD
√n
Z
Z =
X − U
=
SD/√n
0-0
=
0
0/√0
Recommended Articles
This has been a guide to Hypothesis Testing Formula. Here we discuss how to calculate Hypothesis Testing along with practical examples. We also provide a Hypothesis Testing calculator with a downloadable exceExcelplate. You may also look at the following articles to learn more –
Examples of T Distribution Formula
Calculator For Consumer Surplus Formula
How To Calculate Equity Multiplier Formula
Guide To Net Realizable Value Formula
Altman Z Score (With Excel Template)
*Please provide your correct email id. Login details for this Free course will be emailed to you
By signing up, you agree to our Terms of Use and Privacy Policy .
Download Hypothesis Testing Formula Excel Template
Corporate Valuation, Investment Banking, Accounting, CFA Calculator & others
Forgot Password?
डाउनलोड Hypothesis Testing Formula Excel Template
This website or its third-party tools use cookies, which are necessary to its functioning and required to achieve the purposes illustrated in the cookie policy. By closing this banner, scrolling this page, clicking a link or continuing to browse otherwise, you agree to our Privacy Policy
Explore 1000+ varieties of Mock tests View more
Submit Next Question
🚀 Limited Time Offer! - 🎁 ENROLL NOW
What is Hypothesis Testing in Statistics? Types and Examples
Varun Saharawat is a seasoned professional in the fields of SEO and content writing. With a profound knowledge of the intricate aspects of these disciplines, Varun has established himself as a valuable asset in the world of digital marketing and online content creation.
Hypothesis testing in statistics involves testing an assumption about a population parameter using sample data. Learners can download Hypothesis Testing PDF to get instant access to all information!
What exactly is hypothesis testing, and how does it work in statistics? Can I find practical examples and understand the different types from this blog?
Hypothesis Testing : Ever wonder how researchers determine if a new medicine actually works or if a new marketing campaign effectively drives sales? They use hypothesis testing! It is at the core of how scientific studies, business experiments and surveys determine if their results are statistically significant or just due to chance.
Hypothesis testing allows us to make evidence-based decisions by quantifying uncertainty and providing a structured process to make data-driven conclusions rather than guessing. In this post, we will discuss hypothesis testing types, examples, and processes!
Table of Contents
Hypothesis Testing
Hypothesis testing is a statistical method used to evaluate the validity of a hypothesis using sample data. It involves assessing whether observed data provide enough evidence to reject a specific hypothesis about a population parameter.
Hypothesis Testing in Data Science
Hypothesis testing in data science is a statistical method used to evaluate two mutually exclusive population statements based on sample data. The primary goal is to determine which statement is more supported by the observed data.
Hypothesis testing assists in supporting the certainty of findings in research and data science projects. This statistical inference aids in making decisions about population parameters using sample data. For those who are looking to deepen their knowledge in data science and expand their skillset, we highly recommend checking out Master Generative AI: Data Science Course by Physics Wallah .
Also Read: What is Encapsulation Explain in Details
What is the Hypothesis Testing Procedure in Data Science?
The hypothesis testing procedure in data science involves a structured approach to evaluating hypotheses using statistical methods. Here’s a step-by-step breakdown of the typical procedure:
1) State the Hypotheses:
Null Hypothesis (H0): This is the default assumption or a statement of no effect or difference. It represents what you aim to test against.
Alternative Hypothesis (Ha): This is the opposite of the null hypothesis and represents what you want to prove.
2) Choose a Significance Level (α):
Decide on a threshold (commonly 0.05) beyond which you will reject the null hypothesis. This is your significance level.
3) Select the Appropriate Test:
Depending on your data type (e.g., continuous, categorical) and the nature of your research question, choose the appropriate statistical test (e.g., t-test, chi-square test, ANOVA, etc.).
4) Collect Data:
Gather data from your sample or population, ensuring that it’s representative and sufficiently large (or as per your experimental design).
5)Compute the Test Statistic:
Using your data and the chosen statistical test, compute the test statistic that summarizes the evidence against the null hypothesis.
6) Determine the Critical Value or P-value:
Based on your significance level and the test statistic’s distribution, determine the critical value from a statistical table or compute the p-value.
7) Make a Decision:
If the p-value is less than α: Reject the null hypothesis.
If the p-value is greater than or equal to α: Fail to reject the null hypothesis.
8) Draw Conclusions:
Based on your decision, draw conclusions about your research question or hypothesis. Remember, failing to reject the null hypothesis doesn’t prove it true; it merely suggests that you don’t have sufficient evidence to reject it.
9) Report Findings:
Document your findings, including the test statistic, p-value, conclusion, and any other relevant details. Ensure clarity so that others can understand and potentially replicate your analysis.
Also Read: Binary Search Algorithm
How Hypothesis Testing Works?
Hypothesis testing is a fundamental concept in statistics that aids analysts in making informed decisions based on sample data about a larger population. The process involves setting up two contrasting hypotheses, the null hypothesis and the alternative hypothesis, and then using statistical methods to determine which hypothesis provides a more plausible explanation for the observed data.
The Core Principles:
The Null Hypothesis (H0): This serves as the default assumption or status quo. Typically, it posits that there is no effect or no difference, often represented by an equality statement regarding population parameters. For instance, it might state that a new drug’s effect is no different from a placebo.
The Alternative Hypothesis (H1 or Ha): This is the counter assumption or what researchers aim to prove. It’s the opposite of the null hypothesis, indicating that there is an effect, a change, or a difference in the population parameters. Using the drug example, the alternative hypothesis would suggest that the new drug has a different effect than the placebo.
Testing the Hypotheses:
Once these hypotheses are established, analysts gather data from a sample and conduct statistical tests. The objective is to determine whether the observed results are statistically significant enough to reject the null hypothesis in favor of the alternative.
Examples to Clarify the Concept:
Null Hypothesis (H0): The sanitizer’s average efficacy is 95%.
By conducting tests, if evidence suggests that the sanitizer’s efficacy is significantly less than 95%, we reject the null hypothesis.
Null Hypothesis (H0): The coin is fair, meaning the probability of heads and tails is equal.
Through experimental trials, if results consistently show a skewed outcome, indicating a significantly different probability for heads and tails, the null hypothesis might be rejected.
What are the 3 types of Hypothesis Test?
Hypothesis testing is a cornerstone in statistical analysis, providing a framework to evaluate the validity of assumptions or claims made about a population based on sample data. Within this framework, several specific tests are utilized based on the nature of the data and the question at hand. Here’s a closer look at the three fundamental types of hypothesis tests:
The z-test is a statistical method primarily employed when comparing means from two datasets, particularly when the population standard deviation is known. Its main objective is to ascertain if the means are statistically equivalent.
A crucial prerequisite for the z-test is that the sample size should be relatively large, typically 30 data points or more. This test aids researchers and analysts in determining the significance of a relationship or discovery, especially in scenarios where the data’s characteristics align with the assumptions of the z-test.
The t-test is a versatile statistical tool used extensively in research and various fields to compare means between two groups. It’s particularly valuable when the population standard deviation is unknown or when dealing with smaller sample sizes.
By evaluating the means of two groups, the t-test helps ascertain if a particular treatment, intervention, or variable significantly impacts the population under study. Its flexibility and robustness make it a go-to method in scenarios ranging from medical research to business analytics.
3. Chi-Square Test:
The Chi-Square test stands distinct from the previous tests, primarily focusing on categorical data rather than means. This statistical test is instrumental when analyzing categorical variables to determine if observed data aligns with expected outcomes as posited by the null hypothesis.
By assessing the differences between observed and expected frequencies within categorical data, the Chi-Square test offers insights into whether discrepancies are statistically significant. Whether used in social sciences to evaluate survey responses or in quality control to assess product defects, the Chi-Square test remains pivotal for hypothesis testing in diverse scenarios.
Also Read: Python vs Java: Which is Best for Machine learning algorithm
Hypothesis Testing in Statistics
Hypothesis testing is a fundamental concept in statistics used to make decisions or inferences about a population based on a sample of data. The process involves setting up two competing hypotheses, the null hypothesis H 0 and the alternative hypothesis H 1.
Through various statistical tests, such as the t-test, z-test, or Chi-square test, analysts evaluate sample data to determine whether there’s enough evidence to reject the null hypothesis in favor of the alternative. The aim is to draw conclusions about population parameters or to test theories, claims, or hypotheses.
Hypothesis Testing in Research
In research, hypothesis testing serves as a structured approach to validate or refute theories or claims. Researchers formulate a clear hypothesis based on existing literature or preliminary observations. They then collect data through experiments, surveys, or observational studies.
Using statistical methods, researchers analyze this data to determine if there’s sufficient evidence to reject the null hypothesis. By doing so, they can draw meaningful conclusions, make predictions, or recommend actions based on empirical evidence rather than mere speculation.
Hypothesis Testing in R
R, a powerful programming language and environment for statistical computing and graphics, offers a wide array of functions and packages specifically designed for hypothesis testing. Here’s how hypothesis testing is conducted in R:
Data Collection : Before conducting any test, you need to gather your data and ensure it’s appropriately structured in R.
Choose the Right Test : Depending on your research question and data type, select the appropriate hypothesis test. For instance, use the t.test() function for a t-test or chisq.test() for a Chi-square test.
Set Hypotheses : Define your null and alternative hypotheses. Using R’s syntax, you can specify these hypotheses and run the corresponding test.
Execute the Test : Utilize built-in functions in R to perform the hypothesis test on your data. For instance, if you want to compare two means, you can use the t.test() function, providing the necessary arguments like the data vectors and type of t-test (one-sample, two-sample, paired, etc.).
Interpret Results : Once the test is executed, R will provide output, including test statistics, p-values, and confidence intervals. Based on these results and a predetermined significance level (often 0.05), you can decide whether to reject the null hypothesis.
Visualization : R’s graphical capabilities allow users to visualize data distributions, confidence intervals, or test statistics, aiding in the interpretation and presentation of results.
Hypothesis testing is an integral part of statistics and research, offering a systematic approach to validate hypotheses. Leveraging R’s capabilities, researchers and analysts can efficiently conduct and interpret various hypothesis tests, ensuring robust and reliable conclusions from their data.
Do Data Scientists do Hypothesis Testing?
Yes, data scientists frequently engage in hypothesis testing as part of their analytical toolkit. Hypothesis testing is a foundational statistical technique used to make data-driven decisions, validate assumptions, and draw conclusions from data. Here’s how data scientists utilize hypothesis testing:
Validating Assumptions : Before diving into complex analyses or building predictive models, data scientists often need to verify certain assumptions about the data. Hypothesis testing provides a structured approach to test these assumptions, ensuring that subsequent analyses or models are valid.
Feature Selection : In machine learning and predictive modeling, data scientists use hypothesis tests to determine which features (or variables) are most relevant or significant in predicting a particular outcome. By testing hypotheses related to feature importance or correlation, they can streamline the modeling process and enhance prediction accuracy.
A/B Testing : A/B testing is a common technique in marketing, product development, and user experience design. Data scientists employ hypothesis testing to compare two versions (A and B) of a product, feature, or marketing strategy to determine which performs better in terms of a specified metric (e.g., conversion rate, user engagement).
Research and Exploration : In exploratory data analysis (EDA) or when investigating specific research questions, data scientists formulate hypotheses to test certain relationships or patterns within the data. By conducting hypothesis tests, they can validate these relationships, uncover insights, and drive data-driven decision-making.
Model Evaluation : After building machine learning or statistical models, data scientists use hypothesis testing to evaluate the model’s performance, assess its predictive power, or compare different models. For instance, hypothesis tests like the t-test or F-test can help determine if a new model significantly outperforms an existing one based on certain metrics.
Business Decision-making : Beyond technical analyses, data scientists employ hypothesis testing to support business decisions. Whether it’s evaluating the effectiveness of a marketing campaign, assessing customer preferences, or optimizing operational processes, hypothesis testing provides a rigorous framework to validate assumptions and guide strategic initiatives.
Hypothesis Testing Examples and Solutions
Let’s delve into some common examples of hypothesis testing and provide solutions or interpretations for each scenario.
Example: Testing the Mean
Scenario : A coffee shop owner believes that the average waiting time for customers during peak hours is 5 minutes. To test this, the owner takes a random sample of 30 customer waiting times and wants to determine if the average waiting time is indeed 5 minutes.
Hypotheses :
H 0 (Null Hypothesis): 5 μ =5 minutes (The average waiting time is 5 minutes)
H 1 (Alternative Hypothesis): 5 μ =5 minutes (The average waiting time is not 5 minutes)
Solution : Using a t-test (assuming population variance is unknown), calculate the t-statistic based on the sample mean, sample standard deviation, and sample size. Then, determine the p-value and compare it with a significance level (e.g., 0.05) to decide whether to reject the null hypothesis.
Example: A/B Testing in Marketing
Scenario : An e-commerce company wants to determine if changing the color of a “Buy Now” button from blue to green increases the conversion rate.
H 0: Changing the button color does not affect the conversion rate.
H 1: Changing the button color affects the conversion rate.
Solution : Split website visitors into two groups: one sees the blue button (control group), and the other sees the green button (test group). Track the conversion rates for both groups over a specified period. Then, use a chi-square test or z-test (for large sample sizes) to determine if there’s a statistically significant difference in conversion rates between the two groups.
Hypothesis Testing Formula
The formula for hypothesis testing typically depends on the type of test (e.g., z-test, t-test, chi-square test) and the nature of the data (e.g., mean, proportion, variance). Below are the basic formulas for some common hypothesis tests:
Z-Test for Population Mean :
Z=(σ/n)(xˉ−μ0)
ˉ x ˉ = Sample mean
0 μ 0 = Population mean under the null hypothesis
σ = Population standard deviation
n = Sample size
T-Test for Population Mean :
t= (s/ n ) ( x ˉ −μ 0 )
s = Sample standard deviation
Chi-Square Test for Goodness of Fit :
χ2=∑Ei(Oi−Ei)2
Oi = Observed frequency
Ei = Expected frequency
Also Read: Full Form of OOPS
Hypothesis Testing Calculator
While you can perform hypothesis testing manually using the above formulas and statistical tables, many online tools and software packages simplify this process. Here’s how you might use a calculator or software:
Z-Test and T-Test Calculators : These tools typically require you to input sample statistics (like sample mean, population mean, standard deviation, and sample size). Once you input these values, the calculator will provide you with the test statistic (Z or t) and a p-value.
Chi-Square Calculator : For chi-square tests, you’d input observed and expected frequencies for different categories or groups. The calculator then computes the chi-square statistic and provides a p-value.
Software Packages (e.g., R, Python with libraries like scipy, or statistical software like SPSS) : These platforms offer more comprehensive tools for hypothesis testing. You can run various tests, get detailed outputs, and even perform advanced analyses, including regression models, ANOVA, and more.
When using any calculator or software, always ensure you understand the underlying assumptions of the test, interpret the results correctly, and consider the broader context of your research or analysis.
Hypothesis Testing FAQs
What are the key components of a hypothesis test.
The key components include: Null Hypothesis (H0): A statement of no effect or no difference. Alternative Hypothesis (H1 or Ha): A statement that contradicts the null hypothesis. Test Statistic: A value computed from the sample data to test the null hypothesis. Significance Level (α): The threshold for rejecting the null hypothesis. P-value: The probability of observing the given data, assuming the null hypothesis is true.
What is the significance level in hypothesis testing?
The significance level (often denoted as α) is the probability threshold used to determine whether to reject the null hypothesis. Commonly used values for α include 0.05, 0.01, and 0.10, representing a 5%, 1%, or 10% chance of rejecting the null hypothesis when it's actually true.
How do I choose between a one-tailed and two-tailed test?
The choice between one-tailed and two-tailed tests depends on your research question and hypothesis. Use a one-tailed test when you're specifically interested in one direction of an effect (e.g., greater than or less than). Use a two-tailed test when you want to determine if there's a significant difference in either direction.
What is a p-value, and how is it interpreted?
The p-value is a probability value that helps determine the strength of evidence against the null hypothesis. A low p-value (typically ≤ 0.05) suggests that the observed data is inconsistent with the null hypothesis, leading to its rejection. Conversely, a high p-value suggests that the data is consistent with the null hypothesis, leading to no rejection.
Can hypothesis testing prove a hypothesis true?
No, hypothesis testing cannot prove a hypothesis true. Instead, it helps assess the likelihood of observing a given set of data under the assumption that the null hypothesis is true. Based on this assessment, you either reject or fail to reject the null hypothesis.
Data Governance Framework, Tools, Principles, Benefits
In the digital age, data is the lifeblood of modern organizations. From customer insights to operational efficiency, data drives key…
Benefits of Big Data Analytics – With Examples
Big data, with its vast datasets, goes beyond the capabilities of traditional software. Companies like Google and Amazon have used…
C++ vs Java: Check Key Difference Between C++ and Java
Find all the differences between C++ and Java. Both programming languages are popular and extremely useful to advance your skills…
Related Articles
Digital Marketing Jobs Description, Responsibilities, Skills
What is Data Science Lifecycle, Applications, Prerequisites and Tools
Diwali Bonanza: Get 50% OFF On 4 Exciting Tech Courses At PW Skills!
Competitive Coding Programming – A Complete Guide
Data Science for Social Good: Tackling Global Challenges
Understanding The System Design Architecture of Tinder: How Dating Apps Work?
Top 10 Books for Data Engineering [Beginners to Advanced]
Math Formulas
Hypothesis Testing Formula
We run a hypothesis test that helps statisticians determine if the evidence are enough in a sample data to conclude that a research condition is true or false for the entire population. For finding out hypothesis of a given sample, we conduct a Z-test. Usually, in Hypothesis testing, we compare two sets by comparing against a synthetic data set and idealized model.
The Z test formula is given as:
Where, \(\begin{array}{l}\overline{x}\end{array} \) is the sample mean \(\begin{array}{l}\mu\end{array} \) is the population mean \(\begin{array}{l}\sigma\end{array} \) is the standard deviation and n is the sample size.
Solved Examples
Question: What will be the z value when the given parameters are sample mean = 600, population mean = 585, the standard deviation is 100 and the sample size is 150?
Given parameters are, Sample mean, \(\begin{array}{l}\bar{x}\end{array} \) = 600 Population mean, \(\begin{array}{l}\mu\end{array} \) = 585 , Standard deviation, \(\begin{array}{l}\sigma\end{array} \) = 100 Sample size, n = 150
The formula for hypothesis testing is given as,
FORMULAS Related Links
Leave a Comment Cancel reply
Your Mobile number and Email id will not be published. Required fields are marked *
Request OTP on Voice Call
Post My Comment
Register with BYJU'S & Download Free PDFs
Register with byju's & watch live videos.
Article Categories
Book categories, collections.
Academics & The Arts Articles
Math Articles
Statistics Articles
Finding the Power of a Hypothesis Test
Statistics ii for dummies.
Sign up for the Dummies Beta Program to try Dummies' newest way to learn.
The probability of correctly rejecting H 0 when it is false is known as the power of the test . The larger it is, the better.
The previously claimed value of
in the null hypothesis,
The one-sided inequality of the alternative hypothesis (either < or >), for example,
The mean of the observed values
The population standard deviation
The sample size (denoted n )
The level of significance
Assume that H 0 is true, and
Find the percentile value corresponding to
sitting in the tail(s) corresponding to H a . That is, if
then find b where
Assume that H 0 is false, and instead H a is true. Since
under this assumption, then let
in the next step.
Find the power by calculating the probability of getting a value more extreme than b from Step 2 in the direction of H a . This process is similar to finding the p -value in a test of a single population mean, but instead of using
You conduct a random sample of 100 working mothers and find they spend an average of 11.5 minutes per day talking with their children. Assume prior research suggests the population standard deviation is 2.3 minutes.
When conducting this hypothesis test for a population mean, you find that the p -value = 0.015, and with a level of significance of
you reject the null hypothesis. But there are a lot of different values of
(not just 11.5) that would lead you to reject H 0 . So how strong is this specific test? Find the power.
sitting in the upper tail. If p ( Z > z b ) = 0.05, then z b = 1.645. Further,
Assume that H 0 is false, and instead
Find the power by calculating the probability of getting a value more extreme than b from Step 2 in the direction of H a . Here, you need to find p ( Z > z) where
Using the Z -table, you find that
of 0.05. Further, you found that Power = 0.6985, meaning that there was nearly a 70 percent chance of correctly rejecting a false null hypothesis.
This is just one power calculation based on a single sample generating a mean of 11.5. Statisticians often calculate a “power curve” based on many likely alternative values. Also, there are some unique considerations to take into account if
but this gives you the gist of things.
About This Article
This article can be found in the category:.
Statistics ,
10 Steps to a Better Math Grade with Statistics
Statistics and Histograms
What is Categorical Data and How is It Summarized?
Statistics II For Dummies Cheat Sheet
SPSS For Dummies Cheat Sheet
View All Articles From Category
Statistics Made Easy
Two Sample t-test: Definition, Formula, and Example
A two sample t-test is used to determine whether or not two population means are equal.
This tutorial explains the following:
The motivation for performing a two sample t-test.
The formula to perform a two sample t-test.
The assumptions that should be met to perform a two sample t-test.
An example of how to perform a two sample t-test.
Two Sample t-test: Motivation
Suppose we want to know whether or not the mean weight between two different species of turtles is equal. Since there are thousands of turtles in each population, it would be too time-consuming and costly to go around and weigh each individual turtle.
Instead, we might take a simple random sample of 15 turtles from each population and use the mean weight in each sample to determine if the mean weight is equal between the two populations:
However, it’s virtually guaranteed that the mean weight between the two samples will be at least a little different. The question is whether or not this difference is statistically significant . Fortunately, a two sample t-test allows us to answer this question.
Two Sample t-test: Formula
A two-sample t-test always uses the following null hypothesis:
H 0 : μ 1 = μ 2 (the two population means are equal)
The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed:
H 1 (two-tailed): μ 1 ≠ μ 2 (the two population means are not equal)
H 1 (left-tailed): μ 1 < μ 2 (population 1 mean is less than population 2 mean)
H 1 (right-tailed): μ 1 > μ 2 (population 1 mean is greater than population 2 mean)
We use the following formula to calculate the test statistic t:
Test statistic: ( x 1 – x 2 ) / s p (√ 1/n 1 + 1/n 2 )
where x 1 and x 2 are the sample means, n 1 and n 2 are the sample sizes, and where s p is calculated as:
If the p-value that corresponds to the test statistic t with (n 1 +n 2 -1) degrees of freedom is less than your chosen significance level (common choices are 0.10, 0.05, and 0.01) then you can reject the null hypothesis.
Two Sample t-test: Assumptions
For the results of a two sample t-test to be valid, the following assumptions should be met:
The observations in one sample should be independent of the observations in the other sample.
The data should be approximately normally distributed.
The two samples should have approximately the same variance. If this assumption is not met, you should instead perform Welch’s t-test .
The data in both samples was obtained using a random sampling method .
Two Sample t-test : Example
Suppose we want to know whether or not the mean weight between two different species of turtles is equal. To test this, will perform a two sample t-test at significance level α = 0.05 using the following steps:
Step 1: Gather the sample data.
Suppose we collect a random sample of turtles from each population with the following information:
Sample size n 1 = 40
Sample mean weight x 1 = 300
Sample standard deviation s 1 = 18.5
Sample size n 2 = 38
Sample mean weight x 2 = 305
Sample standard deviation s 2 = 16.7
Step 2: Define the hypotheses.
We will perform the two sample t-test with the following hypotheses:
H 0 : μ 1 = μ 2 (the two population means are equal)
H 1 : μ 1 ≠ μ 2 (the two population means are not equal)
Step 3: Calculate the test statistic t .
First, we will calculate the pooled standard deviation s p :
t = ( x 1 – x 2 ) / s p (√ 1/n 1 + 1/n 2 ) = (300-305) / 17.647(√ 1/40 + 1/38 ) = -1.2508
Step 4: Calculate the p-value of the test statistic t .
According to the T Score to P Value Calculator , the p-value associated with t = -1.2508 and degrees of freedom = n 1 +n 2 -2 = 40+38-2 = 76 is 0.21484 .
Step 5: Draw a conclusion.
Since this p-value is not less than our significance level α = 0.05, we fail to reject the null hypothesis. We do not have sufficient evidence to say that the mean weight of turtles between these two populations is different.
Note: You can also perform this entire two sample t-test by simply using the Two Sample t-test Calculator .
Additional Resources
The following tutorials explain how to perform a two-sample t-test using different statistical programs:
How to Perform a Two Sample t-test in Excel How to Perform a Two Sample t-test in SPSS How to Perform a Two Sample t-test in Stata How to Perform a Two Sample t-test in R How to Perform a Two Sample t-test in Python How to Perform a Two Sample t-test on a TI-84 Calculator
Featured Posts
Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.
2 Replies to “Two Sample t-test: Definition, Formula, and Example”
I like the detailed information and simplified in the way I can understand and relate easily. Thank you
It seems a couple of parenthesis is missed at the pooled standard deviation formula. Under square root you have (n1-1)s12 + (n2-1)s22 / (n1+n2-2) but it should be [(n1-1)s12 + (n2-1)s22] / (n1+n2-2) I used square bracket
Leave a Reply Cancel reply
Your email address will not be published. Required fields are marked *
Join the Statology Community
Sign up to receive Statology's exclusive study resource: 100 practice problems with step-by-step solutions. Plus, get our latest insights, tutorials, and data analysis tips straight to your inbox!
By subscribing you accept Statology's Privacy Policy.
Help | Advanced Search
Mathematics > Statistics Theory
Title: second maximum of a gaussian random field and exact (t-)spacing test.
Abstract: In this article, we introduce the novel concept of the second maximum of a Gaussian random field on a Riemannian submanifold. This second maximum serves as a powerful tool for characterizing the distribution of the maximum. By utilizing an ad-hoc Kac Rice formula, we derive the explicit form of the maximum's distribution, conditioned on the second maximum and some regressed component of the Riemannian Hessian. This approach results in an exact test, based on the evaluation of spacing between these maxima, which we refer to as the spacing test. We investigate the applicability of this test in detecting sparse alternatives within Gaussian symmetric tensors, continuous sparse deconvolution, and two-layered neural networks with smooth rectifiers. Our theoretical results are supported by numerical experiments, which illustrate the calibration and power of the proposed tests. More generally, this test can be applied to any Gaussian random field on a Riemannian manifold, and we provide a general framework for the application of the spacing test in continuous sparse kernel regression. Furthermore, when the variance-covariance function of the Gaussian random field is known up to a scaling factor, we derive an exact Studentized version of our test, coined the $t$-spacing test. This test is perfectly calibrated under the null hypothesis and has high power for detecting sparse alternatives.
Comments:
5 figures, 22 pages main document, 2 pages supplements
Subjects:
Statistics Theory (math.ST); Machine Learning (cs.LG); Differential Geometry (math.DG); Probability (math.PR); Machine Learning (stat.ML)
Code, data and media associated with this article, recommenders and search tools.
Institution
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .
Hypothesis Testing. Hypothesis testing is a statistical…
PPT
Hypothesis Testing
Hypothesis Testing Solved Problems
VIDEO
25
T test Part 1 Hypothesis Set Up and Formula Discussion MBS First Semester Statistics Solution
Steps in Hypothesis Testing
Statistics 101: Single Sample Hypothesis t-test Examples
Formulating the Hypothesis of the Study||Null Hypothesis and Alternative Hypothesis
Hypothesis Testing Explained with Solved Numerical in Hindi l Machine Learning Course
COMMENTS
Hypothesis Testing
Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test. Step 4: Decide whether to reject or fail to reject your null hypothesis. Step 5: Present your findings. Other interesting articles. Frequently asked questions about hypothesis testing.
Hypothesis Testing
Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant. It involves the setting up of a null hypothesis and an alternate hypothesis. There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
Test Statistic: Definition, Types & Formulas
A test statistic assesses how consistent your sample data are with the null hypothesis in a hypothesis test. Test statistic calculations take your sample data and boil them down to a single number that quantifies how much your sample diverges from the null hypothesis. As a test statistic value becomes more extreme, it indicates larger ...
6a.2
Below these are summarized into six such steps to conducting a test of a hypothesis. Set up the hypotheses and check conditions: Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as H 0, which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is ...
7.4.1
Here, we'll be using the formula below for the general form of the test statistic. Determine the p-value. The p-value is the area under the standard normal distribution that is more extreme than the test statistic in the direction of the alternative hypothesis. Make a decision. If \(p \leq \alpha\) reject the null hypothesis.
9.1: Introduction to Hypothesis Testing
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted \(H_0\) while the alternative hypothesis is usually denoted \(H_1\). An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...
7.1: Basics of Hypothesis Testing
Test Statistic: z = x¯¯¯ −μo σ/ n−−√ z = x ¯ − μ o σ / n since it is calculated as part of the testing of the hypothesis. Definition 7.1.4 7.1. 4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true.
Introduction to Hypothesis Testing
A hypothesis test consists of five steps: 1. State the hypotheses. State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false. 2. Determine a significance level to use for the hypothesis. Decide on a significance level.
Test statistics
The test statistic is a number calculated from a statistical test of a hypothesis. It shows how closely your observed data match the distribution expected under the null hypothesis of that statistical test. The test statistic is used to calculate the p value of your results, helping to decide whether to reject your null hypothesis.
Significance tests (hypothesis testing)
Unit 12: Significance tests (hypothesis testing) Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values ...
What is Hypothesis Testing in Statistics? Types and Examples
Hypothesis Testing Formula. Z = ( x̅ - μ0 ) / (σ /√n) Here, x̅ is the sample mean, μ0 is the population mean, σ is the standard deviation, n is the sample size. How Hypothesis Testing Works? An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis.
Statistical hypothesis test
A statistical hypothesis test is a method of statistical inference used to decide whether the data sufficiently support a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. Then a decision is made, either by comparing the test statistic to a critical value or equivalently by evaluating a p ...
T-test and Hypothesis Testing (Explained Simply)
These problems with intuition can lead to problems with decision-making while testing hypotheses. So, besides knowing what values to paste into the formula and how to use t-tests, it is necessary to know when to use it, why to use it, and the meaning of all that stuff. This article is intended to explain two concepts: t-test and hypothesis testing.
Hypothesis Testing Calculator with Steps
Hypothesis Testing Calculator. The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is ...
Z Test: Uses, Formula & Examples
Related posts: Null Hypothesis: Definition, Rejecting & Examples and Understanding Significance Levels. Two-Sample Z Test Hypotheses. Null hypothesis (H 0): Two population means are equal (µ 1 = µ 2).; Alternative hypothesis (H A): Two population means are not equal (µ 1 ≠ µ 2).; Again, when the p-value is less than or equal to your significance level, reject the null hypothesis.
5.2
5.2 - Writing Hypotheses. The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ( H 0) and an alternative hypothesis ( H a ). When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the ...
7.4.1
Here, we'll be using the formula below for the general form of the test statistic. Determine the p-value. The p-value is the area under the standard normal distribution that is more extreme than the test statistic in the direction of the alternative hypothesis. Make a decision. If \(p \leq \alpha\) reject the null hypothesis.
Hypothesis Testing Formula
H0 (null hypothesis): Mean value > 0; For this, Alternate Hypothesis (Ha): Mean < 0; Step 2: Next thing we have to do is that we need to find out the level of significance.Generally, its value is 0.05 or 0.01. Step 3: Find the z-test value, also called test statistic, as stated in the above formula. Step 4: Find the z score from the z table given the significance level and mean.
What is Hypothesis Testing in Statistics? Types and Examples
Hypothesis Testing Formula. The formula for hypothesis testing typically depends on the type of test (e.g., z-test, t-test, chi-square test) and the nature of the data (e.g., mean, proportion, variance). Below are the basic formulas for some common hypothesis tests: Z-Test for Population Mean: Z=(σ/n )(xˉ−μ0 ) Where: ˉ x ˉ = Sample mean
PDF Harold's Statistics Hypothesis Testing Cheat Sheet
Hypothesis A premise or claim that we want to test. Null Hypothesis: H 0 Currently accepted value for a parameter (middle of the distribution). Is assumed true for the purpose of carrying out the hypothesis test. • Always contains an "=" {=, , } • The null value implies a specific sampling distribution for the test statistic • H 0
10.10: Formula Review
10.4 Test for Differences in Means: Assuming Equal Population Variances \[t_c=\frac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{S_p^2\left(\frac{1}{n_1}+\frac{1 ...
Hypothesis testing formula Hypothesis testing example
Hypothesis Testing Formula. We run a hypothesis test that helps statisticians determine if the evidence are enough in a sample data to conclude that a research condition is true or false for the entire population. For finding out hypothesis of a given sample, we conduct a Z-test. Usually, in Hypothesis testing, we compare two sets by comparing ...
Finding the Power of a Hypothesis Test
To calculate power, you basically work two problems back-to-back. First, find a percentile assuming that H 0 is true. Then, turn it around and find the probability that you'd get that value assuming H 0 is false (and instead H a is true). Assume that H 0 is true, and. Find the percentile value corresponding to.
Two Sample t-test: Definition, Formula, and Example
If the p-value that corresponds to the test statistic t with (n 1 +n 2-1) degrees of freedom is less than your chosen significance level (common choices are 0.10, 0.05, and 0.01) then you can reject the null hypothesis. Two Sample t-test: Assumptions. For the results of a two sample t-test to be valid, the following assumptions should be met:
Second Maximum of a Gaussian Random Field and Exact (t-)Spacing test
By utilizing an ad-hoc Kac Rice formula, we derive the explicit form of the maximum's distribution, conditioned on the second maximum and some regressed component of the Riemannian Hessian. ... This test is perfectly calibrated under the null hypothesis and has high power for detecting sparse alternatives. Comments: 5 figures, 22 pages main ...
IMAGES
VIDEO
COMMENTS
Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test. Step 4: Decide whether to reject or fail to reject your null hypothesis. Step 5: Present your findings. Other interesting articles. Frequently asked questions about hypothesis testing.
Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant. It involves the setting up of a null hypothesis and an alternate hypothesis. There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
A test statistic assesses how consistent your sample data are with the null hypothesis in a hypothesis test. Test statistic calculations take your sample data and boil them down to a single number that quantifies how much your sample diverges from the null hypothesis. As a test statistic value becomes more extreme, it indicates larger ...
Below these are summarized into six such steps to conducting a test of a hypothesis. Set up the hypotheses and check conditions: Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as H 0, which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is ...
Here, we'll be using the formula below for the general form of the test statistic. Determine the p-value. The p-value is the area under the standard normal distribution that is more extreme than the test statistic in the direction of the alternative hypothesis. Make a decision. If \(p \leq \alpha\) reject the null hypothesis.
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted \(H_0\) while the alternative hypothesis is usually denoted \(H_1\). An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...
Test Statistic: z = x¯¯¯ −μo σ/ n−−√ z = x ¯ − μ o σ / n since it is calculated as part of the testing of the hypothesis. Definition 7.1.4 7.1. 4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true.
A hypothesis test consists of five steps: 1. State the hypotheses. State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false. 2. Determine a significance level to use for the hypothesis. Decide on a significance level.
The test statistic is a number calculated from a statistical test of a hypothesis. It shows how closely your observed data match the distribution expected under the null hypothesis of that statistical test. The test statistic is used to calculate the p value of your results, helping to decide whether to reject your null hypothesis.
Unit 12: Significance tests (hypothesis testing) Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values ...
Hypothesis Testing Formula. Z = ( x̅ - μ0 ) / (σ /√n) Here, x̅ is the sample mean, μ0 is the population mean, σ is the standard deviation, n is the sample size. How Hypothesis Testing Works? An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis.
A statistical hypothesis test is a method of statistical inference used to decide whether the data sufficiently support a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. Then a decision is made, either by comparing the test statistic to a critical value or equivalently by evaluating a p ...
These problems with intuition can lead to problems with decision-making while testing hypotheses. So, besides knowing what values to paste into the formula and how to use t-tests, it is necessary to know when to use it, why to use it, and the meaning of all that stuff. This article is intended to explain two concepts: t-test and hypothesis testing.
Hypothesis Testing Calculator. The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is ...
Related posts: Null Hypothesis: Definition, Rejecting & Examples and Understanding Significance Levels. Two-Sample Z Test Hypotheses. Null hypothesis (H 0): Two population means are equal (µ 1 = µ 2).; Alternative hypothesis (H A): Two population means are not equal (µ 1 ≠ µ 2).; Again, when the p-value is less than or equal to your significance level, reject the null hypothesis.
5.2 - Writing Hypotheses. The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ( H 0) and an alternative hypothesis ( H a ). When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the ...
Here, we'll be using the formula below for the general form of the test statistic. Determine the p-value. The p-value is the area under the standard normal distribution that is more extreme than the test statistic in the direction of the alternative hypothesis. Make a decision. If \(p \leq \alpha\) reject the null hypothesis.
H0 (null hypothesis): Mean value > 0; For this, Alternate Hypothesis (Ha): Mean < 0; Step 2: Next thing we have to do is that we need to find out the level of significance.Generally, its value is 0.05 or 0.01. Step 3: Find the z-test value, also called test statistic, as stated in the above formula. Step 4: Find the z score from the z table given the significance level and mean.
Hypothesis Testing Formula. The formula for hypothesis testing typically depends on the type of test (e.g., z-test, t-test, chi-square test) and the nature of the data (e.g., mean, proportion, variance). Below are the basic formulas for some common hypothesis tests: Z-Test for Population Mean: Z=(σ/n )(xˉ−μ0 ) Where: ˉ x ˉ = Sample mean
Hypothesis A premise or claim that we want to test. Null Hypothesis: H 0 Currently accepted value for a parameter (middle of the distribution). Is assumed true for the purpose of carrying out the hypothesis test. • Always contains an "=" {=, , } • The null value implies a specific sampling distribution for the test statistic • H 0
10.4 Test for Differences in Means: Assuming Equal Population Variances \[t_c=\frac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{S_p^2\left(\frac{1}{n_1}+\frac{1 ...
Hypothesis Testing Formula. We run a hypothesis test that helps statisticians determine if the evidence are enough in a sample data to conclude that a research condition is true or false for the entire population. For finding out hypothesis of a given sample, we conduct a Z-test. Usually, in Hypothesis testing, we compare two sets by comparing ...
To calculate power, you basically work two problems back-to-back. First, find a percentile assuming that H 0 is true. Then, turn it around and find the probability that you'd get that value assuming H 0 is false (and instead H a is true). Assume that H 0 is true, and. Find the percentile value corresponding to.
If the p-value that corresponds to the test statistic t with (n 1 +n 2-1) degrees of freedom is less than your chosen significance level (common choices are 0.10, 0.05, and 0.01) then you can reject the null hypothesis. Two Sample t-test: Assumptions. For the results of a two sample t-test to be valid, the following assumptions should be met:
By utilizing an ad-hoc Kac Rice formula, we derive the explicit form of the maximum's distribution, conditioned on the second maximum and some regressed component of the Riemannian Hessian. ... This test is perfectly calibrated under the null hypothesis and has high power for detecting sparse alternatives. Comments: 5 figures, 22 pages main ...