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Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

There are 5 main steps in hypothesis testing:

  • State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a  or H 1 ).
  • Collect data in a way designed to test the hypothesis.
  • Perform an appropriate statistical test .
  • Decide whether to reject or fail to reject your null hypothesis.
  • Present the findings in your results and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

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.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

  • H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

  • an estimate of the difference in average height between the two groups.
  • a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

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The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

  • Normal distribution
  • Descriptive statistics
  • Measures of central tendency
  • Correlation coefficient

Methodology

  • Cluster sampling
  • Stratified sampling
  • Types of interviews
  • Cohort study
  • Thematic analysis

Research bias

  • Implicit bias
  • Cognitive bias
  • Survivorship bias
  • Availability heuristic
  • Nonresponse bias
  • Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

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Teach yourself statistics

Hypothesis Test for a Mean

This lesson explains how to conduct a hypothesis test of a mean, when the following conditions are met:

  • The sampling method is simple random sampling .
  • The sampling distribution is normal or nearly normal.

Generally, the sampling distribution will be approximately normally distributed if any of the following conditions apply.

  • The population distribution is normal.
  • The population distribution is symmetric , unimodal , without outliers , and the sample size is 15 or less.
  • The population distribution is moderately skewed , unimodal, without outliers, and the sample size is between 16 and 40.
  • The sample size is greater than 40, without outliers.

This approach consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.

State the Hypotheses

Every hypothesis test requires the analyst to state a null hypothesis and an alternative hypothesis . The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false; and vice versa.

The table below shows three sets of hypotheses. Each makes a statement about how the population mean μ is related to a specified value M . (In the table, the symbol ≠ means " not equal to ".)

The first set of hypotheses (Set 1) is an example of a two-tailed test , since an extreme value on either side of the sampling distribution would cause a researcher to reject the null hypothesis. The other two sets of hypotheses (Sets 2 and 3) are one-tailed tests , since an extreme value on only one side of the sampling distribution would cause a researcher to reject the null hypothesis.

Formulate an Analysis Plan

The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements.

  • Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
  • Test method. Use the one-sample t-test to determine whether the hypothesized mean differs significantly from the observed sample mean.

Analyze Sample Data

Using sample data, conduct a one-sample t-test. This involves finding the standard error, degrees of freedom, test statistic, and the P-value associated with the test statistic.

SE = s * sqrt{ ( 1/n ) * [ ( N - n ) / ( N - 1 ) ] }

SE = s / sqrt( n )

  • Degrees of freedom. The degrees of freedom (DF) is equal to the sample size (n) minus one. Thus, DF = n - 1.

t = ( x - μ) / SE

  • P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t Distribution Calculator to assess the probability associated with the t statistic, given the degrees of freedom computed above. (See sample problems at the end of this lesson for examples of how this is done.)

Sample Size Calculator

As you probably noticed, the process of hypothesis testing can be complex. When you need to test a hypothesis about a mean score, consider using the Sample Size Calculator. The calculator is fairly easy to use, and it is free. You can find the Sample Size Calculator in Stat Trek's main menu under the Stat Tools tab. Or you can tap the button below.

Interpret Results

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

Test Your Understanding

In this section, two sample problems illustrate how to conduct a hypothesis test of a mean score. The first problem involves a two-tailed test; the second problem, a one-tailed test.

Problem 1: Two-Tailed Test

An inventor has developed a new, energy-efficient lawn mower engine. He claims that the engine will run continuously for 5 hours (300 minutes) on a single gallon of regular gasoline. From his stock of 2000 engines, the inventor selects a simple random sample of 50 engines for testing. The engines run for an average of 295 minutes, with a standard deviation of 20 minutes. Test the null hypothesis that the mean run time is 300 minutes against the alternative hypothesis that the mean run time is not 300 minutes. Use a 0.05 level of significance. (Assume that run times for the population of engines are normally distributed.)

Solution: The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

Null hypothesis: μ = 300

Alternative hypothesis: μ ≠ 300

  • Formulate an analysis plan . For this analysis, the significance level is 0.05. The test method is a one-sample t-test .

SE = s / sqrt(n) = 20 / sqrt(50) = 20/7.07 = 2.83

DF = n - 1 = 50 - 1 = 49

t = ( x - μ) / SE = (295 - 300)/2.83 = -1.77

where s is the standard deviation of the sample, x is the sample mean, μ is the hypothesized population mean, and n is the sample size.

Since we have a two-tailed test , the P-value is the probability that the t statistic having 49 degrees of freedom is less than -1.77 or greater than 1.77. We use the t Distribution Calculator to find P(t < -1.77) is about 0.04.

  • If you enter 1.77 as the sample mean in the t Distribution Calculator, you will find the that the P(t < 1.77) is about 0.04. Therefore, P(t >  1.77) is 1 minus 0.96 or 0.04. Thus, the P-value = 0.04 + 0.04 = 0.08.
  • Interpret results . Since the P-value (0.08) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Note: If you use this approach on an exam, you may also want to mention why this approach is appropriate. Specifically, the approach is appropriate because the sampling method was simple random sampling, the population was normally distributed, and the sample size was small relative to the population size (less than 5%).

Problem 2: One-Tailed Test

Bon Air Elementary School has 1000 students. The principal of the school thinks that the average IQ of students at Bon Air is at least 110. To prove her point, she administers an IQ test to 20 randomly selected students. Among the sampled students, the average IQ is 108 with a standard deviation of 10. Based on these results, should the principal accept or reject her original hypothesis? Assume a significance level of 0.01. (Assume that test scores in the population of engines are normally distributed.)

Null hypothesis: μ >= 110

Alternative hypothesis: μ < 110

  • Formulate an analysis plan . For this analysis, the significance level is 0.01. The test method is a one-sample t-test .

SE = s / sqrt(n) = 10 / sqrt(20) = 10/4.472 = 2.236

DF = n - 1 = 20 - 1 = 19

t = ( x - μ) / SE = (108 - 110)/2.236 = -0.894

Here is the logic of the analysis: Given the alternative hypothesis (μ < 110), we want to know whether the observed sample mean is small enough to cause us to reject the null hypothesis.

The observed sample mean produced a t statistic test statistic of -0.894. We use the t Distribution Calculator to find P(t < -0.894) is about 0.19.

  • This means we would expect to find a sample mean of 108 or smaller in 19 percent of our samples, if the true population IQ were 110. Thus the P-value in this analysis is 0.19.
  • Interpret results . Since the P-value (0.19) is greater than the significance level (0.01), we cannot reject the null hypothesis.

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Introduction to Hypothesis Testing

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In many instances, it is not clear whether a data sample supports a claim about the population from which it came. Some examples include:

  • One might want to see if particulate matter (PM2.5) pollution exceeds healthy concentration levels in economically disadvantaged areas. The population is the air in all economically disadvantaged areas over a long period of time. It is infeasible to collect data for the whole population, so one collects air quality samples from several areas at several different times throughout the year. Hypothesis testing then answers the question: do the sampled data indicate that the average PM2.5 concentration is statistically significantly greater than the limit to be deemed healthy?
  • A popular electric car company has heard reports that some of the individual cars it has made do not meet the mileage range, on a single charge, reported by the company. If the proportion of such defective cars is greater than a certain amount, the company needs to recall that particular vehicle model and pay to repair all the cars. The population is all the individual cars of that model. Since it is impossible to test the entire population to see if the range is met or not, the company randomly samples 100 of the cars and tests them. Hypothesis testing then answers the question: is the proportion of defective cars less than the limit for issuing a recall?
  • A natural gas company has come up with a new well design that they think will boost production. Here there are two populations: the population of wells with the new design and the population of wells with the old design. The population parameter of interest is the difference in natural gas production between the two populations. Since it is financially risky to drill a whole bunch of wells with the new design (enough to constitute a population), they drill a few new wells with this design. The company then samples the amount of production from the new wells and compares the average to the average production from the old wells. Hypothesis testing then answers the question: is average production from the new wells statistically significantly  greater than the average production from the old wells? 

In order to perform hypothesis testing, one needs a unbiased sample from a population that has variability in the measure of interest. If, for example, the PM2.5 concentration was constant across all areas and all times, then one single measurement would equal the population average, and one could make a definitive statement about this population average without hypothesis testing. There would be no uncertainty in this population parameter. Using a biased sample in a hypothesis test may lead to a false conclusion. If, for example, the electric car company only tested the next 100 cars produced from only one of its factories, it may miss that the defect is originating from another factory. 

As we go through this lesson, you will see that all hypothesis testing follows a similar process:

  • Formulate your null and alternative hypotheses
  • Calculate the relevant sample statistic
  • Conduct the hypothesis test: compute the randomization distribution
  • (Optional) Visualize the randomization distribution and sample statistic
  • Calculate the p-value
  • State the conclusion of your hypothesis test

Even before these steps, an incredibly useful, if not necessary, preliminary step is to explore your data . Visualizations and five-number summaries will help you formulate appropriate hypotheses, which is important for interpreting the results of the subsequent steps. 

Throughout this lesson, the hypothesis testing will be conducted via randomization distributions. Besides allowing you to practice your Python coding, this simulation-based approach provides a more intuitive understanding of what a p-value means and how it is determined, as compared to the method traditionally presented in a statistics class. Furthermore, this randomization distribution approach is flexible and widely-adaptable to other statistics besides the means and proportions we'll cover in this lesson. 

This lesson will start by going over the guidelines for formulating your hypotheses, and then will go into the concept of randomization distributions. Data from the Texas Power Crisis of 2021 will be used to exemplify several different common varieties of hypothesis tests, and the lesson will conclude with some important considerations to sample size, significance, and multiple testing.

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9.1: Null and Alternative Hypotheses

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The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

\(H_0\): The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

\(H_a\): The alternative hypothesis: It is a claim about the population that is contradictory to \(H_0\) and what we conclude when we reject \(H_0\). This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject \(H_0\)" if the sample information favors the alternative hypothesis or "do not reject \(H_0\)" or "decline to reject \(H_0\)" if the sample information is insufficient to reject the null hypothesis.

\(H_{0}\) always has a symbol with an equal in it. \(H_{a}\) never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example \(\PageIndex{1}\)

  • \(H_{0}\): No more than 30% of the registered voters in Santa Clara County voted in the primary election. \(p \leq 30\)
  • \(H_{a}\): More than 30% of the registered voters in Santa Clara County voted in the primary election. \(p > 30\)

Exercise \(\PageIndex{1}\)

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

  • \(H_{0}\): The drug reduces cholesterol by 25%. \(p = 0.25\)
  • \(H_{a}\): The drug does not reduce cholesterol by 25%. \(p \neq 0.25\)

Example \(\PageIndex{2}\)

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

  • \(H_{0}: \mu = 2.0\)
  • \(H_{a}: \mu \neq 2.0\)

Exercise \(\PageIndex{2}\)

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.

  • \(H_{0}: \mu \_ 66\)
  • \(H_{a}: \mu \_ 66\)
  • \(H_{0}: \mu = 66\)
  • \(H_{a}: \mu \neq 66\)

Example \(\PageIndex{3}\)

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

  • \(H_{0}: \mu \geq 5\)
  • \(H_{a}: \mu < 5\)

Exercise \(\PageIndex{3}\)

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

  • \(H_{0}: \mu \_ 45\)
  • \(H_{a}: \mu \_ 45\)
  • \(H_{0}: \mu \geq 45\)
  • \(H_{a}: \mu < 45\)

Example \(\PageIndex{4}\)

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

  • \(H_{0}: p \leq 0.066\)
  • \(H_{a}: p > 0.066\)

Exercise \(\PageIndex{4}\)

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (\(=, \neq, \geq, <, \leq, >\)) for the null and alternative hypotheses.

  • \(H_{0}: p \_ 0.40\)
  • \(H_{a}: p \_ 0.40\)
  • \(H_{0}: p = 0.40\)
  • \(H_{a}: p > 0.40\)

COLLABORATIVE EXERCISE

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:

  • Evaluate the null hypothesis , typically denoted with \(H_{0}\). The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality \((=, \leq \text{or} \geq)\)
  • Always write the alternative hypothesis , typically denoted with \(H_{a}\) or \(H_{1}\), using less than, greater than, or not equals symbols, i.e., \((\neq, >, \text{or} <)\).
  • If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
  • Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

\(H_{0}\) and \(H_{a}\) are contradictory.

  • If \(\alpha \leq p\)-value, then do not reject \(H_{0}\).
  • If\(\alpha > p\)-value, then reject \(H_{0}\).

\(\alpha\) is preconceived. Its value is set before the hypothesis test starts. The \(p\)-value is calculated from the data.References

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm .

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6a.2 - steps for hypothesis tests, the logic of hypothesis testing section  .

A hypothesis, in statistics, is a statement about a population parameter, where this statement typically is represented by some specific numerical value. In testing a hypothesis, we use a method where we gather data in an effort to gather evidence about the hypothesis.

How do we decide whether to reject the null hypothesis?

  • If the sample data are consistent with the null hypothesis, then we do not reject it.
  • If the sample data are inconsistent with the null hypothesis, but consistent with the alternative, then we reject the null hypothesis and conclude that the alternative hypothesis is true.

Six Steps for Hypothesis Tests Section  

In hypothesis testing, there are certain steps one must follow. 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 evidence to suggest otherwise. The second hypothesis is called the alternative, or research hypothesis, notated as \(H_a \). The alternative hypothesis is a statement of a range of alternative values in which the parameter may fall. One must also check that any conditions (assumptions) needed to run the test have been satisfied e.g. normality of data, independence, and number of success and failure outcomes.
  • Decide on the significance level, \(\alpha \): This value is used as a probability cutoff for making decisions about the null hypothesis. This alpha value represents the probability we are willing to place on our test for making an incorrect decision in regards to rejecting the null hypothesis. The most common \(\alpha \) value is 0.05 or 5%. Other popular choices are 0.01 (1%) and 0.1 (10%).
  • Calculate the test statistic: Gather sample data and calculate a test statistic where the sample statistic is compared to the parameter value. The test statistic is calculated under the assumption the null hypothesis is true and incorporates a measure of standard error and assumptions (conditions) related to the sampling distribution.
  • Calculate probability value (p-value), or find the rejection region: A p-value is found by using the test statistic to calculate the probability of the sample data producing such a test statistic or one more extreme. The rejection region is found by using alpha to find a critical value; the rejection region is the area that is more extreme than the critical value. We discuss the p-value and rejection region in more detail in the next section.
  • Make a decision about the null hypothesis: In this step, we decide to either reject the null hypothesis or decide to fail to reject the null hypothesis. Notice we do not make a decision where we will accept the null hypothesis.
  • State an overall conclusion : Once we have found the p-value or rejection region, and made a statistical decision about the null hypothesis (i.e. we will reject the null or fail to reject the null), we then want to summarize our results into an overall conclusion for our test.

We will follow these six steps for the remainder of this Lesson. In the future Lessons, the steps will be followed but may not be explained explicitly.

Step 1 is a very important step to set up correctly. If your hypotheses are incorrect, your conclusion will be incorrect. In this next section, we practice with Step 1 for the one sample situations.

Fault Detection in Solar PV Systems Using Hypothesis Testing

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Hypothesis Testing

Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.

A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.

What is Hypothesis Testing in Statistics?

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.

Hypothesis Testing Definition

Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.

Null Hypothesis

The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as \(H_{0}\). Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.

Alternative Hypothesis

The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as \(H_{1}\) or \(H_{a}\). For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.

Hypothesis Testing P Value

In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, \(\alpha\) or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.

Hypothesis Testing Critical region

All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.

Hypothesis Testing Formula

Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:

  • z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation and n is the size of the sample.
  • t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\). s is the sample standard deviation.
  • \(\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}\). \(O_{i}\) is the observed value and \(E_{i}\) is the expected value.

We will learn more about these test statistics in the upcoming section.

Types of Hypothesis Testing

Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.

Hypothesis Testing Z Test

A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:

  • One sample: z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
  • Two samples: z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing t Test

The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.

  • One sample: t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\).
  • Two samples: t = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing Chi Square

The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.

One Tailed Hypothesis Testing

One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.

Right Tailed Hypothesis Testing

The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:

\(H_{0}\): The population parameter is ≤ some value

\(H_{1}\): The population parameter is > some value.

If the test statistic has a greater value than the critical value then the null hypothesis is rejected

Right Tail Hypothesis Testing

Left Tailed Hypothesis Testing

The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:

\(H_{0}\): The population parameter is ≥ some value

\(H_{1}\): The population parameter is < some value.

The null hypothesis is rejected if the test statistic has a value lesser than the critical value.

Left Tail Hypothesis Testing

Two Tailed Hypothesis Testing

In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:

\(H_{0}\): the population parameter = some value

\(H_{1}\): the population parameter ≠ some value

The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.

Two Tail Hypothesis Testing

Hypothesis Testing Steps

Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:

  • Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
  • Step 2: Set up the alternative hypothesis.
  • Step 3: Choose the correct significance level, \(\alpha\), and find the critical value.
  • Step 4: Calculate the correct test statistic (z, t or \(\chi\)) and p-value.
  • Step 5: Compare the test statistic with the critical value or compare the p-value with \(\alpha\) to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.

Hypothesis Testing Example

The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.

Step 1: This is an example of a right-tailed test. Set up the null hypothesis as \(H_{0}\): \(\mu\) = 100.

Step 2: The alternative hypothesis is given by \(H_{1}\): \(\mu\) > 100.

Step 3: As this is a one-tailed test, \(\alpha\) = 100% - 95% = 5%. This can be used to determine the critical value.

1 - \(\alpha\) = 1 - 0.05 = 0.95

0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.

Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.

z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).

\(\mu\) = 100, \(\overline{x}\) = 112.5, n = 30, \(\sigma\) = 15

z = \(\frac{112.5-100}{\frac{15}{\sqrt{30}}}\) = 4.56

Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.

Hypothesis Testing and Confidence Intervals

Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.

Related Articles:

  • Probability and Statistics
  • Data Handling

Important Notes on 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.
  • Hypothesis testing can be classified as right tail, left tail, and two tail tests.

Examples on Hypothesis Testing

  • Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) > 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 5, s = 18. \(\alpha\) = 0.05 Using the t-distribution table, the critical value is 2.132 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
  • Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. \(H_{0}\): \(\mu\) = 80, \(H_{1}\): \(\mu\) ≠ 80 \(\overline{x}\) = 88, \(\mu\) = 80, n = 36, \(\sigma\) = 10. \(\alpha\) = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) z = \(\frac{88-80}{\frac{10}{\sqrt{36}}}\) = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
  • Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) < 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = \(\frac{82-90}{\frac{18}{\sqrt{6}}}\) t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.

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FAQs on Hypothesis Testing

What is hypothesis testing.

Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.

What is the z Test in Hypothesis Testing?

The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.

What is the t Test in Hypothesis Testing?

The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.

What is the formula for z test in Hypothesis Testing?

The formula for a one sample z test in hypothesis testing is z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) and for two samples is z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

What is the p Value in Hypothesis Testing?

The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.

What is One Tail Hypothesis Testing?

When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.

A New Surge in Power Use Is Threatening U.S. Climate Goals

A boom in data centers and factories is straining electric grids and propping up fossil fuels.

By Brad Plumer and Nadja Popovich

Something unusual is happening in America. Demand for electricity, which has stayed largely flat for two decades, has begun to surge.

Over the past year, electric utilities have nearly doubled their forecasts of how much additional power they’ll need by 2028 as they confront an unexpected explosion in the number of data centers, an abrupt resurgence in manufacturing driven by new federal laws, and millions of electric vehicles being plugged in.

Many power companies were already struggling to keep the lights on, especially during extreme weather, and say the strain on grids will only increase. Peak demand in the summer is projected to grow by 38,000 megawatts nationwide in the next five years, according to an analysis by the consulting firm Grid Strategies , which is like adding another California to the grid.

“The numbers we’re seeing are pretty crazy,” said Daniel Brooks, vice president of integrated grid and energy systems at the Electric Power Research Institute, a nonprofit organization.

In an ironic twist, the swelling appetite for more electricity, driven not only by electric cars but also by battery and solar factories and other aspects of the clean-energy transition, could also jeopardize the country’s plans to fight climate change.

An aerial view of three large, rectangular buildings with a two-lane road running in the foreground. The shadows are long and it appears to be early evening.

At least 75 data centers have opened in Virginia since 2019.

Nathan Howard for The New York Times

Four white Tesla cars at a bank of charging stations in an urban setting.

In California, electric vehicles could soon account for 10 percent of peak power demand.

Lauren Justice for The New York Times

To meet spiking demand, utilities in states like Georgia, North Carolina, South Carolina, Tennessee and Virginia are proposing to build dozens of power plants over the next 15 years that would burn natural gas. In Kansas, one utility has postponed the retirement of a coal plant to help power a giant electric-car battery factory.

Burning more gas and coal runs counter to President Biden’s pledge to halve the nation’s planet-warming greenhouse gases and to generate all of America’s electricity from pollution-free sources such as wind, solar and nuclear by 2035.

“I can’t recall the last time I was so alarmed about the country’s energy trajectory,” said Tyler H. Norris, a former solar developer and expert in power systems who is now pursuing a doctorate at Duke University. If a wave of new gas-fired plants gets approved by state regulators, he said, “it is game over for the Biden administration’s 2035 decarbonization goal.”

Some utilities say they need additional fossil fuel capacity because cleaner alternatives like wind or solar power aren’t growing fast enough and can be bogged down by delayed permits and snarled supply chains. While a data center can be built in just one year, it can take five years or longer to connect renewable energy projects to the grid and a decade to build some of the long-distance power lines they require. Utilities also note that data centers and factories need power 24 hours a day, something wind and solar can’t do alone.

Yet many regulated utilities also have financial incentives to build new gas plants, since they can recover their costs to build plants, wires and other equipment from ratepayers and pocket an additional percentage as profit. As a result, critics say, utilities often overlook, or even block, ways to make existing power systems more efficient or to integrate more renewable energy into the grid.

“It is entirely feasible to meet growing electricity demand without so much gas, but it requires regulators to challenge the utilities and push for less-traditional solutions,” Mr. Norris said.

The stakes are high. If more power isn’t brought online relatively soon, large portions of the country could risk blackouts, according to a recent report by the North American Electric Reliability Corporation , which monitors the health of the nation’s electric grids.

“Right now everyone’s getting caught flat-footed” by rising demand for electricity, said John Wilson, a vice president at Grid Strategies.

Why Electricity Demand Is Spiking

In Virginia, power-hungry data centers are being approved at breakneck pace.

For much of the 20th century, America’s electricity use increased steadily and utilities built plenty of coal, gas and nuclear plants in response. But starting in the mid-2000s, demand flattened. The economy and population kept expanding, but factories, lightbulbs and even refrigerators became much more energy efficient.

Now demand is rising again, for several reasons.

The growth of remote work, video streaming and online shopping has led to a frenzied expansion of data centers across the nation. The rise of artificial intelligence is poised to accelerate that trend: By 2030, electricity demand at U.S. data centers could triple , using as much power as 40 million homes, according to Boston Consulting Group.

In Northern Virginia, one of the nation’s largest data center hubs, at least 75 facilities have opened since 2019 and Dominion Energy, the local utility, says data center capacity could double in just five years.

In Georgia, large new manufacturing hubs are looking to hook into the grid.

At the same time, investment in American manufacturing is hitting a 50-year high, fueled by new federal tax breaks to lift microchip and clean-tech production. Since 2021, companies have announced plans to spend at least $525 billion on factories for semiconductors, batteries, solar panels and more.

In Georgia, where dozens of electric vehicle companies and suppliers are setting up shop, the state’s largest utility now expects 16 times as much growth in electricity demand this decade as it did two years ago.

Millions of Americans are also buying plug-in vehicles and electric heat pumps for their homes, spurred by recent federal incentives. In California, one-fifth of new cars sold are electric, and officials estimate that E.V.s could account for 10 percent of power use during peak hours by 2035.

On top of that, record heat fueled by global warming is spurring people to crank up air-conditioning, causing summer demand in Arizona and Texas to rise faster than forecast.

Many worry the grid won’t keep up.

PJM Interconnection, which oversees the nation’s largest regional grid, stretching from Illinois to New Jersey, is now expecting an additional 10,000 megawatts of demand by 2030 that wasn’t forecast last year. That’s akin to adding another New York City to the system.

“To see that come on all of the sudden, even for a system as big as ours, that’s significant,” said Ken Seiler, who leads system planning for PJM.

Finding enough power could be a challenge, since PJM’s process for connecting renewable energy projects to the grid has been afflicted by delays . Utilities in PJM have been preparing to retire roughly 40,000 megawatts of mostly coal, gas and oil-burning power plants this decade as states seek to transition away from fossil fuels. PJM has already approved an additional 40,000 megawatts of mostly wind, solar and batteries as partial replacements. But many of those projects have been stalled by local opposition or trouble getting vital equipment like transformers.

“We have a huge concern about that,” Mr. Seiler said. “Folks aren’t building.”

Nationwide, just 251 miles of high-voltage transmission lines were completed last year, a number that has been declining for a decade.

So far, one state that has kept pace with explosive demand is Texas, where electricity use has risen 29 percent over the past decade, partly driven by things like bitcoin mining , liquefied natural gas terminals and the electrification of oil fields. Texas’s streamlined permitting process allows wind, solar and battery projects to get built and connected faster than almost anywhere else , and the state zoomed past California last year to lead the nation in large-scale solar power.

“Texas still has problems, but there’s a lot to learn from how the state makes it easier to build clean energy,” said Devin Hartman, director of energy and environmental policy at the R Street Institute.

A Challenge for Cutting Emission s

A large metal frame full of heavy wires and insulators. The sky above is clear blue.

A power substation near a CloudHQ data center in Ashburn, Va.

Soaring demand has provoked major fights over the future of natural gas.

In North Carolina, regulators had ordered Duke Energy, the state’s biggest utility, to slash its planet-warming carbon dioxide emissions by 70 percent by 2030.

But in January, Duke warned it could miss that target by at least five years under a new plan to build up to five large gas-burning power plants and five smaller versions by 2033, more than previously proposed. Even though Duke is planning a major expansion of solar and offshore wind power, the company says it needs additional gas plants because demand from industrial customers is rising faster than expected.

“The growth we’re seeing is historic in scale and speed,” said Kendal Bowman, president of Duke Energy’s operations in North Carolina. “But it’s also going to be a challenge, particularly in the near term, to see carbon reduction at the same time we’ve got this unprecedented growth.”

Similar revisions are occurring elsewhere. In Virginia, Dominion Energy has proposed to meet rising demand for data centers with a mix of renewables and gas generation in a plan that could increase its overall emissions. Georgia Power has asked permission to build three new gas- and oil-burning turbines and is evaluating whether to postpone the planned retirement of two older coal plants.

“It’s completely at odds with what we need to do to” to fight climate change, said Greg Buppert, a senior attorney at the Southern Environmental Law Center, which has identified at least 33,000 megawatts worth of gas projects being proposed by utilities across the Southeast, plants that could stick around burning fossil fuels for decades.

Solar panels being built at a factory.

A solar panel plant in Dalton, Ga.

REUTERS/Megan Varner

An employee working on a solar panel inside a solar plant.

Work in progress at the Dalton plant.

AP Photo/Mike Stewart

In interviews, utility executives say gas is needed to back up wind and solar power, which don’t run all the time. Gas plants can sometimes be easier to build than renewables, since they may not require new long-distance transmission lines. Eventually, alternative sources of clean power may emerge (both Duke and Dominion want to build smaller nuclear reactors ) but those are years away.

“We need to meet our customers’ needs at all times, even when renewable resources might not be providing energy,” said Aaron Mitchell, vice president of planning and pricing at Georgia Power. “It’s going to take a diversified fleet.”

Mr. Mitchell noted that Georgia Power was planning a large build-out of solar power and batteries over the next decade and would offer incentives to companies to use less power during times of grid stress. But, he added, “gas has to be a near-term part of our fleet.”

Critics say that regulated utilities often default to building gas plants because it’s a familiar technology and because, in many states, they earn a guaranteed profit from capital projects. They don’t always have the same incentive to adopt energy-efficiency programs that reduce sales or to plan transmission lines that can import cheaper wind power from elsewhere.

“The big utilities are typically most comfortable with one way of doing things: building those big, conventional power plants,” said Heather O’Neill, president of Advanced Energy United, a trade group representing low-carbon technology companies.

There are other ways to meet rising demand that require burning fewer fossil fuels, some experts say. Utilities could get more creative about helping customers use less electricity during peak hours or make better use of batteries, reducing strains on the grid. Advanced sensors and other technologies could push more renewable energy through existing transmission lines. Some utilities are pursuing these options, but many are not.

Over the coming months, environmentalists and other groups aim to challenge utility plans at state regulatory proceedings. In some cases, they’ll argue that the utility has overestimated future demand growth or neglected alternatives to gas . While these debates can get technical, they could have a significant impact on the nation’s energy future.

The tech companies and manufacturers that are driving up electricity demand could also play a big role. Many firms have pledged to use clean electricity for their operations, and it remains to be seen how hard they actually push power companies to provide it.

“A big question,” said Brian Janous, a former vice president of energy at Microsoft who now focuses on ways to clean up the grid, “is how much outside pressure utilities and state regulators will face to do things differently.”

Sources and notes

Top chart: Data via the North American Electric Reliability Corporation . The data reflects annual net energy for load for the United States only, but select years include small portions of Mexico and Canada.

Virginia map: Data center locations were collected by The Piedmont Environmental Council , based on publicly available documents and news articles. Locations are approximate. The map shows existing data centers and new projects that have been approved, are actively being marketed or are seeking approval for development as data center space. The map does not include proposed expansions.

Georgia map: Data courtesy of Georgia Power, with additional research by The New York Times. Projects include factories that manufacture solar panels, electric vehicles and batteries, as well as parts suppliers for those industries and recyclers.

Learn More About Climate Change

Have questions about climate change? Our F.A.Q. will tackle your climate questions, big and small .

MethaneSAT, a washing-machine-sized satellite , is designed to detect emissions of methane, an invisible yet potent gas that is dangerously heating the world.  Here is how it works .

Two friends, both young climate researchers, recently spent hours confronting the choices that will shape their careers, and the world. Their ideas are very different .

New satellite-based research reveals how land along the East Coast is slumping into the ocean, compounding the danger from global sea level rise . A major culprit: overpumping of groundwater.

The planet needs solar power. Can we build it without harming nature ? Today’s decisions about how and where to set up new energy projects will reverberate for generations.

Did you know the ♻ symbol doesn’t mean something is actually recyclable ? Read on about how we got here, and what can be done.

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  1. 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.

  2. How to Find the Power of a Statistical Test

    The power of the test is the probability of rejecting the null hypothesis, assuming that the true population mean is equal to the critical parameter value. Since the region of acceptance is 294.46 to 305.54, the null hypothesis will be rejected when the sampled run time is less than 294.46 or greater than 305.54.

  3. S.3.3 Hypothesis Testing Examples

    If the engineer used the P -value approach to conduct his hypothesis test, he would determine the area under a tn - 1 = t24 curve and to the right of the test statistic t * = 1.22: In the output above, Minitab reports that the P -value is 0.117. Since the P -value, 0.117, is greater than α = 0.05, the engineer fails to reject the null hypothesis.

  4. 7.1: Basics of Hypothesis Testing

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  5. Hypothesis Test for a Mean

    How to conduct a hypothesis test for a mean value, using a one-sample t-test. The test procedure is illustrated with examples for one- and two-tailed tests. ... Problem 1: Two-Tailed Test. An inventor has developed a new, energy-efficient lawn mower engine. He claims that the engine will run continuously for 5 hours (300 minutes) on a single ...

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  7. 5.2

    Alternative Hypothesis. The statement that there is some difference in the population (s), denoted as H a or H 1. When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

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    Photo from StepUp Analytics. Hypothesis testing is a method of statistical inference that considers the null hypothesis H₀ vs. the alternative hypothesis Ha, where we are typically looking to assess evidence against H₀. Such a test is used to compare data sets against one another, or compare a data set against some external standard. The former being a two sample test (independent or ...

  9. Simple hypothesis testing (video)

    Simple hypothesis testing. Examples of null and alternative hypotheses. Writing null and alternative hypotheses. P-values and significance tests. Comparing P-values to different significance levels. Estimating a P-value from a simulation. ... 0 energy points. About About this video Transcript.

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    0 energy points. About About this video Transcript. The lesson explores hypothesis testing in statistics, demonstrating how varied outcomes from a test with a known accuracy rate can influence the acceptance or rejection of a hypothesis. This understanding is key to evaluating the reliability of statistical tests. ... So we test that hypothesis ...

  11. PDF Physics 509: Intro to Hypothesis Testing

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  12. 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 ...

  13. Hypothesis Testing

    Purchase single chapter. 48-Hour online access $10.00. Details. Online-only access $18.00. Details. Single Chapter PDF Download $42.00. Details. Check out.

  14. Introduction to Hypothesis Testing

    In order to perform hypothesis testing, one needs a unbiased sample from a population that has variability in the measure of interest. If, for example, the PM2.5 concentration was constant across all areas and all times, then one single measurement would equal the population average, and one could make a definitive statement about this ...

  15. 9.1: Null and Alternative Hypotheses

    The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. \(H_0\): The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

  16. 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.

  17. 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 ...

  18. PDF 9 Hypothesis*Tests

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  19. Hypothesis Testing and Confidence Intervals

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    A hypothesis test is used to test whether or not some hypothesis about a population parameter is true.. To perform a hypothesis test in the real world, researchers obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:. Null Hypothesis (H 0): The sample data occurs purely from chance.

  21. Fault Detection in Solar PV Systems Using Hypothesis Testing

    The demand for solar energy has rapidly increased throughout the world in recent years. However, anomalies in photovoltaic (PV) plants can reduce performances and result in serious consequences. Developing reliable statistical approaches able to detect anomalies in PV plants is vital to improving the management of these plants. Here, we present a statistical approach for detecting anomalies in ...

  22. 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.

  23. Writing a Hypothesis for Your Science Fair Project

    A hypothesis is a tentative, testable answer to a scientific question. Once a scientist has a scientific question she is interested in, the scientist reads up to find out what is already known on the topic. Then she uses that information to form a tentative answer to her scientific question. Sometimes people refer to the tentative answer as "an ...

  24. A New Surge in Power Use Is Threatening U.S. Climate Goals

    By Brad Plumer and Nadja Popovich. March 14, 2024. Something unusual is happening in America. Demand for electricity, which has stayed largely flat for two decades, has begun to surge. Over the ...

  25. Assessing food limitation for marine juvenile fishes in coastal

    We use a mechanistic approach based on DEB (Dynamic Energy Budget) theory to test the trophic limitation hypothesis from a metabolic point of view. The energy intake of individuals is quantified given the experienced temperatures, and measures of individuals' length-at-age. We reconstruct the food ingested in an "inverse"-DEB modelling.