ho vs ha hypothesis

Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

• Describe hypothesis testing in general and in practice

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 about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

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 .

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 adecision. 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.

Mathematical Symbols Used in  H 0 and H a :

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.

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ 30

H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30

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 ≠ 0.25

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 : μ = 2.0

H a : μ ≠ 2.0

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 (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 66 H a : μ __ 66

• H 0 : μ = 66
• H a : μ ≠ 66

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 : μ ≥ 5

H a : μ < 5

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 : μ __ 45 H a : μ __ 45

• H 0 : μ ≥ 45
• H a : μ < 45

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 ≤ 0.066

H a : p > 0.066

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 (=, ≠, ≥, <, ≤, >) 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

Concept Review

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 (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, 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.

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Choosing $H_0$ and $H_a$ in hypothesis testing

There seems to be some ambiguity or contradiction in how to correctly choose the null and alternative hypotheses, both online and in my instructor's notes. I'm trying to figure out if this stems merely from my lack of understanding or if there actually is a disagreement in the scientific community at large. I've seen the following two ideas on choosing $H_0$ and $H_a$

The null hypothesis is the status quo, the state of things already accepted and/or shown to be true by previous data. We assume it to be true and need convincing evidence to reject it. The alternative hypothesis is the one being proposed based on data from the experiment in question, and is assumed to be false unless the data supporting it can convincingly show otherwise.

The null hypothesis is always the one that includes the equality, and the alternative hypothesis is the complement to it. It doesn't matter whether the equality is the status quo or is being claimed by the researcher, it is always $H_0$ .

An example I made up myself for demonstrative purposes, I'm not looking for an actual solution. Only interested in the following hypotheses:

A researcher believes that children in economically disadvantaged areas are more likely to be raised in single-parent homes. He surveys 1000 children from such an area and finds that 317 of them are raised in a single-parent home. Can we conclude with 95% confidence that 30% or more of the children in economically disadvantaged areas are raised in single-parent homes?

What would be the $H_0$ and $H_a$ in this case and why? My professor provided the correct answer (for an equivalent question but with different numbers) to be

$H_0$ : $p >= 0.3$ ; $H_a$ : $p < 0.3$

With the rationale that H0 must include the equality, which in this case is greater or equal to 30% . Her solution than failed to reject the null hypothesis and concluded that the researcher's claim is therefore correct. To me, this seems like assuming the claim to be true and giving it the benefit of the doubt, which is the opposite of what I thought was the correct approach.

A professor in this related question Difference between "at least" and "more than" in hypothesis testing? seemingly took the same approach.

I wish I could talk to my professor about this, but unfortunately, there's a significant language barrier.

• hypothesis-testing

3 Answers 3

Your null hypothesis is $H_0:p=0.3$

The alternative hypothesis is $H_1:p>0.3$

You need to calculate $$p(X\geq317)$$ using $X\sim Bin(1000,0.3)$

Can you finish?

Just to clarify:

• The null hypothesis always has an equal sign and never an inequality symbol
• In this particular example we conclude that $317$ is not in the critical region.

We conclude that in accepting the null hypothesis there is insufficient evidence that the probability is more than $30$%

• $\begingroup$ The question is purely hypothetical, this isn't an actual homework assignment. I'm trying to understand the overall strategy of picking the null and alternative hypotheses. If you check the related question, you'll see that a professor would have picked H1 : p < 0.3 instead. Why? $\endgroup$ –  Egor Commented Apr 9, 2018 at 17:35
• $\begingroup$ It wouldn't make sense to choose this alternative hypothesis, since 317 is greater than the mean, we should be looking in the upper part of the distribution, not the lower $\endgroup$ –  David Quinn Commented Apr 9, 2018 at 17:38
• $\begingroup$ please see my additional comments to my previous answer. $\endgroup$ –  David Quinn Commented Apr 9, 2018 at 18:08
• $\begingroup$ The way my professor explained it in class, we have two hypotheses: p >= 0.3 (since the question states "30% or more") and p < 0.3 as the complement. We must pick the one that has an equality as the null hypothesis, and the other one as the alternative. $\endgroup$ –  Egor Commented Apr 9, 2018 at 18:13
• $\begingroup$ The null hypothesis always states a particular value, in this case $0.3$. The alternative hypothesis is always either $<$ or $>$ or $\neq$ $\endgroup$ –  David Quinn Commented Apr 9, 2018 at 18:16

Both ideas of the null and alternative hypothesis are true. The null hypothesis must always include an equals sign, whether it be $\geq\text{, } \leq\text{, or just}=$. Usually, however, it's just $=$. The alternative hypothesis is what we wish to show.

The null hypothesis in this case is that the proportion of children in economically disadvantaged areas raised in single-parent homes is $30$%.

The alternative hypothesis is that the proportion of children in economically disadvantaged areas raised in single-parent homes is greater than $30$%.

More formally

$$H_0 : p=0.3$$

$$H_a : p \gt 0.3$$

There are two ways you can test this hypothesis if you so wish. Letting $X$ be the number of children raised in single-parent homes, you can use normal approximation to the binomial:

$$P(X\geq317)=1-P(X\lt317)=1-\Phi\left(\frac{316.5-300}{\sqrt{1000\cdot0.3\cdot0.7}}\right)$$

where I used a continuity correction

In R statistical software

You could also, using software, find the exact probability using the standard binomial distribution:

$$P(X\geq317)=\sum_{k=317}^{1000} {1000 \choose k}\cdot0.3^k\cdot0.7^{1000-k}$$

Since $n$ is large, the normal approximation does very well.

At $\alpha=0.05$ we fail to reject the null hypothesis.

• 1 $\begingroup$ You choice is also the one I would have gone with, since for a continuous distribution >= 30% and >30% are the same thing. However, both my professor, and the professor in the linked related question, picked H0 : p = 0.3; H1 : p < 0.3 . That's what I'm trying to understand. $\endgroup$ –  Egor Commented Apr 9, 2018 at 17:42

You always have to choose $H_a$ so that the sample’s estimation fulfills $H_a$.

The reason is that otherwise the rejection rule will always vote for $H_0$ as in the incorrect choice of your professor.

In your case you want to test a probability against $0.3$, the sample’s estimation was $0.37$, hence $H_a\colon p>0.3$ as $0.37>0.3$. And it does in no way matter where the equal-sign occurs as long as you’re dealing with continuous random variables.

• 1 $\begingroup$ Why should we choose the alternative according to the observede data? $\endgroup$ –  Thomas Commented Jun 20, 2022 at 6:36

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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: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

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.

Mathematical Symbols Used in H 0 and H a :

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 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 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

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

Example 9.2

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 the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

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 (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

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 : μ __ 45
• H a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

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

• 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.

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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 ).

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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• Idea behind hypothesis testing

Examples of null and alternative hypotheses

• Writing null and alternative hypotheses
• P-values and significance tests
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Video transcript

Formulating the Alternate Hypothesis: Guidelines and Examples

Updated: July 5, 2023 by Ken Feldman

Hypothesis testing is a branch of statistics in which, using data from a sample, an inference is made about a population parameter or a population probability distribution . 

First, a hypothesis statement and assumption is made about the population parameter or probability distribution. This initial statement is called the Null Hypothesis and is denoted by H o. 

An alternative or alternate hypothesis (denoted Ha ) is then stated, which will be the opposite of the Null Hypothesis. 

The hypothesis testing process and analysis involves using sample data to determine whether or not you can be statistically confident that you can reject or fail to reject the H o. If the H o is rejected, the statistical conclusion is that the alternative or alternate hypothesis Ha is true.

Overview: What is the Alternate Hypothesis (Ha)? 

Hypothesis testing applies to all forms of statistical inquiry. For example, it can be used to determine whether there are differences between population parameters or an understanding about slopes of regression lines or equality of probability distributions.

In all cases, the first thing you do is state the Null Hypothesis. The word “null” in the context of hypothesis testing means “nothing” or “zero.” 

If we wanted to test whether there was a difference in two population means based on the calculations from two samples, we would state the Null Hypothesis in the form of: 

Ho: mu1 = mu2 or mu1 – mu2 = 0 

In other words, there is no difference, or the difference is zero. Note that the notation is in the form of a population parameter, not a sample statistic. 

The analysis of the Null Hypothesis is designed to test the Null, which will determine whether the Null should be rejected so that the Alternate Hypothesis is defaulted to and assumed to be true, or not to reject the Null so it is assumed to be the true condition. 

A classic example is when you get the results back from your doctor after taking a blood test. The Null is written to state that there is no infection. Remember, the Null is always in the form of “nothingness.” The alternate hypothesis is that you have an infection. Once the test is done, the lab will determine whether the Null can be rejected or not. If the test shows an infection, the Null will be rejected, and the Alternate will be assumed to be true. If the test shows no infection, we cannot reject the Null.

While the Null can only be written in one form — “equal to” or “no difference” — the Alternate can be written for three conditions. For example, a marketing director wants to improve sales.  She designs and launches a new social media campaign, collecting sample data for sales activity prior to the new campaign. After six months, sample sales data was collected to determine whether the campaign was successful. Hypothesis testing was used to statistically confirm whether the campaign was successful or not. The Null Hypothesis was written as: Ho: muBefore = muAfter, where the claim was that the population average sales before the campaign is the same as the population average sales after the campaign. In other words, the campaign had no effect on sales.

The Alternate Hypothesis can now be written in one of three forms:

• muBefore does not equal muAfter : The average of the population average sales before is not equal to the population average sales after, but you don’t know if it was more or less. You would have to look at the actual numbers to understand this result.
• muBefore less than muAfter : In this form, if you reject the Null, your conclusion will be that the sales after the campaign are greater than before the campaign. Put another way, the campaign increased average sales.
• muBefore greater than muAfter : Again, if you reject the Null, the alternate says that average sales before were greater than after the campaign. The campaign seems to have been a failure.

3 benefits of the Alternate Hypothesis  

The stating and testing of the Null and the default to the Alternate hypothesis is the foundation of hypothesis testing. By doing so, you set the parameters for your statistical inference.

1. There can be a statistical assurance of determining differences between population parameters

Just looking at the mathematical difference between the means of two samples and making a decision is woefully inadequate. By statistically testing the Null hypothesis, you will have more confidence in any inferences you want to make about populations based on your samples. If you reject the Null, you’ll know which Alternate is most appropriate.

2. Statistically based estimation of the probability of a population distribution

Many statistical tests require assumptions of specific distributions. Many of these tests assume that the population follows the normal distribution . If it doesn’t, the test may be invalid. Rejecting the Null will establish whether the estimated distribution fulfills any test assumption or not. 

3. You can assess the strength of your conclusions as to what to do with the Alternate hypothesis

Hypothesis testing calculations will provide some relative strength to your decisions as to whether you reject or fail to reject the Null hypothesis and, therefore, the Alternate.

Why is the Alternate Hypothesis important to understand?

It is the interpretation of the statistics relative to the Null and Alternate hypotheses that is important.

Properly write the Alternate hypothesis to capture what you are seeking to prove 

The Alternate can take the form of “not equal,” “less than,” or “greater than.” 

Frame your statement and select an appropriate alpha risk

You don’t want to place too big of a hurdle (or burde)n on your decision-making relative to action on the Null hypothesis by selecting an alpha value that is too high or too low. The Alternate can really reflect the true condition of the population, so failing to reject the Null too often can mask the truth.

Knowing there are decision errors when deciding how to respond to the Null Hypothesis

Since your decision relative to rejecting or not rejecting the Null impacts the Alternate Hypothesis, it’s important to understand how that decision works. 

An industry example of using the Alternate Hypothesis 

The director of manufacturing at a medium-size window manufacturer recently had an older machine retrofitted to increase run speed of the equipment. The supplier, after doing the retrofit, claimed that the machine was now running significantly faster. He showed a comparison of the sample average run speed before the retrofit and the sample average run speed after the retrofit. Looking at the two averages, it appeared that the supplier was correct and that the retrofit did indeed increase the speed.

However, having had some training in Lean Six Sigma, the director asked his local Black Belt for some help in doing a hypothesis test on the data to see if there was truly a statistically significant improvement of at least 100 rpm. The Null hypothesis was written as:

mu1-mu2= -100 rpms

That is the Before speed minus the After speed equaled -100 rpms. This was slightly different than the format he was used to seeing, where the Null would be reflective of “no difference” rather than a value of interest.

The Alternate hypothesis was written to reflect an increase of at least 100 rpms. That form was written as: 

mu1 – mu2  greater -100

That would mean the After speed was more than 100 rpms faster than the Before speed. Fortunately, the results indicated that the Null hypothesis should be rejected and that in actuality, the difference was greater than the 100 rpms he wanted.

3 best practices when thinking about the Alternate Hypothesis 

Using hypothesis testing to help make better data-driven decisions requires that you properly address the Null and Alternate Hypotheses. 

1. Always use the proper nomenclature when stating the Alternate Hypothesis 

The Alternate should be in the form of “not equal to,” “greater than,” “less than,” or “equal to some value of interest.”

2. Be sure that the Alternate statement reflects what you want to learn about the process characteristics

The writing of the Alternate Hypothesis can vary, so be sure you understand exactly what condition you are testing against. 

3. Pick a reasonable alpha risk so you are not always failing to reject the Null Hypothesis

Being too cautious will lead you to not reject the Null enough so you will never learn anything about your population data. 

Frequently Asked Questions (FAQ) about the Alternate Hypothesis

What form should the Alternate Hypothesis be written in?

The Alternate Hypothesis should be in the form of “not equal to,” “greater than,” “less than,” or “equal to some value of interest.”

When do you default to the Alternate Hypothesis? 

If your analysis of the sample data suggests that you should reject the Null Hypothesis, you will default to the statement of the Alternate Hypothesis.

Should I be disappointed if I default to the Alternate Hypothesis? 

Usually no. Typically, you want the actions you took to have had some impact or effect.  Rejecting the Null and defaulting to the Alternate signals that something did, in fact, happen — and that might be a good thing.

Using an Alternate Hypothesis (Ha)

The Alternate Hypothesis is the default should you reject the Null. It is an indication that something has happened that is significant. Often, that is what you want to see if you’re comparing a before and after situation.

While the form of the Null Hypothesis is usually written in a single format, the format of the Alternate can be written a number of different ways. This provides more sensitivity to your interpretation of the population data and will therefore provide a richer insight for decision-making.

About the Author

Ken Feldman

What is a statistical test?

A statistical test is a way to evaluate the evidence the data provides against a hypothesis. This hypothesis is called the null hypothesis and is often referred to as H0 . Under H0, data are generated by random processes. In other words, the controlled processes (the experimental manipulations for example) do not affect the data. Usually, H0 is a statement of equality (equality between averages or between variances or between a correlation coefficient and zero, for example).

H0 is usually opposed to a hypothesis called the alternative hypothesis , referred to as H1 or Ha . Most of the time, the alternative hypothesis is the one the user would like to demonstrate. It involves a statement of difference (difference between averages for example).

If the data does not provide enough evidence against H0, H0 is not rejected. If instead, the data shows strong evidence against H0, H0 is rejected and Ha is considered as true with a quantified (low) risk of being wrong. A statistical test allows to reject / not to reject the H0 hypothesis. Let’s have a look at an example ! Suppose you're comparing two varieties of apples and you're wondering whether the average size of apples from variety 1 differs from the average size of apples from variety 2. Here's how we would write down the null and alternative hypotheses:

H0: average size of apple from variety 1 = average size of apple from variety 2.

Ha: average size of apple from variety 1 ≠ average size of apple from variety 2.

Other examples of null hypotheses versus alternative challenging hypotheses

H0: the insulin rate of the group of patients receiving a placebo is equal to the insulin rate of patients receiving a medication.

Ha: the insulin rate of the group of patients receiving a placebo is different from the insulin rate of patients receiving a medication.

H0: the presence of attribute A does not affect consumer preference toward this product.

Ha: the presence of attribute A affects consumer preference toward this product.

H0: there is no trend in this time series.

Ha: there is a trend in this time series.

H0: Corn fields submitted to fertilizers A, B, C or D produce equivalent yields.

Ha: at least one fertilizer induces a difference in corn yield.

How to interpret the output of a statistical test: the significance level alpha and the p-value

When setting up a study, a risk threshold above which H0 should not be rejected must be specified. This threshold is referred to as the significance level alpha and should lay between 0 and 1. Low alpha’s are more conservative. The choice of alpha should depend on how dangerous it is to reject H0 while it is true. For example, in a study aiming at demonstrating the benefits of a medical treatment, alpha should be low. On the other hand, when screening the effects of many attributes on the appreciation of a product, alpha’s could be more moderate. Very often, alpha is set at 0.05 or 0.01 or 0.001.

The statistical test produces a number called p-value (that is also bounded between 0 and 1). The p-value is the probability of obtaining the data or more extreme data under the null hypothesis.

More practically, the p-value should be compared to alpha:

If p-value < alpha , we reject H0 and accept Ha with a risk proportional to p-value of being wrong.

If p-value > alpha , we do not reject H0, but this does not necessarily imply that we should accept it. It either means that H0 is true, or that H0 is false but our experiment and statistical test were not “strong” enough to lead to a p-value lower than alpha.

What is statistical power and in what case can we accept H0?

Statistically speaking, the ability of an experiment/a test to lead to a rejection of the null hypothesis is called statistical power . The power of an experiment increases with alpha, with the precision of the measurements and with the number of repetitions. Power also changes according to the type of statistical tests being used (see the last section of this tutorial). Power may be computed before or after an experiment. It equals 1 minus the risk of being wrong when accepting H0 (also called risk beta). So the higher the power, the lower the What is the difference between a parametric and a nonparametric test? risk of being wrong when accepting H0 (when p-value > alpha, of course).

In summary, if p > alpha AND if statistical power is high enough (usually higher than 0.95), then we may accept H0 with a risk proportional to (1 – Power) of being wrong.

What are the kinds of statistical tests?

A statistical test can be:

Parametric or nonparametric

two-tailed or one-tailed

Paired or independant samples

How do I know what statistical test to use?

Here is a grid which will help you choose an appropriate test according to your question.[

How can I run a statistical test in XLSTAT?

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Hypothesis testing

• PMID: 8900794
• DOI: 10.1097/00002800-199607000-00009

Hypothesis testing is the process of making a choice between two conflicting hypotheses. The null hypothesis, H0, is a statistical proposition stating that there is no significant difference between a hypothesized value of a population parameter and its value estimated from a sample drawn from that population. The alternative hypothesis, H1 or Ha, is a statistical proposition stating that there is a significant difference between a hypothesized value of a population parameter and its estimated value. When the null hypothesis is tested, a decision is either correct or incorrect. An incorrect decision can be made in two ways: We can reject the null hypothesis when it is true (Type I error) or we can fail to reject the null hypothesis when it is false (Type II error). The probability of making Type I and Type II errors is designated by alpha and beta, respectively. The smallest observed significance level for which the null hypothesis would be rejected is referred to as the p-value. The p-value only has meaning as a measure of confidence when the decision is to reject the null hypothesis. It has no meaning when the decision is that the null hypothesis is true.

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